Repository: stanfordLINQS/SQcircuit (language: python)

Issue:
Merge in QuTiP 5.0.x compatibility updates

[Bug report] The code crashes for coupled fluxoniums
Hello,

I try to simulate 2 capacitively coupled fluxoniums. I do it using the following code:

```
import SQcircuit as sq

loop1 = sq.Loop()
loop2 = sq.Loop()

C1 = sq.Capacitor(1, 'GHz')
L1 = sq.Inductor(0.55,'GHz',loops=[loop1])
JJ1 = sq.Junction(4,'GHz',loops=[loop1])

C2 = sq.Capacitor(1.5, 'GHz')
L2 = sq.Inductor(0.5,'GHz',loops=[loop2])
JJ2 = sq.Junction(5,'GHz',loops=[loop2])

Cg = sq.Capacitor(5, 'GHz')

elements = {(0, 1): [JJ1, L1, C1],
            (1, 2): [Cg],
            (2, 3): [JJ2, L2, C2]}

coupled_fl = sq.Circuit(elements)
```

However, it gives an error:
```
LinAlgError                               Traceback (most recent call last)
/tmp/ipykernel_522532/4206533622.py in <module>
     16             (2, 3): [JJ2, L2, C2]}
     17 
---> 18 coupled_fl = sq.Circuit(elements)

~/miniconda3/envs/quantum/lib/python3.9/site-packages/SQcircuit/circuit.py in __init__(self, elements, flux_dist, random)
    120 
    121         # get the capacitance matrix, inductance matrix, and w matrix
--> 122         self.C, self.L, self.W = self._get_LCWB()
    123 
    124         # the inverse of transformation of coordinates for charge operators

~/miniconda3/envs/quantum/lib/python3.9/site-packages/SQcircuit/circuit.py in _get_LCWB(self)
    339             Y = np.concatenate((np.zeros((count - numLoop, numLoop)),
    340                                 np.eye(numLoop)), axis=0)
--> 341             self.K2 = np.linalg.inv(X) @ Y
    342             self.K2 = np.around(self.K2, 5)
    343 

<__array_function__ internals> in inv(*args, **kwargs)

~/miniconda3/envs/quantum/lib/python3.9/site-packages/numpy/linalg/linalg.py in inv(a)
    538     a, wrap = _makearray(a)
    539     _assert_stacked_2d(a)
--> 540     _assert_stacked_square(a)
    541     t, result_t = _commonType(a)
    542 

~/miniconda3/envs/quantum/lib/python3.9/site-packages/numpy/linalg/linalg.py in _assert_stacked_square(*arrays)
    201         m, n = a.shape[-2:]
    202         if m != n:
--> 203             raise LinAlgError('Last 2 dimensions of the array must be square')
    204 
    205 def _assert_finite(*arrays):

LinAlgError: Last 2 dimensions of the array must be square
```

I use SQcircuit version 0.0.10 and python 3.9.12


Discussion / hints:

Hi @AlesyaSokol thanks for reporting your issue. 

In SQcircuit, the edges that you put in the `elements` should form a connected graph (the node 3 is not connected to anywhere in your code) . In your circuit, two fluxoniums share a common ground which is 0 node. You should change the `elements` dictionary as follow (I changed the (2, 3) edge to (0, 2) edge):
```python
elements = {
    (0, 1): [JJ1, L1, C1],
    (1, 2): [Cg],
    (0, 2): [JJ2, L2, C2]
}
```
Let me know if you have any further questions.

Fix patch:
diff --git a/.github/workflows/pypi.yml b/.github/workflows/pypi.yml
old mode 100644
new mode 100755
diff --git a/.gitignore b/.gitignore
old mode 100644
new mode 100755
index 70d7d42..4c4462a
--- a/.gitignore
+++ b/.gitignore
@@ -16,8 +16,8 @@
 
 /SQcircuit/tests/.ipynb_checkpoints/
 /SQcircuit/tests/jupyter_test.ipynb
+/SQcircuit/tests/data/system_debuging.py
 
 /examples/.ipynb_checkpoints/
 
-
-
+.mypy_cache/
diff --git a/LICENSE b/LICENSE
old mode 100644
new mode 100755
diff --git a/MANIFEST.in b/MANIFEST.in
old mode 100644
new mode 100755
diff --git a/README.md b/README.md
old mode 100644
new mode 100755
index 744884c..a2a3d4a
--- a/README.md
+++ b/README.md
@@ -1,37 +1,46 @@
-<br />
-<p align="center">
-<img src = pics/README_logo.png width= 500px" />
-</p>
+<div align="center"><picture>
+  <source media="(prefers-color-scheme: dark)" srcset="pics/dark_logo_sq.png">
+  <source media="(prefers-color-scheme: light)" srcset="pics/light_logo_sq.png">
+  <img alt="Logo image" src="pics/dark_logo_sq.png" width="350" height="auto">
+</picture></div>
 
 # SQcircuit: Superconducting Quantum Circuit Analyzer
 [**What is SQcircuit?**](#What-is-SQcircuit?)
-|[**Installation**](#Installation)
-|[**Documentation**](#Documentation)
-|[**Examples**](#Examples)
-|[**Contribution**](#Contribution)
+| [**Installation**](#Installation)
+| [**Documentation**](#Documentation)
+| [**Examples**](#Examples)
+| [**Contribution**](#Contribution)
 
 [![license](https://img.shields.io/badge/license-New%20BSD-orange.svg)](https://opensource.org/licenses/BSD-3-Clause)
 [![codecov](https://codecov.io/gh/stanfordLINQS/SQcircuit/branch/main/graph/badge.svg?token=6FT6L9ZPHP)](https://codecov.io/gh/stanfordLINQS/SQcircuit)
+[![Conda-Forge Badge](https://anaconda.org/conda-forge/sqcircuit/badges/downloads.svg)](https://anaconda.org/conda-forge/sqcircuit)
+
+> ⚠️ **Note:** SQcircuit is compatible with QuTip versions 5.0 and above.
+
 ## What is SQcircuit?
 
-SQcircuit is an open-source Python library that is capable of analyzing an 
-arbitrary superconducting quantum circuit. SQcircuit uses the theory discussed 
-in [Rajabzadeh et al., 2022] to describe the Hamiltonian in the appropriate 
-basis and to effectively find the energy spectrum and eigenvectors of the 
-circuit. To design the desired quantum circuit and to discover new qubits, 
-additional functionalities and methods are provided to extract the circuit 
-properties such as matrix elements, dephasing rate, decay rates, etc.
+SQcircuit is an open-source Python package designed to facilitate the 
+analysis and optimization of arbitrary superconducting quantum circuits. 
+Developed by researchers at Stanford University, SQcircuit provides a 
+comprehensive framework to model, analyze, and optimize quantum circuits by 
+constructing and diagonalizing their Hamiltonian from physical descriptions 
+and efficient basis construction. This package supports the calculation of 
+key circuit properties such as energy spectra, coherence times, transition 
+matrix elements, coupling operators, and phase coordinate representations of 
+eigenfunctions. With the integration of automatic differentiation 
+capabilities using PyTorch, SQcircuit enables efficient computation of 
+gradients for all the mentioned properties and custom-made loss functions, 
+making it a powerful tool for optimizing superconducting quantum circuits.
 
-Theory detail of SQcircuit and the introduction to library functionalities 
-are presented in the following paper:
+The detailed theory behind the SQcircuit core code and an introduction to 
+the library's functionalities are provided in the following paper:
 
 > Taha Rajabzadeh, Zhaoyou Wang, Nathan Lee, Takuma Makihara, Yudan Guo, 
 > Amir H. Safavi-Naeini,<br>
 > *Analysis of arbitrary superconducting quantum circuits accompanied by a 
 > Python package: SQcircuit*,<br>
-> arXiv:2206.08319 (2022),<br>
-> https://arxiv.org/abs/2206.08319
-
+> Quantum 7, 1118,<br>
+> https://quantum-journal.org/papers/q-2023-09-25-1118/
 
 ## Installation
 For Python above 3.6, SQcirucit can be simply installed via Conda:
diff --git a/SQcircuit/__init__.py b/SQcircuit/__init__.py
old mode 100644
new mode 100755
index e1404fd..ffcadd0
--- a/SQcircuit/__init__.py
+++ b/SQcircuit/__init__.py
@@ -1,8 +1,27 @@
 """
 Importing all the modules.
 """
+
+import logging
+sq_logger = logging.getLogger(__name__)
+
+def get_logger() -> logging.Logger:
+    """Get the SQcircuit-wide parent logger."""
+    return sq_logger
+
+import qutip
+qutip.settings.core['auto_tidyup'] = True
+qutip.settings.core['auto_tidyup_atol'] = 1e-12 # Make consistent with QuTip 4.7.x
+
 from SQcircuit.elements import *
 from SQcircuit.circuit import *
 from SQcircuit.sweep import *
 from SQcircuit.storage import *
 from SQcircuit.units import *
+from SQcircuit.systems import *
+
+from SQcircuit.settings import (
+    set_engine,
+    get_engine,
+)
+from SQcircuit.torch_extensions import set_max_eigenvector_grad
diff --git a/SQcircuit/circuit.py b/SQcircuit/circuit.py
old mode 100644
new mode 100755
index 297601e..4baeb78
--- a/SQcircuit/circuit.py
+++ b/SQcircuit/circuit.py
@@ -1,46 +1,267 @@
 """circuit.py contains the classes for the circuit and their properties
 """
 
-from typing import Dict, Tuple, List, Sequence, Optional, Union
+from collections import defaultdict, OrderedDict
+from copy import copy, deepcopy
+import logging
+from typing import Any, Dict, List, Optional, Sequence, Tuple, Union, Type
+from typing_extensions import Self
 
 import numpy as np
 import qutip as qt
 import scipy.special
 import scipy.sparse
+import torch
 
+import mpmath
 from numpy import ndarray
-from qutip.qobj import Qobj
+from qutip import Qobj
 from scipy.linalg import sqrtm, block_diag
-from scipy.special import eval_hermite
+from scipy.special import eval_hermitenorm, hyperu
+from scipy.sparse.linalg import ArpackNoConvergence
 
-import SQcircuit.units as unt
+from torch import Tensor
 
-from SQcircuit.elements import (Capacitor, Inductor, Junction, Loop, Charge,
-                                VerySmallCap, VeryLargeCap)
-from SQcircuit.texts import is_notebook, HamilTxt
+import SQcircuit.units as unt
+import SQcircuit.functions as sqf
+import SQcircuit.torch_extensions as sqtorch
+
+from SQcircuit.elements import (
+    Element,
+    Capacitor,
+    Inductor,
+    Junction,
+    Loop,
+    Charge
+)
+from SQcircuit.texts import HamilTxt, is_notebook
+from SQcircuit import symbolic
 from SQcircuit.noise import ENV
-from SQcircuit.settings import ACC
+from SQcircuit.settings import ACC, get_optim_mode
+from SQcircuit.exceptions import raise_optim_error_if_needed, CircuitStateError
+
+logger = logging.getLogger(__name__)
+
+
+class CircuitEdge:
+    """Class that contains the properties of an edge in the circuit.
+
+    Parameters
+    ----------
+        circ:
+            Circuit that edge is part of.
+        edge:
+            The tuple represents the edge of the circuit.
+    """
+
+    def __init__(
+        self,
+        circ: 'Circuit',
+        edge: Tuple[int, int],
+    ) -> None:
+
+        self.circ = circ
+        self.edge = edge
+
+        self.w = self._set_w_at_edge()
+        self.mat_rep = self._set_matrix_rep()
+
+        self.edge_elems_by_type = {
+            Capacitor: [],
+            Inductor: [],
+            Junction: []
+        }
+
+        # check if edge is processed
+        self.processed = False
+
+    def _set_w_at_edge(self) -> list:
+        """Get the w_k vector as list at the edge."""
+
+        # i1 and i2 are the nodes of the edge
+        i1, i2 = self.edge
+
+        w = (self.circ.n + 1) * [0]
+
+        if i1 == 0 or i2 == 0:
+            w[i1 + i2] += 1
+        else:
+            w[i1] += 1
+            w[i2] -= 1
+
+        return w[1:]
+
+    def _set_matrix_rep(self) -> ndarray:
+        """Special form of matrix representation for an edge of a graph.
+        This helps to construct the capacitance and susceptance matrices.
+        """
+
+        edge_mat = np.zeros((self.circ.n, self.circ.n))
+
+        if 0 in self.edge:
+            i = self.edge[0] + self.edge[1] - 1
+            edge_mat[i, i] = 1
+        else:
+            i = self.edge[0] - 1
+            j = self.edge[1] - 1
+
+            edge_mat[i, i] = 1
+            edge_mat[j, j] = 1
+            edge_mat[i, j] = -1
+            edge_mat[j, i] = -1
+
+        return edge_mat
+
+    def update_circuit_loop_from_element(
+        self,
+        el: Union[Inductor, Junction],
+        b_id: int,
+    ) -> None:
+        """Update loop properties related to circuit from element in the edge
+        with its inductive index (b_id).
+
+        Parameters
+        ----------
+            el:
+                Inductive element.
+            b_id:
+                Inductive index.
+        """
+
+        for loop in el.loops:
+
+            self.circ.add_loop(loop)
+            loop.add_index(b_id)
+            loop.add_to_k_mat(self.w)
+
+            if get_optim_mode():
+                self.circ.add_to_parameters(loop)
+
+    def process_edge_and_update_circ(
+        self,
+        b_id: int,
+        w_id: int,
+        k_mat: list,
+        c_edge_mat: list,
+    ) -> Tuple[int, list, list]:
+        """Process the edge and update the related circuit properties.
+
+        Parameters
+        ----------
+            b_id:
+                Point to each row of B matrix of the circuit.
+            w_id:
+                Point to each row of W matrix of the circuit.
+            k_mat:
+                Matrix related to loop calculation
+            c_edge_mat:
+                edge capacitance matrix
+        """
+
+        for el in self.circ.elements[self.edge]:
+
+            self.edge_elems_by_type[el.type].append(el)
+
+            # Case of the inductive element
+            if hasattr(el, 'loops'):
+
+                self.edge_elems_by_type[Capacitor].append(el.cap)
+
+                if get_optim_mode():
+                    self.circ.add_to_parameters(el.cap)
+                    if hasattr(el.cap, 'partial_mat'):
+                        self.circ.partial_mats[el.cap] += (
+                            el.cap.partial_mat(self.mat_rep)
+                        )
+
+                self.circ.elem_keys[el.type].append(
+                    el.get_key(self.edge, b_id, w_id)
+                )
+
+                self.update_circuit_loop_from_element(el, b_id)
+
+                b_id += 1
+
+                k_mat.append(self.w)
+
+                c_edge_mat.append(
+                    el.get_cap_for_flux_dist(self.circ.flux_dist)
+                )
+
+            # Case of L and C
+            if hasattr(el, 'partial_mat'):
+
+                self.circ.partial_mats[el] += el.partial_mat(self.mat_rep)
+
+            if get_optim_mode():
+
+                self.circ.add_to_parameters(el)
+
+        self.processed = True
+
+        return b_id, k_mat, c_edge_mat
+
+    def _check_if_edge_is_processed(self) -> None:
+
+        assert self.processed, 'Edge is not processed yet!'
+
+    def get_eff_cap_value(self) -> float:
+        """Return effective capacitor value of the edge."""
+
+        self._check_if_edge_is_processed()
+
+        return sum(list(map(
+            lambda c: c.get_value(),
+            self.edge_elems_by_type[Capacitor]
+        )))
+
+    def get_eff_ind_value(self) -> float:
+        """Return effective inductor value of the edge."""
+
+        self._check_if_edge_is_processed()
+
+        return sum(list(map(
+            lambda ind: 1/ind.get_value(),
+            self.edge_elems_by_type[Inductor]
+        )))
+
+    def is_junc_in_this_edge(self) -> bool:
+        """Check if the edge contains any JJ."""
+
+        self._check_if_edge_is_processed()
+
+        return len(self.edge_elems_by_type[Junction]) != 0
+
+    def is_junc_without_ind(self) -> bool:
+        """Check if the edge only has JJ and no inductor."""
+
+        self._check_if_edge_is_processed()
+
+        flag = (
+            len(self.edge_elems_by_type[Junction]) != 0
+            and len(self.edge_elems_by_type[Inductor]) == 0
+        )
+
+        return flag
 
 
 class Circuit:
-    """Class that contains the circuit properties and uses the theory discussed
-    in the original paper of the SQcircuit to calculate:
+    """Class that contains circuit properties and builds the Hamiltonian using
+    the theory discussed in the original SQcircuit paper. Provides methods to
+    calculate:
 
         * Eigenvalues and eigenvectors
         * Phase coordinate representation of eigenvectors
         * Coupling operators
         * Matrix elements
         * Decoherence rates
-        * Robustness analysis
+        * Gradients of Hamiltonian, eigenvalues/vectors, and decoherence rates
 
     Parameters
     ----------
         elements:
-            A dictionary that contains the circuit's elements at each branch
+            A dictionary that contains the circuit's elements at each edge
             of the circuit.
-        random:
-            If `True`, each element of the circuit is a random number due to
-            fabrication error. This is necessary for robustness analysis.
         flux_dist:
             Provide the method of distributing the external fluxes. If
             ``flux_dist`` is ``"all"``, SQcircuit assign the external fluxes
@@ -53,46 +274,45 @@ class Circuit:
     """
 
     def __init__(
-            self,
-            elements: Dict[Tuple[int, int],
-                           List[Union[Capacitor, Inductor, Junction]]],
-            flux_dist: str = 'junctions',
-            random: bool = False
+        self,
+        elements: Dict[Tuple[int, int], List[Element]],
+        flux_dist: str = 'junctions',
     ) -> None:
 
         #######################################################################
         # General circuit attributes
         #######################################################################
 
-        self.elements = elements
+        self.elements = OrderedDict(
+            [(key, elements[key]) for key in elements.keys()]
+        )
 
-        error = ("flux_dist option must be either \"junctions\", "
-                 "\"inductors\", or \"all\"")
-        assert flux_dist in ["junctions", "inductors", "all"], error
+        if flux_dist not in ['junctions', 'inductors', 'all']:
+            raise ValueError("flux_dist option must either be 'junctions', "
+                             "'inductors', or 'all'.")
         self.flux_dist = flux_dist
 
-        self.random = random
-
         # circuit inductive loops
         self.loops: List[Loop] = []
 
-        # # charge islands of the circuit
-        # self.charge_islands: Dict[int, Charge] = {}
-
         # number of nodes without ground
         self.n: int = max(max(self.elements))
 
-        # number of branches that contain JJ without parallel inductor.
-        self.countJJnoInd: int = 0
-
-        # inductor element keys: (edge, el, B_idx) B_idx point to
-        # each row of B matrix (external flux distribution of that element)
-        self.inductor_keys: List[tuple, Inductor, int] = []
+        # number of branches that contain junctions without a parallel inductor.
+        self.num_jun_without_ind: int = 0
+
+        self.elem_keys = {
+            # inductor element keys: (edge, el, b_id) b_id point to
+            # each row of B matrix (external flux distribution of that element)
+            Inductor: [],
+            # junction element keys: (edge, el, b_id, w_id) b_id point to
+            # each row of B matrix (external flux distribution of that element)
+            # and w_id point to each row of W matrix
+            Junction: [],
+        }
 
-        # junction element keys: (edge, el, B_idx, W_idx) B_idx point to
-        # each row of B matrix (external flux distribution of that element)
-        # and W_idx point to each row of W matrix
-        self.junction_keys: List[tuple, Junction, int, int] = []
+        # contains the parameters that we want to optimize.
+        self._parameters: OrderedDict[Tuple[Element, Tensor]] = OrderedDict()
 
         #######################################################################
         # Transformation related attributes
@@ -100,16 +320,18 @@ def __init__(
 
         # get the capacitance matrix, sudo-inductance matrix, W matrix,
         # and B matrix (loop distribution over inductive elements)
-        (self.C, self.L, self.W, self.B,
-         self.partial_C, self.partial_L) = self._get_LCWB()
+        self.C, self.L, self.W, self.B = self._get_LCWB()
 
-        # initialize the transformation matrix for charge and flux operators.
+        # initialize the transformation matrices for charge and flux operators.
         self.R, self.S = np.eye(self.n), np.eye(self.n)
 
-        # initialize transformed susceptance, inverse capacitance, and W matrix.
-        self.cInvTrans, self.lTrans, self.wTrans = (np.linalg.inv(self.C),
-                                                    self.L.copy(),
-                                                    self.W.copy())
+        # initialize transformed susceptance, inverse capacitance,
+        # and W matrices.
+        self.cInvTrans, self.lTrans, self.wTrans = (
+            np.linalg.inv(sqf.to_numpy(self.C)),
+            sqf.to_numpy(self.L).copy(),
+            self.W.copy()
+        )
 
         # natural angular frequencies of the circuit for each mode as a numpy
         # array (zero for charge modes)
@@ -118,12 +340,14 @@ def __init__(
         # transform the Hamiltonian of the circuit
         self._transform_hamil()
 
-        if self._is_JJ_in_circuit():
+        if self._is_junction_in_circuit() and get_optim_mode() == False:
             self._update_nmodes_remove_uncoupled_modes()
 
         # charge islands of the circuit
-        self.charge_islands = {i: Charge() for i in range(self.n) if
-                               self._is_charge_mode(i)}
+        self.charge_islands = {
+            i: Charge() for i in range(self.n) if
+            self._is_charge_mode(i)
+        }
 
         #######################################################################
         # Operator and diagonalization related attributes
@@ -137,272 +361,478 @@ def __init__(
 
         self._memory_ops: Dict[str, Union[List[Qobj],
                                           List[List[Qobj]], dict]] = {
-            "Q": [],  # list of charge operators (normalized by 1/sqrt(hbar))
-            "QQ": [[]],  # list of charge times charge operators
-            "phi": [],  # list of flux operators (normalized by 1/sqrt(hbar))
-            "N": [],  # list of number operators
-            "exp": [],  # List of exponential operators
-            "root_exp": [],  # List of square root of exponential operators
-            "cos": {},  # List of cosine operators
-            "sin": {},  # List of sine operators
-            "sin_half": {},  # list of sin(phi/2)
-            "ind_hamil": {},  # list of w^T*phi that appears in Hamiltonian
+            'Q': [],  # list of charge operators (normalized by 1/sqrt(hbar))
+            'QQ': [[]],  # list of charge times charge operators
+            'phi': [],  # list of flux operators (normalized by 1/sqrt(hbar))
+            'N': [],  # list of number operators
+            'exp': [],  # List of exponential operators
+            'root_exp': [],  # List of square root of exponential operators
+            'cos': {},  # List of cosine operators
+            'sin': {},  # List of sine operators
+            'sin_half': {},  # list of sin(phi/2)
+            'ind_hamil': {},  # list of w^T*phi that appears in Hamiltonian
+        }
+
+        # TODO: fix typing; add comments etc.
+        self.descrip_vars: Dict[str, Union[List[float], np.ndarray]] = {
+            'omega': [],
+            'phi_zp': [],
+            'ng': [],
+            'EC': None,
+            'EJ': None
         }
 
         # LC part of the Hamiltonian
-        self._LC_hamil = qt.Qobj()
+        self._LC_hamil = 0
 
         # eigenvalues of the circuit
-        self._efreqs = np.array([])
+        self._efreqs = sqf.array([])
         # eigenvectors of the circuit
         self._evecs = []
 
-    def __getstate__(self):
+        # Toggle whether we need to copy all elements (namely the
+        # _memory_ops and _LC_hamil; see .__getstate__)
+        self._toggle_fullcopy = True
+
+    def update(self) -> None:
+        """Update the circuit Hamiltonian to reflect in-place changes made to
+        the scalar values used for circuit elements (ex. C, L, J...).
+        """
+
+        self.elem_keys = {
+            Inductor: [],
+            Junction: [],
+        }
+        self.loops: List[Loop] = []
+
+        self._parameters = OrderedDict()
+
+        self.C, self.L, self.W, self.B = self._get_LCWB()
+        self.R, self.S = np.eye(self.n), np.eye(self.n)
+        self.cInvTrans, self.lTrans, self.wTrans = (
+            np.linalg.inv(sqf.to_numpy(self.C)),
+            sqf.to_numpy(self.L).copy(),
+            self.W.copy()
+        )
+        self.omega = np.zeros(self.n)
+        self._transform_hamil()
+
+        self._memory_ops: Dict[
+            str, Union[List[Qobj], List[List[Qobj]], dict]
+        ] = {
+            'Q': [],  # list of charge operators (normalized by 1/sqrt(hbar))
+            'QQ': [[]],  # list of charge times charge operators
+            'phi': [],  # list of flux operators (normalized by 1/sqrt(hbar))
+            'N': [],  # list of number operators
+            'exp': [],  # List of exponential operators
+            'root_exp': [],  # List of square root of exponential operators
+            'cos': {},  # List of cosine operators
+            'sin': {},  # List of sine operators
+            'sin_half': {},  # list of sin(phi/2)
+            'ind_hamil': {},  # list of w^T*phi that appears in Hamiltonian
+        }
+
+        self._build_op_memory()
+        self._LC_hamil = self._get_LC_hamil()
+        self._build_exp_ops()
+
+        self._efreqs = sqf.array([])
+        self._evecs = []
+
+    def __getstate__(self) -> dict[str, Any]:
         attrs = self.__dict__
-        # type_attrs = type(self).__dict__
 
-        # Attributes that we are avoiding to store for reducing the size of
-        # the saved file( Qutip objects and Quantum operators usually).
-        avoid_attrs = ["_memory_ops", "_LC_hamil"]
+        # When ``_toggle_fullcopy`` is ``False``, remove attributes of the
+        # circuit which cost a lot of memory but can be reconstructed by calling
+        # ``.update()``. Useful when pickling a copy of the circuit.
+        if self._toggle_fullcopy:
+            avoid_attrs = []
+        else:
+            avoid_attrs = ['_memory_ops', '_LC_hamil']
 
         self_dict = {k: attrs[k] for k in attrs if k not in avoid_attrs}
 
         return self_dict
 
-    def __setstate__(self, state):
+    def __setstate__(self, state) -> None:
         self.__dict__ = state
+        # Set ``_toggle_fullcopy`` back to ``True`` because this is the default
+        # state when initializing a circuit, but is often set to ``False`` when
+        # pickling the circuit.
+        self._toggle_fullcopy = True
+
+    @property
+    def efreqs(self) -> Union[ndarray, Tensor]:
+        """Eigenfrequencies in the chosen frequency unit for SQcircuit. If the
+        SQcircuit engine is ``PyTorch``, the efreqs will be in ``Tensor``
+        format; otherwise, they will be in ``ndarray`` format."""
+        if len(self._efreqs) == 0:
+            raise CircuitStateError('Please diagonalize the circuit first.')
+
+        return self._efreqs / (2 * np.pi * unt.get_unit_freq())
 
     @property
-    def efreqs(self):
+    def evecs(self) -> Union[List[Qobj], Tensor]:
+        """List of circuit eigenvectors. If the SQcircuit engine is ``PyTorch``,
+        each eigenvector will be in ``Tensor`` format; otherwise, they will be
+        in ``Qutip.Qobj`` format."""
+        if len(self._evecs) == 0:
+            raise CircuitStateError('Please diagonalize the circuit first.')
 
-        assert len(self._efreqs) != 0, "Please diagonalize the circuit first."
+        return self._evecs
 
-        return self._efreqs / (2*np.pi*unt.get_unit_freq())
+    @property
+    def trunc_nums(self) -> List[int]:
+        """List of truncation numbers of the circuit. For harmonic modes, these
+        are N where the Hilbert space is 0, 1, …, (N-1) and for charge modes
+        these are N where the Hilbert space is -(N-1), …, 0, …, (N-1).
+        """
+        trunc_nums = []
+        for i in range(self.n):
+            if self._is_charge_mode(i):
+                trunc_nums.append(int((self.m[i] + 1)/2))
+            else:
+                trunc_nums.append(self.m[i])
+        return trunc_nums
 
-    def _add_loop(self, loop: Loop) -> None:
-        """Add loop to the circuit loops.
+    @property
+    def parameters(self) -> List[Tensor]:
+        """The values of the elements in the circuit which require gradients
+        to be computed (either leaf tensors wtih ``requires_grad == True``, or
+        non-leaf tensors). The parameters can be set by a list of new values
+        for each of the elements.
+    
+        Only available when using the ``PyTorch`` engine of ``SQcircuit``.
         """
-        if loop not in self.loops:
-            loop.reset()
-            self.loops.append(loop)
+        raise_optim_error_if_needed()
 
-    def _get_w_at_edge(self, edge: Tuple[int, int]) -> list:
-        """Get the w_k vector as list at the edge.
+        return list(self._parameters.values())
 
-        Parameters
-        ----------
-            edge:
-                Tuple of int which specifies an edge.
+    @parameters.setter
+    def parameters(self, new_params: Union[Tensor, List[Tensor]]) -> None:
+        raise_optim_error_if_needed()
+
+        try:
+            for i, element in enumerate(self._parameters.keys()):
+                element.internal_value = new_params[i]
+        except IndexError as e:
+            raise ValueError('Shape of new parameters does not match.') from e
+
+        self.update()
+
+    @property
+    def parameters_grad(self) -> Union[List[Optional[Tensor]], Tensor]:
+        """Return the gradients of the tensors in ``.parameters``. If all values
+        are not ``None``, it is returned as a stacked ``Tensor``, otherwise as a
+        list of individual values.
         """
+        raise_optim_error_if_needed()
 
-        # i1 and i2 are the nodes of the edge
-        i1, i2 = edge
+        grad_list = []
+        for val in self.parameters:
+            grad_list.append(val.grad)
 
-        w = (self.n + 1) * [0]
+        if None in grad_list:
+            return grad_list
 
-        if i1 == 0 or i2 == 0:
-            w[i1 + i2] += 1
-        else:
-            w[i1] += 1
-            w[i2] -= 1
+        return torch.stack(grad_list).detach().clone()
 
-        return w[1:]
+    @property
+    def parameters_dict(
+            self
+    ) -> OrderedDict[Tuple[Union[Element, Loop], Tensor]]:
+        """The dictionary of (element, value) pairs for the elements in
+        the circuit which require gradient.
+        """
+        raise_optim_error_if_needed()
 
-    def _edge_matrix_rep(self, edge: Tuple[int, int]) -> ndarray:
+        return self._parameters
 
-        """Special form of matrix representation for an edge of a graph.
-        This helps to construct the capacitance and susceptance matrices.
+    @property
+    def parameters_elems(self) -> List[Union[Element, Loop]]:
+        """The elements in the circuit which require gradient.
+        """
+        raise_optim_error_if_needed()
 
-        Parameters
-        ----------
-            edge:
-                Tuple of int which specifies an edge.
+        return list(self._parameters.keys())
+
+    def get_params_type(self) -> List[Union[Type[Element], Type[Loop]]]:
+        """List of the types for each element in the circuit's parameters.
         """
+        raise_optim_error_if_needed()
 
-        A = np.zeros((self.n, self.n))
+        elements_flattened = list(self._parameters.keys())
 
-        if 0 in edge:
-            i = edge[0] + edge[1] - 1
-            A[i, i] = 1
-        else:
-            i = edge[0] - 1
-            j = edge[1] - 1
+        params_type = [type(element) for element in elements_flattened]
+
+        return params_type
+
+    def zero_grad(self) -> None:
+        """Set the gradient of all values in ``self.parameters`` to ``None``.
+        """
+        raise_optim_error_if_needed()
 
-            A[i, i] = 1
-            A[j, j] = 1
-            A[i, j] = -1
-            A[j, i] = -1
+        for val in self.parameters:
+            val.grad=None
 
-        return A
+    def add_to_parameters(self, el: Element) -> None:
+        """Add an element which requires gradient computation to ``.parameters``.
+        Either
+            - ``requires_grad`` is ``True``; or
+            - the element is not a leaf tensor.
 
-    def _get_LCWB(self):
+        Parameters
+        ----------
+            el:
+                An element to add to ``.parameters``, if the element requires
+                gradient and is not already present.
         """
-        calculate the capacitance matrix, inductance matrix, W matrix,
-        and the flux distribution over inductive elements B.
-        outputs:
-            -- cMat: capacitance matrix (self.n,self.n)
-            -- lMat: inductance matrix (self.n,self.n)
-            -- wMat:  W matrix(linear combination of the flux node operators
-            in the JJ cosine (n_J,self.n)
+        if (el.requires_grad or not el.is_leaf) and el not in self._parameters:
+            self._parameters[el] = el.internal_value
+
+    def add_loop(self, loop: Loop) -> None:
+        """Add loop to the circuit loops. Should only be called when
+        initializing the circuit.
+
+        Parameters
+        ----------
+            loop:
+                Loop in the circuit to add to ``.loops``.
         """
+        if loop not in self.loops:
+            loop.reset()
+            self.loops.append(loop)
 
-        cMat = np.zeros((self.n, self.n))
-        lMat = np.zeros((self.n, self.n))
-        wMat = []
-        bMat = np.array([])
+    def safecopy(self, save_eigs=False) -> Self:
+        """Return a copy of ``self``, explicitly detaching and cloning all
+        tensor values in the circuit (which are element and loop values).
+        Eigenvalues/vectors are either discarded or detached and cloned based
+        on the value of ``save_eigs``.
 
-        partial_cMats: Dict[Tuple[Capacitor, ndarray]] = {}
+        Parameters
+        ----------
+            save_eigs:
+                Whether to retain the eigenvalues/vectors in the copied version
+                of the circuit.
 
-        partial_lMats: Dict[Tuple[Inductor, ndarray]] = {}
+        Returns
+        ----------
+            Deepcopy of `self`.
+        """
+        # Instantiate new container
+        new_circuit = copy(self)
+
+        # When using the PyTorch engine, SQcircuit contains many tensor values,
+        # which may not be leafs. These don't implement a deepcopy method, so
+        # need to be explicitly detached/cloned.
+        if get_optim_mode():
+            # Capacitance and inductance matrices are constructed from
+            # element values
+            new_circuit.C = new_circuit.C.detach().clone()
+            new_circuit.L = new_circuit.L.detach().clone()
+
+            # Replace all loops in circuit with identically-valued copies
+            # whose `.internal_value`s are cloned.
+            new_loops: List[Loop] = []
+            replacement_dict: Dict[
+                Union[Loop, Element], Union[Loop, Element]
+            ] = {}
+            for loop in self.loops:
+                new_loop = copy(loop)
+                new_loop.internal_value = loop.internal_value.detach().clone()
+                new_loops.append(new_loop)
+                replacement_dict[loop] = new_loop
+            new_circuit.loops = new_loops
+
+            # Replace all elements in circuit with identically valued copies
+            # whose `.internal_value`s are cloned.
+            new_elements = defaultdict(list)
+            for edge in self.elements:
+                for el in self.elements[edge]:
+                    new_el = copy(el)
+                    new_el.internal_value = el.internal_value.detach().clone()
+                    # Need to also replace loops associated with circuit
+                    # with the new copies.
+                    if hasattr(el, 'loops'):
+                        new_loops = []
+                        for loop in el.loops:
+                            new_loops.append(replacement_dict[loop])
+                        new_el.loops = new_loops
+                    new_elements[edge].append(new_el)
+
+                    replacement_dict[el] = new_el
+            new_circuit.elements = new_elements
+
+            # Replace the parameters dict with the copied elements
+            new_circuit._parameters = OrderedDict()
+            for el in self._parameters:
+                new_el = replacement_dict[el]
+                new_circuit._parameters[new_el] = new_el.internal_value
+
+            # Several operators in SQcircuit are saved in dictionaries
+            # indexed by element. These keys all need to be updated to use
+            # the copied elements.
+
+            # Update the `.elem_keys` dictionary
+            new_circuit.elem_keys = {
+                Inductor: [],
+                Junction: [],
+            }
+            for edge, el, b_id, w_id in self.elem_keys[Junction]:
+                new_el = replacement_dict[el]
+                new_circuit.elem_keys[Junction].append((edge, new_el, b_id, w_id))
+            for edge, el, b_id in self.elem_keys[Inductor]:
+                new_el = replacement_dict[el]
+                new_circuit.elem_keys[Inductor].append((edge, new_el, b_id))
+
+            # Update the `.partial_mats` dictionary
+            new_circuit.partial_mats = defaultdict(lambda: 0)
+            for el, partial_mat in self.partial_mats.items():
+                try:
+                    new_circuit.partial_mats[replacement_dict[el]] = partial_mat
+                except KeyError:
+                    new_circuit.partial_mats[el] = partial_mat
+
+            # Update the `.memory_ops` dictionary.
+            new_circuit._memory_ops = {}
+            # Some operators contain dictionaries indexed by elements; the
+            # others are fine.
+            problem_types = ['cos', 'sin', 'sin_half', 'ind_hamil']
+            for op_type in self._memory_ops:
+                if op_type not in problem_types:
+                    new_circuit._memory_ops[op_type] = self._memory_ops[op_type]
+                else:
+                    new_circuit._memory_ops[op_type] = {}
+                    for el, b_id in self._memory_ops[op_type].keys():
+                        new_circuit._memory_ops[op_type][(replacement_dict[el], b_id)] = (
+                            self._memory_ops[op_type][(el, b_id)]
+                        )
+
+        # Remove old eigenvectors/values, if desired
+        if save_eigs:
+            new_circuit._efreqs = self._efreqs.detach().clone()
+            new_circuit._evecs = self._evecs.detach().clone()
+        else:
+            new_circuit._efreqs = sqf.array([])
+            new_circuit._evecs = []
 
-        # point to each row of B matrix (external flux distribution of that
-        # element) or count the number of inductive elements.
-        B_idx = 0
+        # Delete the `.descrip_vars`, because bug in the `ExplicitSymbol` class
+        # prevents native `deepcopy`ing.
+        new_circuit.descrip_vars = None
 
-        # W_idx point to each row of W matrix for junctions or count the
-        # number of edges contain JJ
-        W_idx = 0
+        # Deepcopy the whole thing, now that problematic attributes have been
+        # explicitly taken care of.
+        return deepcopy(new_circuit)
 
-        # number of branches that contain JJ without parallel inductor.
-        countJJnoInd = 0
+    def picklecopy(self) -> Self:
+        """Helper function which returns a shallow copy of ``self`` with
+        ``._toggle_fullcopy = False``. Use for pickling circuit to save memory.
 
-        # K1 is a matrix that transfer node coordinates to edge phase drop
-        # for inductive elements
-        K1 = []
+        Returns
+        ----------
+            Copy of self with ``._toggle_fullcopy = False``.
+        """
+        # Instantiate new container
+        new_circuit = copy(self)
 
-        # capacitor at each inductive elements
-        cEd = []
+        # Remove large objects when saving
+        new_circuit._toggle_fullcopy = False
 
-        for edge in self.elements.keys():
+        return new_circuit
 
-            # w vector at the edge
-            edge_w = self._get_w_at_edge(edge)
+    def _get_LCWB(self) -> Tuple[ndarray, ndarray, ndarray, ndarray]:
+        """Calculate the capacitance matrix, sustenance matrix, W matrix,
+        and the flux distribution over inductive elements B matrix.
 
-            # matrix representation of an edge
-            edge_mat = self._edge_matrix_rep(edge)
+        Returns
+        ----------
+            A tuple of the (capacitance, susceptance, W, B) matrices.
+        """
 
-            # list of capacitors of the edge.
-            edge_caps = []
-            # list of inductors of the edge
-            edge_inds = []
-            # list of Josephson Junction of the edge.
-            edge_JJs = []
+        c_mat = sqf.zeros((self.n, self.n), dtype=torch.float64)
+        l_mat = sqf.zeros((self.n, self.n), dtype=torch.float64)
+        w_mat = []
+        b_mat = sqf.array([])
 
-            for el in self.elements[edge]:
+        self.partial_mats = defaultdict(lambda: 0)
 
-                if isinstance(el, Capacitor):
-                    edge_caps.append(el)
+        # point to each row of B matrix (external flux distribution of that
+        # element) or count the number of inductive elements.
+        b_id = 0
 
-                    if el in partial_cMats:
-                        partial_cMats[el] += edge_mat
-                    else:
-                        partial_cMats[el] = edge_mat
-
-                elif isinstance(el, Inductor):
-                    # if el.loops:
-                    self.inductor_keys.append((edge, el, B_idx))
-                    # else:
-                    #     self.inductor_keys.append((edge, el, None))
-                    edge_inds.append(el)
-                    # capacitor of inductor
-                    edge_caps.append(el.cap)
-
-                    for loop in el.loops:
-                        self._add_loop(loop)
-                        loop.add_index(B_idx)
-                        loop.addK1(edge_w)
-
-                    B_idx += 1
-                    K1.append(edge_w)
-
-                    if self.flux_dist == 'all':
-                        cEd.append(el.cap.value())
-                    elif self.flux_dist == "junctions":
-                        cEd.append(VeryLargeCap().value())
-                    elif self.flux_dist == "inductors":
-                        cEd.append(VerySmallCap().value())
-
-                    if el in partial_lMats:
-                        partial_lMats[el] += edge_mat / el.value()**2
-                    else:
-                        partial_lMats[el] = edge_mat / el.value()**2
+        # w_id point to each row of W matrix for junctions or count the
+        # number of edges contain JJ
+        w_id = 0
 
-                elif isinstance(el, Junction):
-                    # if el.loops:
-                    self.junction_keys.append((edge, el, B_idx, W_idx))
-                    # else:
-                    #     self.junction_keys.append((edge, el, None, W_idx))
-                    edge_JJs.append(el)
-                    # capacitor of JJ
-                    edge_caps.append(el.cap)
+        # number of branches that contain JJ without a parallel inductor.
+        num_jun_without_ind = 0
 
-                    for loop in el.loops:
-                        self._add_loop(loop)
-                        loop.add_index(B_idx)
-                        loop.addK1(edge_w)
+        # k_mat is a matrix that transfers node coordinates to the edge phase
+        # drop for inductive elements (In the paper Appendix D we referred to it
+        # as W matrix)
+        k_mat = []
 
-                    B_idx += 1
-                    K1.append(edge_w)
+        # capacitor at each inductive element
+        c_edge_mat = []
 
-                    if self.flux_dist == 'all':
-                        cEd.append(el.cap.value())
-                    elif self.flux_dist == "junctions":
-                        cEd.append(VerySmallCap().value())
-                    elif self.flux_dist == "inductors":
-                        cEd.append(VeryLargeCap().value())
+        for edge in self.elements.keys():
 
-            if len(edge_inds) == 0 and len(edge_JJs) != 0:
-                countJJnoInd += 1
+            circ_edge = CircuitEdge(self, edge)
 
-            # summation of the capacitor values.
-            cap = sum(list(map(lambda c: c.value(self.random), edge_caps)))
+            b_id, k_mat, c_edge_mat = circ_edge.process_edge_and_update_circ(
+                b_id,
+                w_id,
+                k_mat,
+                c_edge_mat
+            )
 
-            # summation of the one over inductor values.
-            x = np.sum(1 / np.array(list(map(lambda l: l.value(self.random),
-                                             edge_inds))))
+            if circ_edge.is_junc_without_ind():
+                num_jun_without_ind += 1
 
-            cMat += cap * edge_mat
+            c_mat += sqf.array(circ_edge.get_eff_cap_value()) * sqf.array(
+                circ_edge.mat_rep)
 
-            lMat += x * edge_mat
+            l_mat += sqf.array(circ_edge.get_eff_ind_value()) * sqf.array(
+                circ_edge.mat_rep)
 
-            if len(edge_JJs) != 0:
-                wMat.append(edge_w)
-                W_idx += 1
+            if circ_edge.is_junc_in_this_edge():
+                w_mat.append(circ_edge.w)
+                w_id += 1
 
-        wMat = np.array(wMat)
+        w_mat = np.array(w_mat)
 
         try:
-            K1 = np.array(K1)
-            a = np.zeros_like(K1)
-            select = np.sum(K1 != a, axis=0) != 0
-            # eliminate the zero columns
-            K1 = K1[:, select]
-            if K1.shape[0] == K1.shape[1]:
-                K1 = K1[:, 0:-1]
-
-            X = K1.T @ np.diag(cEd)
+            k_mat = np.array(k_mat)
+
+            k_mat = sqf.remove_dependent_columns(k_mat)
+
+            x_mat = k_mat.T @ np.diag([sqf.to_numpy(c) for c in c_edge_mat])
+
             for loop in self.loops:
-                p = np.zeros((1, B_idx))
-                p[0, loop.indices] = loop.getP()
-                X = np.concatenate((X, p), axis=0)
+                g = np.zeros((1, b_id))
+                g[0, loop.indices] = loop.get_g()
+                x_mat = np.concatenate((x_mat, g), axis=0)
 
             # number of inductive loops of the circuit
             n_loops = len(self.loops)
 
             if n_loops != 0:
-                Y = np.concatenate((np.zeros((B_idx - n_loops, n_loops)),
-                                    np.eye(n_loops)), axis=0)
-                bMat = np.linalg.inv(X) @ Y
-                bMat = np.around(bMat, 5)
+                i_mat = np.concatenate(
+                    (np.zeros((b_id - n_loops, n_loops)), np.eye(n_loops)),
+                    axis=0
+                )
+                b_mat = np.linalg.inv(x_mat) @ i_mat
+                b_mat = np.around(b_mat, 5)
 
         except ValueError:
+            raise ValueError(
+                'The edge list does not specify a connected graph or all '
+                'inductive loops of the circuit are not specified.'
+            ) from None
 
-            print("The edge list does not specify a connected graph or "
-                  "all inductive loops of the circuit are not specified.")
-
-        self.countJJnoInd = countJJnoInd
+        self.num_jun_without_ind = num_jun_without_ind
 
-        return cMat, lMat, wMat, bMat, partial_cMats, partial_lMats
+        return c_mat, l_mat, w_mat, b_mat
 
     def _get_uncoupled_indices(self) -> list:
         """Return the indices for the uncoupled mode as list"""
@@ -439,206 +869,267 @@ def _is_charge_mode(self, i: int) -> bool:
         ----------
             i:
                 index of the mode. (starts from zero for the first mode)
+
+        Returns
+        ----------
+            Whether the ``i``th mode is a charge mode.
         """
+        if i >= self.n:
+            raise ValueError(f'The circuit only has {self.n} modes!')
 
         return self.omega[i] == 0
 
-    def _apply_transformation(self, S: ndarray, R: ndarray) -> None:
+    def _apply_transformation(self, s_mat: ndarray, r_mat: ndarray) -> None:
         """Apply S and R transformation on transformed C, L, and W matrix.
 
         Parameters
         ----------
-            S:
+            s_mat:
                 Transformation matrices related to flux operators.
-            R:
+            r_mat:
                 Transformation matrices related to charge operators.
         """
 
-        self.cInvTrans = R.T @ self.cInvTrans @ R
-        self.lTrans = S.T @ self.lTrans @ S
+        self.cInvTrans = r_mat.T @ self.cInvTrans @ r_mat
+        self.lTrans = s_mat.T @ self.lTrans @ s_mat
 
         if len(self.W) != 0:
-            self.wTrans = self.wTrans @ S
+            self.wTrans = self.wTrans @ s_mat
 
-        self.S = self.S @ S
-        self.R = self.R @ R
+        self.S = self.S @ s_mat
+        self.R = self.R @ r_mat
 
     def _get_and_apply_transformation_1(self) -> Tuple[ndarray, ndarray]:
-        """Get and apply Second transformation of the coordinates that
+        """Get and apply the first transformation of the coordinates that
         simultaneously diagonalizes the capacitance and susceptance matrices.
+
+        Returns
+        ----------
+            A tuple of the (S1, R1) matrices.
         """
 
-        cMatRoot = sqrtm(self.C)
-        cMatRootInv = np.linalg.inv(cMatRoot)
-        lMatRoot = sqrtm(self.L)
+        c_mat_root = sqrtm(sqf.to_numpy(self.C))
+        c_mat_root_inv = np.linalg.inv(c_mat_root)
+        l_mat_root = sqrtm(sqf.to_numpy(self.L))
 
-        V, D, U = np.linalg.svd(lMatRoot @ cMatRootInv)
+        _, D, U = np.linalg.svd(l_mat_root @ c_mat_root_inv)
 
         # the case that there is not any inductor in the circuit
         if np.max(D) == 0:
             D = np.diag(np.eye(self.n))
-            singLoc = list(range(0, self.n))
+            sing_locs = list(range(0, self.n))
         else:
-            # find the number of singularity in the circuit
-            lEig, _ = np.linalg.eig(self.L)
-            numSing = len(lEig[lEig / np.max(lEig) < ACC["sing_mode_detect"]])
-            singLoc = list(range(self.n - numSing, self.n))
-            D[singLoc] = np.max(D)
+            # find the number of singularities in the circuit
+
+            l_mat_eigs, _ = np.linalg.eig(sqf.to_numpy(self.L))
+            num_sings = len(l_mat_eigs[
+                    l_mat_eigs / np.max(l_mat_eigs) < ACC['sing_mode_detect']
+            ])
+            sing_locs = list(range(self.n - num_sings, self.n))
+            D[sing_locs] = np.max(D)
 
         # build S1 and R1 matrix
-        S1 = cMatRootInv @ U.T @ np.diag(np.sqrt(D))
-        R1 = np.linalg.inv(S1).T
+        s_1 = c_mat_root_inv @ U.T @ np.diag(np.sqrt(D))
+        r_1 = np.linalg.inv(s_1).T
 
-        self._apply_transformation(S1, R1)
+        self._apply_transformation(s_1, r_1)
 
-        self.lTrans[singLoc, singLoc] = 0
+        self.lTrans[sing_locs, sing_locs] = 0
 
-        return S1, R1
+        return s_1, r_1
 
     @staticmethod
-    def _independentRows(A: ndarray) -> Tuple[List[int], List[ndarray]]:
-        """Use Gram–Schmidt to find the linear independent rows of matrix A
-        and return the list of row indices of A and list of the rows.
+    def _independent_rows(a_mat: ndarray) -> Tuple[List[int], List[ndarray]]:
+        """Use the Gram-Schmidt process to find the linear independent rows of 
+        matrix A and return the list of row indices of A and list of the rows.
 
         Parameters
         ----------
-            A:
-                ``Numpy.ndarray`` matrix that we try to find its independent
+            a_mat:
+                ``np.ndarray`` matrix that we try to find its independent
                 rows.
+
+        Returns
+        ----------
+            The list of indices of linearly independent rows of ``A`` and a
+            basis for the row space of ``A``.
         """
 
-        # normalize the row of matrix A
-        A_norm = A / np.linalg.norm(A, axis=1).reshape(A.shape[0], 1)
+        # normalize the rows of matrix A
+        a_mat_norm = (
+            a_mat
+            / (np.linalg.norm(a_mat, axis=1).reshape(a_mat.shape[0], 1) + 1e-9)
+        )
+
+        # Get the row of each A that has the highest norm in descending order.
+        # This is important for the case of JJ capacitively coupled to JL.
+        sorted_index = np.argsort(-np.linalg.norm(a_mat, axis=1))
 
         basis = []
         idx_list = []
 
-        for i, a in enumerate(A_norm):
+        for i in sorted_index:
+            a = a_mat_norm[i, :]
             a_prime = a - sum([np.dot(a, e) * e for e in basis])
-            if (np.abs(a_prime) > ACC["Gram–Schmidt"]).any():
+            if (np.abs(a_prime) > ACC['Gram-Schmidt']).any():
                 idx_list.append(i)
                 basis.append(a_prime / np.linalg.norm(a_prime))
 
         return idx_list, basis
 
-    def _round_to_zero_one(self, W: ndarray) -> ndarray:
+    def _round_to_zero_one(self, w_mat: ndarray) -> ndarray:
         """Round the charge mode elements of W or transformed W matrix that
         are close to 0, -1, and 1 to the exact value of 0, -1, and 1
         respectively.
 
         Parameters
         ----------
-            W:
-                ``Numpy.ndarray`` that can be either W or transformed W matrix.
+            w_mat:
+                ``np.ndarray`` that can be either W or transformed W matrix.
+
+        Returns
+        ----------
+            A rounded copy of ``W``.
         """
 
-        rounded_W = W.copy()
+        rounded_w_mat = w_mat.copy()
+
+        if self.num_jun_without_ind == 0:
+            rounded_w_mat[:, self.omega == 0] = 0
+
+        charge_w_mat = rounded_w_mat[:, self.omega == 0]
 
-        if self.countJJnoInd == 0:
-            rounded_W[:, self.omega == 0] = 0
+        charge_w_mat[np.abs(charge_w_mat) < ACC['Gram-Schmidt']] = 0
+        charge_w_mat[np.abs(charge_w_mat - 1) < ACC['Gram-Schmidt']] = 1
+        charge_w_mat[np.abs(charge_w_mat + 1) < ACC['Gram-Schmidt']] = -1
 
-        charge_only_W = rounded_W[:, self.omega == 0]
+        rounded_w_mat[:, self.omega == 0] = charge_w_mat
 
-        charge_only_W[np.abs(charge_only_W) < ACC["Gram–Schmidt"]] = 0
-        charge_only_W[np.abs(charge_only_W - 1) < ACC["Gram–Schmidt"]] = 1
-        charge_only_W[np.abs(charge_only_W + 1) < ACC["Gram–Schmidt"]] = -1
+        return rounded_w_mat
 
-        rounded_W[:, self.omega == 0] = charge_only_W
+    def _is_gram_schmidt_successful(self, S) -> bool:
+        """Check if the Gram_Schmidt process has the sufficient accuracy for
+        the ``S`` transformation matrix.
 
-        return rounded_W
+        Parameters
+        ----------
+            S:
+                Transformation matrix related to flux operators.
+
+        Returns
+        ----------
+            True if the Gram-Schmidt succeeded
+        """
+
+        is_successful = True
+
+        # absolute value of the current wTrans
+        cur_w_trans = self._round_to_zero_one(self.wTrans @ S)
+
+        for j in range(self.n):
+            if self._is_charge_mode(j):
+                for abs_w in np.abs(cur_w_trans[:, j]):
+                    if abs_w != 0 and abs_w != 1:
+                        is_successful = False
+
+        return is_successful
 
-    def _is_JJ_in_circuit(self) -> bool:
-        """Check if there is any Josephson junction in the circuit."""
+    def _is_junction_in_circuit(self) -> bool:
+        """Check if there are any Josephson junctions in the circuit.
+        
+        Returns
+        ----------
+            True if the circuit contains Josephson junctions, false otherwise.
+        """
 
         return len(self.W) != 0
 
     def _get_and_apply_transformation_2(self) -> Tuple[ndarray, ndarray]:
-        """ Get and apply Second transformation of the coordinates that
+        """Get and apply the second transformation of the coordinates that
         transforms the subspace of the charge operators in order to have the
         reciprocal primitive vectors in Cartesian direction.
+
+        Returns
+        ----------
+            A tuple of the (S2, R2) transformation matrices.
         """
 
         if len(self.W) != 0:
+            # remove independent rows
+            w_t = sqf.remove_dependent_columns(self.W.T)
+            w_reduced = w_t.T
+            w_reduced_transformed = w_reduced @ self.S1
+
+            # charge part of the reduced w_mat
+            w_charge = w_reduced_transformed[:, self.omega == 0].copy()
+
             # get the charge basis part of the wTrans matrix
-            wQ = self.wTrans[:, self.omega == 0].copy()
+            # w_charge = self.wTrans[:, self.omega == 0].copy()
+
             # number of operators represented in charge bases
-            nq = wQ.shape[1]
+            nq = w_charge.shape[1]
         else:
             nq = 0
-            wQ = np.array([])
+            w_charge = np.array([])
 
-        # if we need to represent an operator in charge basis
-        if nq != 0 and self.countJJnoInd != 0:
+        # if we need to represent an operator in the charge basis
+        if nq != 0 and self.num_jun_without_ind != 0:
 
             # list of indices of w vectors that are independent
-            indList = []
+            ind_lst = []
+
+            # random vectors to complete the basis
+            rnd_vecs = []
 
-            X = []
             # use Gram–Schmidt to find the linear independent rows of
-            # normalized wQ (wQ_norm)
+            # normalized w_charge (w_charge_norm)
             basis = []
+
             while len(basis) != nq:
                 if len(basis) == 0:
-                    indList, basis = self._independentRows(wQ)
+                    ind_lst, basis = self._independent_rows(w_charge)
                 else:
-                    # to complete the basis
-                    X = list(np.random.randn(nq - len(basis), nq))
-                    basisComplete = np.array(basis + X)
-                    _, basis = self._independentRows(basisComplete)
+                    # random vectors to complete the basis
+                    rnd_vecs = list(
+                        np.linalg.norm(w_charge, 'fro')
+                        * np.random.randn(nq - len(basis), nq)
+                    )
+                    completed_basis = np.array(basis + rnd_vecs)
+                    _, basis = self._independent_rows(completed_basis)
 
             # the second S and R matrix are:
-            F = np.array(list(wQ[indList, :]) + X)
-            S2 = block_diag(np.eye(self.n - nq), np.linalg.inv(F))
+            f_mat = np.array(list(w_charge[ind_lst, :]) + rnd_vecs)
+            s_2 = block_diag(np.eye(self.n - nq), np.linalg.inv(f_mat))
 
-            R2 = np.linalg.inv(S2.T)
+            r_2 = np.linalg.inv(s_2.T)
 
         else:
-            S2 = np.eye(self.n, self.n)
-            R2 = S2
+            s_2 = np.eye(self.n, self.n)
+            r_2 = s_2
 
-        if self._is_Gram_Schmidt_successful(S2):
+        if self._is_gram_schmidt_successful(s_2):
 
-            self._apply_transformation(S2, R2)
+            self._apply_transformation(s_2, r_2)
 
             self.wTrans = self._round_to_zero_one(self.wTrans)
 
-            return S2, R2
+            return s_2, r_2
 
         else:
-            print("Gram_Schmidt process failed. Retrying...")
+            logger.warning('Gram_Schmidt process failed. Retrying...')
 
             return self._get_and_apply_transformation_2()
 
-    def _is_Gram_Schmidt_successful(self, S) -> bool:
-        """Check if the Gram_Schmidt process has the sufficient accuracy.
+    def _get_and_apply_transformation_3(self) -> Tuple[ndarray, ndarray]:
+        """ Get and apply the third transformation of the coordinates 
+        that scales the modes.
 
-        Parameters
+        Returns
         ----------
-            S:
-                Transformation matrices related to flux operators.
+            A tuple of the (S3, R3) transformation matrices.
         """
 
-        is_successful = True
-
-        # absolute value of the current wTrans
-        cur_wTrans = self.wTrans @ S
-
-        cur_wTrans = self._round_to_zero_one(cur_wTrans)
-
-        for j in range(self.n):
-            if self._is_charge_mode(j):
-                for abs_w in np.abs(cur_wTrans[:, j]):
-                    if abs_w != 0 and abs_w != 1:
-                        is_successful = False
-
-        return is_successful
-
-    def _get_and_apply_transformation_3(self) -> Tuple[ndarray, ndarray]:
-        """ Get and apply Third transformation of the coordinates that scales
-        the modes.
-        """
-
-        S3 = np.eye(self.n)
+        s_3 = np.eye(self.n)
 
         for j in range(self.n):
 
@@ -647,38 +1138,38 @@ def _get_and_apply_transformation_3(self) -> Tuple[ndarray, ndarray]:
                 continue
 
             # for harmonic modes
-            elif self._is_JJ_in_circuit():
+            elif self._is_junction_in_circuit():
 
                 # note: alpha here is absolute value of alpha (alpha is pure
                 # imaginary)
                 # get alpha for j-th mode
                 jth_alphas = np.abs(self.alpha(range(self.wTrans.shape[0]), j))
-                self.wTrans[:, j][jth_alphas < ACC["har_mode_elim"]] = 0
+                self.wTrans[:, j][jth_alphas < ACC['har_mode_elim']] = 0
 
-                if np.max(jth_alphas) > ACC["har_mode_elim"]:
+                if np.max(jth_alphas) > ACC['har_mode_elim']:
                     # find the coefficient in wTrans for j-th mode that
                     # has maximum alpha
                     s = np.abs(self.wTrans[np.argmax(jth_alphas), j])
-                    S3[j, j] = 1 / s
+                    s_3[j, j] = 1 / s
                 else:
                     # scale the uncoupled mode
                     s = np.max(np.abs(self.S[:, j]))
-                    S3[j, j] = 1 / s
+                    s_3[j, j] = 1 / s
 
             else:
                 # scale the uncoupled mode
                 s = np.max(np.abs(self.S[:, j]))
-                S3[j, j] = 1 / s
+                s_3[j, j] = 1 / s
 
-        R3 = np.linalg.inv(S3.T)
+        r_3 = np.linalg.inv(s_3.T)
 
-        self._apply_transformation(S3, R3)
+        self._apply_transformation(s_3, r_3)
 
-        return S3, R3
+        return s_3, r_3
 
-    def _transform_hamil(self):
-        """transform the Hamiltonian of the circuit that can be expressed
-        in charge and Fock bases
+    def _transform_hamil(self) -> None:
+        """Transform the Hamiltonian of the circuit into the charge and Fock
+        bases.
         """
 
         # get the first transformation
@@ -687,230 +1178,164 @@ def _transform_hamil(self):
         # natural frequencies of the circuit(zero for modes in charge basis)
         self.omega = np.sqrt(np.diag(self.cInvTrans) * np.diag(self.lTrans))
 
-        if self._is_JJ_in_circuit():
+        if self._is_junction_in_circuit():
             # get the second transformation
             self.S2, self.R2 = self._get_and_apply_transformation_2()
 
         # scaling the modes by third transformation
         self.S3, self.R3 = self._get_and_apply_transformation_3()
 
+    def _compute_descrip_vars(self) -> None:
+        """Compute parameters of transformed Hamiltonian to reduced number
+        of significant figures and in the frequency units of ``SQcircuit``;
+        store in the ``.descrip_vars`` dictionary for easy access.
+        """
+
+        # Dimensions of modes of circuit
+        self.descrip_vars['n_modes'] = self.n
+        self.descrip_vars['har_dim'] = np.sum(self.omega != 0)
+        self.descrip_vars['charge_dim'] = np.sum(self.omega == 0)
+        self.descrip_vars['n_loops'] = len(self.loops)
+
+        # Rounded matrices describing circuit layout
+        self.descrip_vars['W'] = np.round(self.wTrans, 6)
+        self.descrip_vars['S'] = np.round(self.S, 3)
+        if self.loops:
+            self.descrip_vars['B'] = np.round(self.B, 2)
+        else:
+            self.descrip_vars['B'] = np.zeros((
+                len(self.elem_keys[Junction]) + len(self.elem_keys[Inductor]), 1
+            ))
+
+        # Harmonic mode variables
+        self.descrip_vars['omega'] = (
+            self.omega / (2 * np.pi * unt.get_unit_freq())
+        )
+        self.descrip_vars['phi_zp'] = [
+            self._phi_zp(i) for i in range(self.descrip_vars['har_dim'])
+        ]
+
+        # Compute the element values in the correct units
+        # Junction energies
+        self.descrip_vars['EJ'] = []
+        for _, el, _, _ in self.elem_keys[Junction]:
+            self.descrip_vars['EJ'].append(
+                sqf.to_numpy(el.get_value('Hz'))
+                / (2 * np.pi * unt.get_unit_freq())
+            )
+
+        # Inductive energies
+        self.descrip_vars['EL'] = []
+        self.descrip_vars['EL_has_ext_flux'] = []
+        for _, el, b_id in self.elem_keys[Inductor]:
+            self.descrip_vars['EL'].append(
+                sqf.to_numpy(el.get_value('Hz')) / unt.get_unit_freq()
+            )
+            # Check if the element has any external flux associated with it
+            # (corresponding row of B is nonzero)
+            self.descrip_vars['EL_has_ext_flux'].append(
+                np.sum(np.abs(self.descrip_vars['B'][b_id, :])) != 0
+            )
+
+        # Charging energies
+        self.descrip_vars['EC'] = (
+            (2 * unt.e) **2
+            / (unt.hbar * 2 * np.pi * unt.get_unit_freq())
+            * self._get_quadratic_cInvTrans()
+        )
+
+        # Values of external flux and gate charge
+        self.descrip_vars['loops'] = [
+            sqf.to_numpy(self.loops[i].value()) / 2 / np.pi
+            for i in range(self.descrip_vars['n_loops'])
+        ]
+        self.descrip_vars['ng'] = [
+            self.charge_islands[i].value()
+            for i in range(self.descrip_vars['har_dim'], self.n)
+        ]
+
     def description(
-            self,
-            tp: Optional[str] = None,
-            _test: bool = False,
+        self,
+        tp: Optional[str] = None,
+        _test: bool = False,
     ) -> Optional[str]:
-        """
-        Print out Hamiltonian and a listing of the modes (whether they are
-        harmonic or charge modes with the frequency for each harmonic mode),
+        """Print out Hamiltonian and a listing of the modes (whether they are
+        harmonic or charge modes, with the frequency for each harmonic mode),
         Hamiltonian parameters, and external flux values.
 
+        Values printed in the description can be accessed using the
+        ``.descrip_vars`` dictionary, including a symbolic Hamiltonian in
+        ``.descrip_vars['H']``.
+
         Parameters
         ----------
             tp:
-                If ``None`` prints out the output as Latex if SQcircuit is
+                If ``None`` prints out the output as LaTeX if SQcircuit is
                 running in a Jupyter notebook and as text if SQcircuit is
-                running in Python terminal. If ``tp`` is ``"ltx"``,
-                the output is in Latex format if ``tp`` is ``"txt"`` the
+                running in Python terminal. If ``tp`` is ``'ltx'``,
+                the output is in LaTeX format, and if ``tp`` is ``'txt'`` the
                 output is in text format.
             _test:
                 if True, return the entire description as string
-                text. (use only for testing the function)
+                text (use only for testing the function).
+
+        Returns
+        ----------
+            The text of the description as a string, if ``_test`` is ``True``.
         """
         if tp is None:
             if is_notebook():
-                txt = HamilTxt('ltx')
+                txt = HamilTxt('ltx', _test=_test)
             else:
-                txt = HamilTxt('txt')
+                txt = HamilTxt('txt', _test=_test)
         else:
-            txt = HamilTxt(tp)
+            txt = HamilTxt(tp, _test=_test)
 
-        hamilTxt = txt.H()
-        harDim = np.sum(self.omega != 0)
-        chDim = np.sum(self.omega == 0)
-        W = np.round(self.wTrans, 6)
-        S = np.round(self.S, 3)
+        self._compute_descrip_vars()
+        self.descrip_vars['H'] = symbolic.construct_hamiltonian(self)
 
-        # If circuit has any loop:
-        if self.loops:
-            B = np.round(self.B, 2)
-        else:
-            B = np.zeros((len(self.junction_keys) + len(self.inductor_keys), 1))
-        EJLst = []
-        ELLst = []
-
-        for i in range(harDim):
-            hamilTxt += txt.omega(i + 1) + txt.ad(i + 1) + \
-                        txt.a(i + 1) + txt.p()
-
-        for i in range(chDim):
-            for j in range(chDim):
-                if j >= i:
-                    hamilTxt += txt.Ec(harDim + i + 1, harDim + j + 1) + \
-                                txt.n(harDim + i + 1, harDim + j + 1) + txt.p()
-
-        if '+' in hamilTxt[-3:-1]:
-            hamilTxt = hamilTxt[0:-2]
-
-        JJHamilTxt = ""
-        indHamilTxt = ""
-
-        for i, (edge, el, B_idx, W_idx) in enumerate(self.junction_keys):
-            EJLst.append(el.value() / 2 / np.pi / unt.get_unit_freq())
-            junTxt = txt.neg() + txt.Ej(i + 1) + txt.cos() + "("
-            # if B_idx is not None:
-            junTxt += txt.linear(txt.phi, W[W_idx, :]) + \
-                      txt.linear(txt.phiExt, B[B_idx, :], st=False)
-            # else:
-            #     junTxt += txt.linear(txt.phi, W[W_idx, :])
-            JJHamilTxt += junTxt + ")"
-
-        for i, (edge, el, B_idx) in enumerate(self.inductor_keys):
-
-            # if np.sum(np.abs(B[B_idx, :])) == 0 or B_idx is None:
-            if np.sum(np.abs(B[B_idx, :])) == 0:
-                continue
-            ELLst.append(el.energy())
-            indTxt = txt.p() + txt.El(i + 1) + "("
-            if 0 in edge:
-                w = S[edge[0] + edge[1] - 1, :]
-            else:
-                w = S[edge[0] - 1, :] - S[edge[1] - 1, :]
-            w = np.round(w[:harDim], 3)
-
-            indTxt += txt.linear(txt.phi, w) + ")(" + \
-                      txt.linear(txt.phiExt, B[B_idx, :])
-            indHamilTxt += indTxt + ")"
-
-        hamilTxt += indHamilTxt + JJHamilTxt + '\n'
-
-        # if '+' in hamilTxt[-3:-1] or '-' in hamilTxt[-3:-1]:
-        #     hamilTxt = hamilTxt[0:-2] + '\n'
-
-        modeTxt = ''
-        for i in range(harDim):
-            modeTxt += txt.mode(i + 1) + txt.tab() + txt.har()
-
-            modeTxt += txt.tab() + txt.phi(i + 1) + txt.eq() + txt.zp(i + 1) \
-                       + "(" + txt.a(i + 1) + "+" + txt.ad(i + 1) + ")"
-
-            omega = np.round(self.omega[i] / 2 / np.pi / unt.get_unit_freq(), 5)
-            zp = 2 * np.pi / unt.Phi0 * np.sqrt(unt.hbar / 2 * np.sqrt(
-                self.cInvTrans[i, i] / self.lTrans[i, i]))
-            zpTxt = "{:.2e}".format(zp)
-
-            modeTxt += txt.tab() + txt.omega(i + 1, False) + txt.eq() + str(
-                omega) + txt.tab() + txt.zp(i + 1) + txt.eq() + zpTxt
-
-            modeTxt += '\n'
-        for i in range(chDim):
-            modeTxt += txt.mode(harDim + i + 1) + txt.tab() + txt.ch()
-            ng = np.round(self.charge_islands[harDim + i].value(), 3)
-            modeTxt += txt.tab() + txt.ng(harDim + i + 1) + txt.eq() + str(ng)
-            modeTxt += '\n'
-
-        paramTxt = txt.param() + txt.tab()
-        for i in range(chDim):
-            for j in range(chDim):
-                if j >= i:
-                    paramTxt += txt.Ec(harDim + i + 1,
-                                       harDim + j + 1) + txt.eq()
-
-                    if i == j:
-                        Ec = (2 * unt.e) ** 2 / (
-                                unt.hbar * 2 * np.pi * unt.get_unit_freq()) * \
-                             self.cInvTrans[
-                                 harDim + i, harDim + j] / 2
-                    else:
-                        Ec = (2 * unt.e) ** 2 / (
-                                unt.hbar * 2 * np.pi * unt.get_unit_freq()) * \
-                             self.cInvTrans[
-                                 harDim + i, harDim + j]
-
-                    paramTxt += str(np.round(Ec, 3)) + txt.tab()
-        for i in range(len(ELLst)):
-            paramTxt += txt.El(i + 1) + txt.eq() + str(
-                np.round(ELLst[i], 3)) + txt.tab()
-        for i in range(len(EJLst)):
-            paramTxt += txt.Ej(i + 1) + txt.eq() + str(
-                np.round(EJLst[i], 3)) + txt.tab()
-        paramTxt += '\n'
-
-        loopTxt = txt.loops() + txt.tab()
-        for i in range(len(self.loops)):
-            phiExt = self.loops[i].value() / 2 / np.pi
-            loopTxt += txt.phiExt(i + 1) + txt.tPi() + txt.eq() + str(
-                phiExt) + txt.tab()
-
-        finalTxt = hamilTxt + txt.line + modeTxt + txt.line + paramTxt + loopTxt
-
-        txt.display(finalTxt)
+        final_txt = txt.print_circuit_description(self.descrip_vars)
 
         if _test:
-            return finalTxt
+            return final_txt
 
     def loop_description(self, _test: bool = False) -> Optional[str]:
-        """
-        Print out the external flux distribution over inductive elements.
+        """Print out the external flux distribution over inductive elements.
 
         Parameters
         ----------
             _test:
                 if True, return the entire description as string
-                text. (use only for testing the function)
+                text (use only for testing the function).
 
+        Returns
+        ----------
+            The text of the external flux distribution, if ``_test`` is
+            ``True``.
         """
+        loop_description_txt = HamilTxt.print_loop_description(self)
 
-        # maximum length of element ID strings
-        nr = max(
-            [len(el.id_str) for _, el, _, _ in self.junction_keys]
-            + [len(el.id_str) for _, el, _ in self.inductor_keys]
-        )
-
-        # maximum length of loop ID strings
-        nh = max([len(lp.id_str) for lp in self.loops])
-
-        # number of loops
-        nl = len(self.loops)
-
-        # space between elements in rows
-        ns = 5
-
-        loop_description_txt = ''
-
-        header = (nr + ns + len(", b1:")) * " "
-        for i in range(nl):
-            lp = self.loops[i]
-            header += ("{}" + (nh + 10 - len(lp.id_str)) * " ").format(
-                lp.id_str)
-
-        loop_description_txt += header + '\n'
-
-        # add line under header
-        loop_description_txt += "-" * len(header) + '\n'
-        for i in range(self.B.shape[0]):
-
-            el = None
-            for _, el_ind, B_idx in self.inductor_keys:
-                if i == B_idx:
-                    el = el_ind
-            for _, el_ind, B_idx, W_idx in self.junction_keys:
-                if i == B_idx:
-                    el = el_ind
+        if _test:
+            return loop_description_txt
+        
+    def element_description(self, _test: bool = False) -> Optional[str]:
+        """Print out the element
+        
+        Parameters
+        ----------
+            _test:
+                if True, return the entire description as string
+                text (use only for testing the function).
 
-            id = el.id_str
-            row = id + (nr - len(id)) * " "
-            bStr = f", b{i + 1}:"
-            row += bStr
-            row += (ns + len(", b1:") - len(bStr)) * " "
-            for j in range(nl):
-                b = np.round(np.abs(self.B[i, j]), 2)
-                row += ("{}" + (nh + 10 - len(str(b))) * " ").format(b)
-            loop_description_txt += row + '\n'
+        Returns
+        ----------
+            The text of the external flux distribution, if ``_test`` is
+            ``True``.
+        """
+        el_description_txt = HamilTxt.print_el_description(self.elements)
 
         if _test:
-            return loop_description_txt
-        else:
-            print(loop_description_txt)
+            return el_description_txt
 
     def set_trunc_nums(self, nums: List[int]) -> None:
         """Set the truncation numbers for each mode.
@@ -924,10 +1349,15 @@ def set_trunc_nums(self, nums: List[int]) -> None:
                 ..., 0, ..., (N-1).
         """
 
-        error1 = "The input must be be a python list"
-        assert isinstance(nums, list), error1
-        error2 = "Length of the input must be equal to the number of modes."
-        assert len(nums) == self.n, error2
+        logger.info('set_trunc_nums called')
+
+        if not isinstance(nums, list):
+            raise ValueError('The input must be be a python list')
+        if len(nums) != self.n:
+            raise ValueError(
+                'The number of modes (length of the input) must be equal to the'
+                ' number of nodes.'
+            )
 
         self.m = self.n*[1]
 
@@ -949,7 +1379,7 @@ def set_trunc_nums(self, nums: List[int]) -> None:
         self._build_exp_ops()
 
     def set_charge_offset(self, mode: int, ng: float) -> None:
-        """set the charge offset for each charge mode.
+        """Set the charge offset for each charge mode.
 
         Parameters
         ----------
@@ -959,36 +1389,47 @@ def set_charge_offset(self, mode: int, ng: float) -> None:
             ng:
                 The charge offset.
         """
-        assert isinstance(mode, int), "Mode number should be an integer"
+        if not isinstance(mode, int):
+            raise ValueError('Mode number should be an integer.')
+        if mode - 1 not in self.charge_islands:
+            raise ValueError('The specified mode is not a charge mode.')
 
-        error = "The specified mode is not a charge mode."
-        assert mode - 1 in self.charge_islands, error
         if len(self.m) == 0:
-            self.charge_islands[mode - 1].setOffset(ng)
+            self.charge_islands[mode - 1].set_offset(ng)
         else:
-            self.charge_islands[mode - 1].setOffset(ng)
+            self.charge_islands[mode - 1].set_offset(ng)
 
             self._build_op_memory()
 
             self._LC_hamil = self._get_LC_hamil()
 
-    def set_charge_noise(self, mode: int, A: float) -> None:
-        """set the charge noise for each charge mode.
+    def set_charge_noise(self, mode: int, amp: float) -> None:
+        """Set the charge noise for each charge mode.
 
         Parameters
         ----------
             mode:
                 An integer that specifies the charge mode. To see which mode
                 is a charge mode, we can use ``description()`` method.
-            A:
-                The charge noise.
+            amp:
+                The charge noise amplitude.
         """
-        assert isinstance(mode, int), "Mode number should be an integer"
+        if not isinstance(mode, int):
+            raise ValueError('The mode number should be an integer.')
+        if mode - 1 not in self.charge_islands:
+            raise ValueError('The specified mode is not a charge mode.')
+
+        self.charge_islands[mode - 1].setNoise(amp)
+
+    def _get_op_dims(self) -> List[list]:
+        """Return the operator dims related to ``Qutip.Qobj``."""
+
+        return [self.ms, self.ms]
 
-        assert mode - 1 in self.charge_islands, "The specified mode " \
-                                                "is not a charge mode."
+    def _get_state_dims(self) -> List[list]:
+        """Return the state dims related to ``Qutip.Qobj``."""
 
-        self.charge_islands[mode - 1].setNoise(A)
+        return [self.ms, len(self.ms) * [1]]
 
     def _squeeze_op(self, op: Qobj) -> Qobj:
         """
@@ -998,14 +1439,35 @@ def _squeeze_op(self, op: Qobj) -> Qobj:
         ----------
             op:
                 Any quantum operator in qutip.Qobj format
-        """
 
-        op_sq = op.copy()
+        Returns
+        ----------
+            A squeezed copy of ``op``.
+        """
 
-        op_sq.dims = [self.ms, self.ms]
+        if isinstance(op, Qobj):
+            return op.contract()
 
+        op_sq = sqf.copy(op)
+        op_sq.dims = self._get_op_dims()
         return op_sq
 
+    def _impedance(self, i: int) -> float:
+        """Compute the impedance of the ``i``th mode.
+
+        Parameters
+        ----------
+            i:
+                Index of the mode (starts from zero for the first mode).
+        
+        Returns
+        ----------
+            The impedance Z of mode ``i``.
+        """
+        return np.sqrt(
+            self.cInvTrans[i, i] / self.lTrans[i, i]
+        )
+
     def _charge_op_isolated(self, i: int) -> Qobj:
         """Return charge operator for each isolated mode normalized by
         square root of hbar. By isolated, we mean that the operator is not in
@@ -1014,17 +1476,23 @@ def _charge_op_isolated(self, i: int) -> Qobj:
         Parameters
         ----------
             i:
-                Index of the mode. (starts from zero for the first mode)
+                Index of the mode (starts from zero for the first mode).
+
+        Returns
+        ----------
+            The isolated charge operator for the ``i``th mode.
         """
 
         if self._is_charge_mode(i):
             ng = self.charge_islands[i].value()
             op = (2*unt.e/np.sqrt(unt.hbar)) * (qt.charge((self.m[i]-1)/2)-ng)
-
         else:
-            Z = np.sqrt(self.cInvTrans[i, i] / self.lTrans[i, i])
-            Q_zp = -1j * np.sqrt(0.5/Z)
-            op = Q_zp * (qt.destroy(self.m[i]) - qt.create(self.m[i]))
+            Q_zp = -1j * np.sqrt(0.5 / self._impedance(i))
+            if self.m[i] == 1:
+                # QuTiP 5.0.x no longer supports create/destroy with N = 1
+                op = Qobj([[0.]]).to('csr')
+            else:
+                op = Q_zp * (qt.destroy(self.m[i]) - qt.create(self.m[i]))
 
         return op
 
@@ -1037,14 +1505,21 @@ def _flux_op_isolated(self, i: int) -> Qobj:
         ----------
             i:
                 Index of the mode. (starts from zero for the first mode)
+
+        Returns
+        ----------
+            The isolated flux operator for the ``i``th mode.
         """
 
         if self._is_charge_mode(i):
             op = qt.qeye(self.m[i])
-
         else:
-            Z = np.sqrt(self.cInvTrans[i, i] / self.lTrans[i, i])
-            op = np.sqrt(0.5*Z) * (qt.destroy(self.m[i])+qt.create(self.m[i]))
+            if self.m[i] == 1:
+                op = Qobj([[0.]]).to('csr')
+            else:
+                op = np.sqrt(0.5 * self._impedance(i)) * (
+                    qt.destroy(self.m[i]) + qt.create(self.m[i])
+                )
 
         return op
 
@@ -1057,6 +1532,10 @@ def _num_op_isolated(self, i: int) -> Qobj:
         ----------
             i:
                 Index of the mode. (starts from zero for the first mode)
+    
+        Returns
+        ----------
+            The isolated number operator for the ``i``th mode.
         """
 
         if self._is_charge_mode(i):
@@ -1067,7 +1546,7 @@ def _num_op_isolated(self, i: int) -> Qobj:
 
         return op
 
-    def _d_op_isolated(self, i: int, w: float, qp: bool = False) -> Qobj:
+    def _d_op_isolated(self, i: int, w: float) -> Qobj:
         """Return charge displacement operator for each isolated mode. By
         isolated, we mean that the operator is not in the general tensor
         product states of the overall system.
@@ -1079,118 +1558,68 @@ def _d_op_isolated(self, i: int, w: float, qp: bool = False) -> Qobj:
             w:
                 Represent the power of the displacement operator, d^w. Right
                 now w should be only 0, 1, and -1.
-            qp:
-                boolean flag shows if we calculate the displacement operator
-                for quasiparticle loss calculation.
-        """
-
-        if qp:
-            N = 2*self.m[i] - 1
-        else:
-            N = self.m[i]
-
-        if w == 0:
-            return qt.qeye(N)
-
-        d = np.zeros((N, N))
-
-        for k in range(N):
-            for j in range(N):
-                if j - 1 == k:
-                    d[k, j] = 1
-        d = qt.Qobj(d)
-
-        if w < 0:
-            return d
-
-        elif w > 0:
-            return d.dag()
 
-    def _T_even_isolated(self, i: int) -> Qobj:
-        """Return transformations related to even parity for quasiparticle
-        loss calculation for i-th mode.
-
-        Parameters
+        Returns
         ----------
-            i:
-                Index of the mode. (starts from zero for the first mode)
+            The isolated charge displacement operator for the ``i``th mode, to 
+            the power of ``w``.
         """
 
-        T_even = np.zeros((2*self.m[i]-1, self.m[i]))
-        indices = np.arange(self.m[i])
-        T_even[2*indices, indices] = 1
-
-        return qt.Qobj(T_even)
-
-    def _T_odd_isolated(self, i: int) -> Qobj:
-        """Return transformations related to odd parity for quasiparticle
-        loss calculation for i-th mode.
-
-        Parameters
-        ----------
-            i:
-                Index of the mode. (starts from zero for the first mode)
-        """
-
-        T_odd = np.zeros((2*self.m[i]-1, self.m[i]))
-        indices = np.arange(self.m[i]-1)
-
-        T_odd[2*indices+1, indices] = 1
-
-        return qt.Qobj(T_odd)
-
-    def _T_even(self):
-        """Return transformations related to even parity for quasiparticle
-        loss calculation.
-        """
-
-        # list of transformation for each mode
-        T_list = []
+        if w == 0:
+            return qt.qeye(self.m[i])
 
-        for i in range(self.n):
-            if self._is_charge_mode(i):
-                T_list.append(self._T_even_isolated(i))
-            else:
-                T_list.append(qt.qeye(self.m[i]))
+        d = np.zeros((self.m[i], self.m[i]))
 
-        return self._squeeze_op(qt.tensor(*T_list))
+        for k in range(self.m[i]):
+            for j in range(self.m[i]):
+                if j - 1 == k:
+                    d[k, j] = 1
+        d = Qobj(d).to('csr')
 
-    def _T_odd(self):
-        """Return transformations related to odd parity for quasiparticle
-        loss calculation.
-        """
+        if w < 0:
+            return d
 
-        # list of transformation for each mode
-        T_list = []
+        elif w > 0:
+            return d.dag()
 
-        for i in range(self.n):
-            if self._is_charge_mode(i):
-                T_list.append(self._T_odd_isolated(i))
-            else:
-                T_list.append(qt.qeye(self.m[i]))
+    def _phi_zp(self, i: int) -> float:
+        """Return the zero-point fluctuations for the ``i``th mode's
+        flux operator.
+
+        Parameters
+        ----------
+            i:
+                Index of the mode (starts from zero for the first mode).
 
-        return self._squeeze_op(qt.tensor(*T_list))
+        Returns
+        ----------
+            The zero-point fluctuations for mode ``i``.
+        """
+        return (
+            (2 * np.pi / unt.Phi0)
+            * np.sqrt(unt.hbar / 2 * self._impedance(i))
+        )
 
-    def alpha(self, i: Union[int, range], j: int) -> float:
+    def alpha(self, i: int, j: int) -> float:
         """Return the alpha, amount of displacement, for the bosonic
         displacement operator for junction i and mode j.
 
         Parameters
         ----------
             i:
-                Index of the Junction. (starts from zero for the first mode)
+                Index of the Junction (starts from zero for the first mode)
             j:
-               Index of the mode. (starts from zero for the first mode)
-        """
-
-        Z = np.sqrt(self.cInvTrans[j, j] / self.lTrans[j, j])
+                Index of the mode (starts from zero for the first mode).
 
-        coef = 2 * np.pi / unt.Phi0 * 1j
+        Returns
+        ----------
+            The value of alpha for the junction ``i`` and mode ``j``.
+        """
 
-        return coef * np.sqrt(unt.hbar/2*Z) * self.wTrans[i, j]
+        return 1j * self._phi_zp(j) * self.wTrans[i, j]
 
     def _build_op_memory(self) -> None:
-        """build the charge, flux, number, and cross multiplication of charge
+        """Build the charge, flux, number, and cross multiplication of charge
         operators and store them in memory related to operators.
         """
 
@@ -1208,7 +1637,11 @@ def _build_op_memory(self) -> None:
             for j in range(self.n):
                 if i == j:
                     Q_iso = self._charge_op_isolated(j)
-                    Q2 = Q + [Q_iso * Q_iso]
+                    if Q_iso.shape == (1, 1):
+                        # QuTip 5.0.x converts (1x1 matrix)^2 to scalar
+                        Q2 = Q + [Qobj(Q_iso * Q_iso).to('csr')]
+                    else:
+                        Q2 = Q + [Q_iso * Q_iso]
                     # append the rest with qeye.
                     Q2 += [qt.qeye(self.m[k]) for k in range(j+1, self.n)]
                     charges_row.append(self._squeeze_op(qt.tensor(*Q2)))
@@ -1232,10 +1665,10 @@ def _build_op_memory(self) -> None:
             flux_ops.append(self._squeeze_op(qt.tensor(*flux)))
             charge_by_charge_ops.append(charges_row)
 
-        self._memory_ops["Q"] = charge_ops
-        self._memory_ops["QQ"] = charge_by_charge_ops
-        self._memory_ops["phi"] = flux_ops
-        self._memory_ops["N"] = num_ops
+        self._memory_ops['Q'] = charge_ops
+        self._memory_ops['QQ'] = charge_by_charge_ops
+        self._memory_ops['phi'] = flux_ops
+        self._memory_ops['N'] = num_ops
 
     def _build_exp_ops(self) -> None:
         """Build exponential operators needed to construct cosine potential of
@@ -1265,101 +1698,137 @@ def _build_exp_ops(self) -> None:
 
                 if self._is_charge_mode(j):
                     exp.append(self._d_op_isolated(j, self.wTrans[i, j]))
-                    exp_h.append(self._d_op_isolated(j, self.wTrans[i, j]
-                                                     , qp=True))
+                    exp_h.append(qt.qeye(self.m[j]))
                 else:
-                    exp.append(qt.displace(self.m[j], self.alpha(i, j)))
-                    exp_h.append(qt.displace(self.m[j], self.alpha(i, j)/2))
+                    if self.m[j] == 1:
+                        exp.append(qt.qeye(1).to('csr'))
+                        exp_h.append(qt.qeye(1).to('csr'))
+                    else:
+                        exp.append(qt.displace(self.m[j], self.alpha(i, j)).to('csr'))
+                        exp_h.append(qt.displace(self.m[j], self.alpha(i, j) / 2).to('csr'))
 
             exp_ops.append(self._squeeze_op(qt.tensor(*exp)))
             root_exp_ops.append(self._squeeze_op(qt.tensor(*exp_h)))
 
-        self._memory_ops["exp"] = exp_ops
-        self._memory_ops["root_exp"] = root_exp_ops
+        self._memory_ops['exp'] = exp_ops
+        self._memory_ops['root_exp'] = root_exp_ops
 
-    def _get_LC_hamil(self) -> Qobj:
+    def _get_quadratic_cInvTrans(self) -> ndarray:
+        """Compute a modified ``cInvTrans`` for use when computing a
+        quadratic form like
+            ``0.5 Q^T @ cInvTrans @ Q``
+        by summing over the upper/lower triangle.
+
+        Returns
+        ----------
+            ``self.cInvTrans`` with the diagonal divided by 2.
         """
-        get the LC part of the Hamiltonian
-        outputs:
-            -- HLC: LC part of the Hamiltonian (qutip Object)
+        q_cInvTrans = self.cInvTrans.copy()
+        q_cInvTrans[np.diag_indices_from(q_cInvTrans)] /= 2
+
+        return q_cInvTrans
+
+    def _get_LC_hamil(self) -> Qobj:
+        """Construct the LC part of the circuit Hamiltonian.
+
+        Returns
+        ----------
+            The LC part of the Hamiltonian.
         """
+        LC_hamil = 0
 
-        LC_hamil = qt.Qobj()
+        q_cInvTrans = self._get_quadratic_cInvTrans()
 
         for i in range(self.n):
             # we write j in this form because of "_memory_ops["QQ"]" shape
             for j in range(self.n - i):
                 if j == 0:
                     if self._is_charge_mode(i):
-                        LC_hamil += (0.5 * self.cInvTrans[i, i]
-                                     * self._memory_ops["QQ"][i][j])
+                        LC_hamil += (q_cInvTrans[i, i]
+                                     * self._memory_ops['QQ'][i][j])
                     else:
-                        LC_hamil += self.omega[i] * self._memory_ops["N"][i]
+                        LC_hamil += self.omega[i] * self._memory_ops['N'][i]
 
                 elif j > 0:
                     if self.cInvTrans[i, i + j] != 0:
-                        LC_hamil += (self.cInvTrans[i, i + j]
-                                     * self._memory_ops["QQ"][i][j])
+                        LC_hamil += (q_cInvTrans[i, i + j]
+                                     * self._memory_ops['QQ'][i][j])
 
         return LC_hamil
 
-    def _get_external_flux_at_element(self, B_idx: int) -> float:
+    def _get_external_flux_at_element(
+        self,
+        b_id: int,
+        torch = False
+    ) -> Union[float, Tensor]:
         """
         Return the external flux at an inductive element.
 
         Parameters
         ----------
-            B_idx:
+            b_id:
                 An integer point to each row of B matrix (external flux
                 distribution of that element)
+            torch:
+                If ``True``, cast loop values to floats always
+
+        Returns
+        ----------
+            The external flux at the element referenced by ``b_id.``
         """
-        phi_ext = 0.0
+        phi_ext = sqf.zero()
         for i, loop in enumerate(self.loops):
-            phi_ext += loop.value(self.random) * self.B[B_idx, i]
+            phi_ext += loop.value() * self.B[b_id, i]
 
-        return phi_ext
+        if isinstance(phi_ext, Tensor) and not torch:
+            return sqf.to_numpy(phi_ext)
+        else:
+            return phi_ext
 
     def _get_inductive_hamil(self) -> Qobj:
+        """Construct the inductive part of the circuit Hamiltonian.
 
-        H = qt.Qobj()
+        Returns
+        ----------
+            The inductive part of the Hamiltonian.
+        """
 
-        for edge, el, B_idx in self.inductor_keys:
+        H = 0
+        for edge, el, b_id in self.elem_keys[Inductor]:
             # phi = 0
-            # if B_idx is not None:
-            phi = self._get_external_flux_at_element(B_idx)
+            # if b_id is not None:
+            phi = self._get_external_flux_at_element(b_id)
 
             # summation of the 1 over inductor values.
-            x = 1 / el.value(self.random)
-            op = self.coupling_op("inductive", edge)
+            x = np.squeeze(sqf.to_numpy(1 / el.get_value()))
+            op = self.coupling_op('inductive', edge, force_use_qutip=True)
             H += x * phi * (unt.Phi0 / 2 / np.pi) * op / np.sqrt(unt.hbar)
 
             # save the operators for loss calculation
-            self._memory_ops["ind_hamil"][(el, B_idx)] = op
-
-        for _, el, B_idx, W_idx in self.junction_keys:
+            self._memory_ops['ind_hamil'][(el, b_id)] = op
+        for _, el, b_id, w_id in self.elem_keys[Junction]:
             # phi = 0
-            # if B_idx is not None:
-            phi = self._get_external_flux_at_element(B_idx)
-
-            EJ = el.value(self.random)
+            # if b_id is not None:
+            phi = self._get_external_flux_at_element(b_id)
 
-            exp = np.exp(1j * phi) * self._memory_ops["exp"][W_idx]
-            root_exp = np.exp(1j * phi / 2) * self._memory_ops["root_exp"][
-                W_idx]
+            EJ = sqf.to_numpy(el.get_value())
 
+            exp = np.exp(1j * phi) * self._memory_ops['exp'][w_id]
+            root_exp = np.exp(1j * phi / 2) * self._memory_ops['root_exp'][
+                w_id]
             cos = (exp + exp.dag()) / 2
             sin = (exp - exp.dag()) / 2j
             sin_half = (root_exp - root_exp.dag()) / 2j
 
-            self._memory_ops["cos"][el, B_idx] = self._squeeze_op(cos)
-            self._memory_ops["sin"][el, B_idx] = self._squeeze_op(sin)
-            self._memory_ops["sin_half"][el, B_idx] = sin_half
+            self._memory_ops['cos'][el, b_id] = self._squeeze_op(cos)
+            self._memory_ops['sin'][el, b_id] = self._squeeze_op(sin)
+            self._memory_ops['sin_half'][el, b_id] = self._squeeze_op(sin_half)
 
             H += -EJ * cos
 
         return H
 
-    def charge_op(self, mode: int, basis: str = 'FC') -> Qobj:
+    def charge_op(self, mode: int, basis: str = 'original') -> Qobj:
         """Return charge operator for specific mode in the Fock/Charge basis or
         the eigenbasis.
 
@@ -1368,63 +1837,136 @@ def charge_op(self, mode: int, basis: str = 'FC') -> Qobj:
             mode:
                 Integer that specifies the mode number.
             basis:
-                String that specifies the basis. It can be either ``"FC"``
-                for original Fock/Charge basis or ``"eig"`` for eigenbasis.
-        """
+                String that specifies the basis. It can be either ``'original'``
+                for original Fock/Charge basis or ``'eig'`` for eigenbasis.
 
-        error1 = "Please specify the truncation number for each mode."
-        assert len(self.m) != 0, error1
+        Returns
+        ----------
+            The charge operator for the ``i``th mode in the basis specified by
+            ``basis``.
+        """
+        if len(self.m) == 0:
+            raise CircuitStateError('Please specify the truncation number for each mode.')
+        if basis not in ['original', 'eig']:
+            raise ValueError(f'Invalid basis \'{basis}\' passed. Permitted '
+                             'bases are \'original\' and \'eig\'.')
 
         # charge operator in Fock/Charge basis
-        Q_FC = self._memory_ops["Q"][mode-1]
+        Q_FC = self._memory_ops['Q'][mode-1]
+
+        if basis == 'original':
+            if get_optim_mode():
+                return sqf.qobj_to_tensor(Q_FC)
+            else:
+                return Q_FC
+
+        elif basis == 'eig':
+            if get_optim_mode():
+                mat_evecs = self._evecs.T
+                Q = sqf.qobj_to_tensor(Q_FC)
+                Q_eig = torch.conj(mat_evecs.T) @ Q @ mat_evecs
 
-        if basis == "FC":
+                return Q_eig
+            else:
+                mat_evecs = np.concatenate(list(map(
+                    lambda v: v.full(), self._evecs)), axis=1)
+                Q_eig = np.conj(mat_evecs.T) @ Q_FC.full() @ mat_evecs
+
+                return Qobj(Q_eig)
 
-            return Q_FC
+    def _get_w_id(self, el: Junction, b_id: int) -> Optional[int]:
+        """"
+        Find the corresponding ``W`` matrix index given an junction and its
+        ``B`` matrix index.
+        
+        Parameters
+        ----------
+            el:
+                Josephson junction in circuit.
+            b_id:
+                Index of B matrix corresponding to ``el``.
 
-        elif basis == "eig":
+        Returns
+        ----------
+            The corresponding ``W`` matrix index, if it exists.
+        """
+        for _, o_el, o_b_id, w_id in self.elem_keys[Junction]:
+            if o_el == el and o_b_id == b_id:
+                return w_id
 
-            # number of eigenvalues
-            n_eig = len(self.efreqs)
+        return None
 
-            Q_eig = np.zeros((n_eig, n_eig), dtype=complex)
+    def op(self, typ: str, keywords: Dict) -> Union[Qobj, Tensor]:
+        """Get a saved circuit operator of type ``typ``, specified by keywords
+        given in the ``keywords`` dict, as a backpropagatable ``Tensor`` object 
+        when using the ``'PyTorch'`` engine. Currently supports the
+        following operators:
 
-            for i in range(n_eig):
-                for j in range(n_eig):
-                    Q_eig[i, j] = (self._evecs[i].dag()
-                                   * Q_FC * self._evecs[j]).data[0, 0]
+        * ``'sin_half'``
 
-            return qt.Qobj(Q_eig)
+        Parameters
+        ----------
+            typ:
+                Type of saved operator.
+            keywords:
+                Dictionary specifying which operator of type ``typ`` to return.
 
-    def diag(self, n_eig: int) -> Tuple[ndarray, List[Qobj]]:
+        Returns
+        ----------
+            The `typ` operator of the circuit specified by ``keywords``.
         """
-        Diagonalize the Hamiltonian of the circuit and return the
-        eigenfrequencies and eigenvectors of the circuit up to specified
-        number of eigenvalues.
+        if typ == 'sin_half':
+            b_id = keywords['b_id']
+            el = keywords['el']
+            if get_optim_mode():
+                w_id = self._get_w_id(el, b_id)
+
+                phi = self._get_external_flux_at_element(b_id, torch=True)
+                root_exp = (
+                    torch.exp(1j * phi / 2)
+                    * sqf.qobj_to_tensor(self._memory_ops['root_exp'][w_id])
+                )
 
+                sin_half = (root_exp - sqf.dag(root_exp)) / 2j
+                # ToDo: need to squeeze?
+                return sin_half
+            else:
+                return self._memory_ops['sin_half'][el, b_id]
+        else:
+            raise ValueError('The operator \'{typ}\' is not supported.')
+
+    def _diag_np(
+        self,
+        n_eig: int
+    ) -> Tuple[Union[ndarray, Tensor], List[Union[Qobj, Tensor]]]:
+        """Perform the diagonalization of the circuit Hailtonian using SciPy's
+        sparse eigensolver. 
+        
         Parameters
         ----------
             n_eig:
-                Number of eigenvalues to output. The lower ``n_eig``, the
-                faster ``SQcircuit`` finds the eigenvalues.
+                Number of eigenvalues to compute.
+
         Returns
         ----------
-            efreq:
-                ndarray of eigenfrequencies in frequency unit of SQcircuit
-                (gigahertz by default)
+            efreqs:
+                Array of eigenfrequencies in frequency unit of SQcircuit.
             evecs:
-                List of eigenvectors in qutip.Qobj format.
+                List of eigenvectors in qutip.Qobj or Tensor format, depending
+                on numerical engine.
         """
-        error1 = "Please specify the truncation number for each mode."
-        assert len(self.m) != 0, error1
-        error2 = "n_eig (number of eigenvalues) should be an integer."
-        assert isinstance(n_eig, int), error2
+        hamil = self.hamiltonian()
 
-        H = self.hamiltonian()
-
-        # get the data out of qutip variable and use sparse scipy eigen
-        # solver which is faster.
-        efreqs, evecs = scipy.sparse.linalg.eigs(H.data, n_eig, which='SR')
+        # get the data out of qutip variable and use sparse SciPy 
+        # eigensolver which is faster.
+        try:
+            efreqs, evecs = scipy.sparse.linalg.eigs(
+                hamil.data_as('csr_matrix'), k=n_eig, which='SR'
+            )
+        except ArpackNoConvergence:
+            efreqs, evecs = scipy.sparse.linalg.eigs(
+                hamil.data_as('csr_matrix'), k=n_eig, ncv=10*n_eig, which='SR'
+            )
         # the output of eigen solver is not sorted
         efreqs_sorted = np.sort(efreqs.real)
 
@@ -1433,7 +1975,7 @@ def diag(self, n_eig: int) -> Tuple[ndarray, List[Qobj]]:
             sort_arg = [sort_arg]
 
         evecs_sorted = [
-            qt.Qobj(evecs[:, ind], dims=[self.ms, len(self.ms) * [1]])
+            Qobj(evecs[:, ind], dims=self._get_state_dims())
             for ind in sort_arg
         ]
 
@@ -1441,7 +1983,139 @@ def diag(self, n_eig: int) -> Tuple[ndarray, List[Qobj]]:
         self._efreqs = efreqs_sorted
         self._evecs = evecs_sorted
 
-        return efreqs_sorted / (2*np.pi*unt.get_unit_freq()), evecs_sorted
+        return efreqs_sorted / (2 * np.pi * unt.get_unit_freq()), evecs_sorted
+
+    def _diag_torch(self, n_eig: int) -> Tuple[Tensor, Tensor]:
+        """Diagonalize the circuit using a Torch Function, so that the 
+        calculated eigenvalues/vectors are backpropagatable.
+
+        To restrict the number ``n`` of eigenvectors for which the gradient is
+        computed, call ``set_max_eigenvector_grad(n)`` before
+        diagonalizing.
+        
+        Parameters
+        ----------
+            n_eig:
+                Number of eigenvalues to compute.
+
+        Returns
+        ----------
+            efreqs:
+                Tensor of eigenfrequencies.
+            evecs:
+                Tensor of eigenvectors.
+        """
+        eigen_solution = sqtorch.eigencircuit(self, n_eig)
+        eigenvalues = torch.real(eigen_solution[:, 0])
+        eigenvectors = eigen_solution[:, 1:]
+        self._efreqs = eigenvalues
+        self._evecs = eigenvectors
+
+        return eigenvalues / (2 * np.pi * unt.get_unit_freq()), eigenvectors
+
+    def diag(
+        self,
+        n_eig: int
+    ) -> Tuple[Union[ndarray, Tensor], List[Union[Qobj, Tensor]]]:
+        """
+        Diagonalize the Hamiltonian of the circuit and return the
+        eigenfrequencies and eigenvectors of the circuit up to specified
+        number of eigenvalues.
+
+        Parameters
+        ----------
+            n_eig:
+                Number of eigenvalues to output. The lower ``n_eig``, the
+                faster ``SQcircuit`` finds the eigenvalues.
+
+        Returns
+        ----------
+            efreqs:
+                ndarray of eigenfrequencies in frequency unit of SQcircuit
+                (gigahertz by default).
+            evecs:
+                List of eigenvectors in qutip.Qobj or Tensor format, depending
+                on numerical engine.
+        """
+        if len(self.m) == 0:
+            raise CircuitStateError('Please specify the truncation number for each mode.')
+        if not isinstance(n_eig, int):
+            raise ValueError('n_eig (number of eigenvalues) should be an integer.')
+
+        logger.info('diag called')
+        if get_optim_mode():
+            return self._diag_torch(n_eig)
+        else:
+            return self._diag_np(n_eig)
+
+    def truncate_circuit(self, K: int) -> List[int]:
+        """Set Hilbert space dimensionality of circuit to ``k=floor(K^{1/n})``
+        for all modes, where ``n`` is the number of modes in the circuit.
+        Note that charge modes have dimensionality 2*m+1, where m is the
+        assigned truncation number. Consequently, to ensure equal Hilbert
+        space sizing among charge and harmonic modes, a truncation number
+        of ``(1 / 2) * (k + 1)`` is assigned to charge modes.
+
+
+        Parameters
+        ----------
+            K:
+                Total truncation number
+
+        Returns
+        ----------
+            trunc_nums:
+                List of truncation numbers for each mode of the circuit
+        """
+        if not isinstance(K, int):
+            raise ValueError('The total truncation number must be an integer.')
+        if K < 1:
+            raise ValueError('The total truncation number must be >=1.')
+
+        trunc_num_average = K ** (1 / len(self.omega))
+        charge_cutoff = (1 / 2) * (trunc_num_average + 1)
+
+        trunc_nums = []
+        for mode_idx in range(len(self.omega)):
+            # Harmonic mode
+            if self.omega[mode_idx] != 0:
+                trunc_nums.append(int(np.floor(trunc_num_average)))
+            # Charge mode
+            else:
+                trunc_nums.append(int(np.floor(charge_cutoff)))
+
+        self.set_trunc_nums(trunc_nums)
+        return trunc_nums
+
+    def check_convergence(self, eig_vec_idx=1, t=10, threshold=1e-5):
+        """
+        Check whether the diagonalization of the circuit has converged.
+
+        Parameters
+        ----------
+            eig_vec_idx:
+                Index of eigenvector to use to test convergence.
+            t:
+                Number of entries of eigenvector to use to test convergence.
+            threshold:
+                Cutoff for convergence.
+
+        Returns
+        ----------
+            convergence_succeeded:
+                Truthy value of whether the circuit converged
+            epsilon:
+                Calculated value for convergence test
+        """
+        if (self._efreqs.shape[0] == 0) or (len(self._evecs) == 0):
+            raise CircuitStateError('Must call circuit.diag before testing convergence.')
+
+        reshaped_evec = sqf.to_numpy(self.evecs[eig_vec_idx]).reshape(self.m)
+        restricted_evec = reshaped_evec[(slice(-t),)*len(self.m)]
+
+        epsilon = 1 - np.sum(np.abs(restricted_evec)**2)
+
+        return (epsilon < threshold), epsilon
 
     ###########################################################################
     # Methods that calculate circuit properties
@@ -1457,37 +2131,53 @@ def coord_transform(self, var_type: str) -> ndarray:
             var_type:
                 The type of the variables that can be either ``"charge"`` or
                 ``"flux"``.
+
+        Returns
+        ----------
+            Matrix giving coordinate transformation for ``var_type`` coordinates.
         """
-        if var_type == "charge" or var_type == "Charge":
+        var_type = var_type.lower()
+        if var_type == 'charge':
             return np.linalg.inv(self.R)
-        elif var_type == "flux" or var_type == "Flux":
+        elif var_type == 'flux':
             return np.linalg.inv(self.S)
         else:
-            raise ValueError("The input must be either \"charge\" or \"flux\".")
+            raise ValueError("The input must be either 'charge' or 'flux'.")
 
     def hamiltonian(self) -> Qobj:
         """
         Returns the transformed hamiltonian of the circuit as
-        qutip.Qobj format.
+        ``qutip.Qobj`` format.
+
+        Returns
+        ----------
+            Circuit Hamiltonian.
         """
-        error = "Please specify the truncation number for each mode."
-        assert len(self.m) != 0, error
+        if len(self.m) == 0:
+            raise CircuitStateError('Please specify the truncation number for each mode.')
 
         Hind = self._get_inductive_hamil()
 
         H = Hind + self._LC_hamil
 
-        return H
+        # TODO: if diagonal matrix, could diagonalize instantantly
+        return H.to('csr')
 
     def _tensor_to_modes(self, tensorIndex: int) -> List[int]:
         """
-        decomposes the tensor product space index to each mode indices. For
+        Decomposes the tensor product space index to each mode indices. For
         example index 5 of the tensor product space can be decomposed to [1,
         0,1] modes if the truncation number for each mode is 2.
-        inputs:
-            -- tensorIndex: Index of tensor product space
-        outputs:
-            -- indList: a list of mode indices (self.n)
+
+        Parameters
+        ----------
+            tensorIndex:
+                Index of tensor product space
+
+        Returns
+        ----------
+            ind_lst:
+                A list of mode indices (self.n)
         """
 
         # i-th mP element is the multiplication of the self.m elements until
@@ -1499,16 +2189,16 @@ def _tensor_to_modes(self, tensorIndex: int) -> List[int]:
             else:
                 mP = [mP[0] * self.m[-1 - i]] + mP
 
-        indList = []
+        ind_lst = []
         indexP = tensorIndex
         for i in range(self.n):
             if i == self.n - 1:
-                indList.append(indexP)
+                ind_lst.append(indexP)
                 continue
-            indList.append(int(indexP / mP[i]))
+            ind_lst.append(int(indexP / mP[i]))
             indexP = indexP % mP[i]
 
-        return indList
+        return ind_lst
 
     def eig_phase_coord(self, k: int, grid: Sequence[ndarray]) -> ndarray:
         """
@@ -1523,10 +2213,16 @@ def eig_phase_coord(self, k: int, grid: Sequence[ndarray]) -> ndarray:
             grid:
                 A list that contains the range of values of phase φ for which
                 we want to evaluate the wavefunction.
-        """
 
-        assert isinstance(k, int), ("The k (index of eigenstate) should be "
-                                    "an integer.")
+        Returns
+        ----------
+            Phase coordinate representation of the ``k``th eigenvector over
+            the values of φ provided in ``grid``.
+        """
+        if len(self._evecs) == 0:
+            raise CircuitStateError('Please diagonalize the circuit first.')
+        if not isinstance(k, int):
+            raise ValueError('The eigenstate index must be an integer.')
 
         phi_list = [*np.meshgrid(*grid, indexing='ij')]
 
@@ -1539,14 +2235,17 @@ def eig_phase_coord(self, k: int, grid: Sequence[ndarray]) -> ndarray:
 
             # decomposes the tensor product space index (i) to each mode
             # indices as a list
-            indList = self._tensor_to_modes(i)
+            ind_lst = self._tensor_to_modes(i)
 
-            term = self._evecs[k][i][0, 0]
+            if get_optim_mode():
+                term = self._evecs[k][i].item()
+            else:
+                term = self._evecs[k][i][0]
 
             for mode in range(self.n):
 
                 # mode number related to that node
-                n = indList[mode]
+                n = ind_lst[mode]
 
                 # For charge basis
                 if self._is_charge_mode(mode):
@@ -1554,16 +2253,50 @@ def eig_phase_coord(self, k: int, grid: Sequence[ndarray]) -> ndarray:
                         1j * phi_list[mode] * n)
                 # For harmonic basis
                 else:
+                    # compute in log-space due to large magnitude variation
                     x0 = np.sqrt(unt.hbar * np.sqrt(
                         self.cInvTrans[mode, mode] / self.lTrans[mode, mode]))
 
-                    coef = 1 / np.sqrt(np.sqrt(np.pi) * (2 ** n) *
-                                       scipy.special.factorial(
-                                           n) * x0 / unt.Phi0)
-
-                    term *= coef * np.exp(
-                        -(phi_list[mode]*unt.Phi0/x0)**2/2) * \
-                            eval_hermite(n, phi_list[mode] * unt.Phi0 / x0)
+                    coeff_log = (
+                        - 0.25 * np.log(np.pi)
+                        - 0.5 * sum(np.log(np.arange(1, n + 1)))
+                        - 0.5 * np.log(x0)
+                        + 0.5 * np.log(unt.Phi0)
+                    )
+
+                    if n < 250:
+                        term_hermitenorm = eval_hermitenorm(n, np.sqrt(2) * phi_list[mode] * unt.Phi0 / x0)
+                        term_hermite_signs = np.where(term_hermitenorm != 0, np.sign(term_hermitenorm), 0)
+                        term_hermitenorm_log = np.where(term_hermitenorm != 0, np.log(np.abs(term_hermitenorm)), 0)
+                    else:
+                        term_hyper = hyperu(-0.5 * n,
+                                            -0.5,
+                                            (phi_list[mode] * unt.Phi0 / x0)**2)
+                        term_hermite_signs = np.where(term_hyper != 0, np.power(np.sign(phi_list[mode]), n), 0)
+                        term_hermitenorm_log = np.where(term_hyper != 0, -(n/2) * np.log(2) * np.log(np.abs(term_hyper)), 0)
+
+                    # Resort to mpmath library if vectorized SciPy code fails
+                    if not np.all(np.isfinite(term_hermitenorm_log)):
+                        bad_pos = ~np.isfinite(term_hermitenorm_log)
+                        it = np.nditer(term_hermitenorm_log, flags=['multi_index'])
+                        for _ in it:
+                            idx = it.multi_index
+                            if bad_pos[idx]:
+                                hermite_val = mpmath.hermite(n, phi_list[mode][idx] * unt.Phi0 / x0)
+                                if hermite_val == 0:
+                                    term_hermite_signs[idx] = 0
+                                    term_hermitenorm_log[idx] = 0
+                                else:
+                                    term_hermite_signs[idx] = mpmath.sign(hermite_val)
+                                    term_hermitenorm_log[idx] = mpmath.log(mpmath.fabs(hermite_val)) - (n/2) * np.log(2)
+
+                    term_log = (
+                        coeff_log
+                        + (-(phi_list[mode]*unt.Phi0/x0)**2/2)
+                        + term_hermitenorm_log
+                    )
+
+                    term *= term_hermite_signs * np.exp(term_log)
 
             state += term
 
@@ -1579,13 +2312,14 @@ def eig_phase_coord(self, k: int, grid: Sequence[ndarray]) -> ndarray:
         return state
 
     def coupling_op(
-            self,
-            ctype: str,
-            nodes: Tuple[int, int]
-    ) -> Qobj:
-        """
-        Return the capacitive or inductive coupling operator related to the
-        specified nodes. The output has the `qutip.Qobj` format.
+        self,
+        ctype: str,
+        nodes: Tuple[int, int],
+        force_use_qutip = False,
+    ) -> Union[Qobj, Tensor]:
+        """Return the capacitive or inductive coupling operator related to the
+        specified nodes. The output is in ``qutip.Qobj`` when using the
+        ``'NumPy'`` engine and as a ``torch.Tensor`` in the ``PyTorch`` engine.
 
         Parameters
         ----------
@@ -1594,51 +2328,77 @@ def coupling_op(
                 ``"inductive"``.
             nodes:
                 A tuple of circuit nodes to which we want to couple.
+
+        Returns
+        ----------
+            Coupling operator of type ``ctype`` between nodes in ``nodes``.
         """
-        error1 = ("The coupling type must be either \"capacitive\" or "
-                  "\"inductive\"")
-        assert ctype in ["capacitive", "inductive"], error1
-        error2 = "Nodes must be a tuple of int"
-        assert isinstance(nodes, tuple) or isinstance(nodes, list), error2
 
-        op = qt.Qobj()
+        if ctype not in ['capacitive', 'inductive']:
+            raise ValueError("The coupling type must be either 'capacitive' or "
+                             "'inductive'.")
+        if not (isinstance(nodes, tuple) or isinstance(nodes, list)):
+            raise ValueError('Nodes must be a tuple of integers.')
+
+        def conditional_cast(op):
+            if get_optim_mode() and not force_use_qutip:
+                return sqf.qobj_to_tensor(op)
+            return op
+
+        def sp_add(a, b):
+            if a == 0:
+                return b
+            return a + b
 
         node1 = nodes[0]
         node2 = nodes[1]
 
+        if ctype == 'capacitive':
+            K = sqf.mat_mul(sqf.mat_inv(self.C), self.R)
+        elif ctype == 'inductive':
+            K = self.S
+
+        if force_use_qutip:
+            K = sqf.to_numpy(K)
+
+        op = 0
         # for the case that we have ground in the edge
         if 0 in nodes:
             node = node1 + node2
-            if ctype == "capacitive":
-                # K = np.linalg.inv(self.getMatC()) @ self.R
-                K = np.linalg.inv(self.C) @ self.R
+            if ctype == 'capacitive':
                 for i in range(self.n):
-                    op += K[node - 1, i] * self._memory_ops["Q"][i]
-            if ctype == "inductive":
-                K = self.S
+                    op = sp_add(
+                        op, 
+                        K[node - 1, i] * conditional_cast(self._memory_ops['Q'][i])
+                    )
+            if ctype == 'inductive':
                 for i in range(self.n):
-                    op += K[node - 1, i] * self._memory_ops["phi"][i]
+                    op = sp_add(
+                        op,
+                        K[node - 1, i] * conditional_cast(self._memory_ops['phi'][i])
+                    )
 
         else:
-            if ctype == "capacitive":
-                # K = np.linalg.inv(self.getMatC()) @ self.R
-                K = np.linalg.inv(self.C) @ self.R
+            if ctype == 'capacitive':
                 for i in range(self.n):
-                    op += (K[node2 - 1, i] - K[node1 - 1, i]) * \
-                          self._memory_ops["Q"][i]
-            if ctype == "inductive":
-                K = self.S
+                    op = sp_add(
+                        op,
+                        (K[node2 - 1, i] - K[node1 - 1, i]) * conditional_cast(self._memory_ops['Q'][i])
+                    )
+            if ctype == 'inductive':
                 for i in range(self.n):
-                    op += ((K[node1 - 1, i] - K[node2 - 1, i])
-                           * self._memory_ops["phi"][i])
+                    op = sp_add(
+                        op,
+                        (K[node1 - 1, i] - K[node2 - 1, i])* conditional_cast(self._memory_ops['phi'][i])
+                    )
 
         return self._squeeze_op(op)
 
     def matrix_elements(
-            self,
-            ctype: str,
-            nodes: Tuple[int, int],
-            states: Tuple[int, int],
+        self,
+        ctype: str,
+        nodes: Tuple[int, int],
+        states: Tuple[int, int],
     ) -> float:
         """
         Return the matrix element of two eigenstates for either capacitive
@@ -1654,7 +2414,14 @@ def matrix_elements(
             states:
                 A tuple of indices of eigenstates for which we want to
                 calculate the matrix element.
+
+        Returns
+        ----------
+            Matrix element between eigenstates in ``states`` for coupling
+            operator of type ``ctype`` between nodes in ``nodes``.
         """
+        if len(self._evecs) == 0:
+            raise CircuitStateError('Please diagonalize the circuit first.')
 
         state1 = self._evecs[states[0]]
         state2 = self._evecs[states[1]]
@@ -1662,12 +2429,14 @@ def matrix_elements(
         # get the coupling operator
         op = self.coupling_op(ctype, nodes)
 
-        return (state1.dag() * op * state2).data[0, 0]
+        # return (state1.dag() * op * state2).data[0, 0] (original)
+        return sqf.operator_inner_product(state1, op, state2) # (modified)
 
     @staticmethod
     def _dephasing(A: float, partial_omega: float) -> float:
         """
-        calculate dephasing rate.
+        Calculate a dephasing rate of arbitrary type given a noise amplitude
+        and eigenfrequency derivative.
 
         Parameters
         ----------
@@ -1676,16 +2445,92 @@ def _dephasing(A: float, partial_omega: float) -> float:
             partial_omega:
                 The derivatives of angular frequency with respect to the
                 noisy parameter
+
+        Returns
+        ----------
+            Dephasing rate specified by ``A`` and ``partial_omega``.
+        """
+        return (sqf.abs(partial_omega * A)
+                * np.sqrt(2 * np.abs(np.log(ENV['omega_low'] * ENV['t_exp']))))
+
+    def _dec_rate_flux_np(self, states: Tuple[int, int]) -> float:
+        """
+        Calculate dephasing rate due to flux noise.
+
+        Parameters
+        ----------
+            states:
+                A tuple of state to calculate the decoherence rate
+
+        Returns
+        ----------
+            Flux dephasing rate between ``states``.
+        """
+        decay = 0
+        for loop in self.loops:
+            partial_omega = self._get_partial_omega_mn(loop, states=states)
+            decay = decay + self._dephasing(loop.A, partial_omega)
+
+        return decay
+
+    def _dec_rate_charge_np(self, states: Tuple[int, int]) -> float:
+        """
+        Calculate dephasing rate due to charge noise.
+
+        Parameters
+        ----------
+            states:
+                A tuple of state to calculate the decoherence rate
+
+        Returns
+        ----------
+            Charge dephasing rate between ``states``.
+        """
+        state_m= self._evecs[states[0]]
+        state_n = self._evecs[states[1]]
+
+        decay = 0
+        for i in range(self.n):
+            op = 0
+            if self._is_charge_mode(i):
+                for j in range(self.n):
+                    op += (self.cInvTrans[i, j] * self._memory_ops['Q'][j] / np.sqrt(unt.hbar))
+
+                partial_omega = sqf.abs(sqf.operator_inner_product(state_m, op, state_m)
+                                        - sqf.operator_inner_product(state_n, op, state_n))
+                A = self.charge_islands[i].A * 2 * unt.e
+                decay = decay + self._dephasing(A, partial_omega)
+
+        return decay
+
+    def _dec_rate_cc_np(self, states: Tuple[int, int]) -> float:
+        """
+        Calculate dephasing rate due to critical current noise.
+
+        Parameters
+        ----------
+            states:
+                A tuple of state to calculate the decoherence rate
+
+        Returns
+        ----------
+            Critical current dephasing rate between ``states``.
         """
+        decay = 0
+        for el, b_id in self._memory_ops['cos']:
+            partial_omega = self._get_partial_omega_mn(
+                el, states=states, _b_id=b_id
+            )
+            A = el.A * el.get_value()
+            decay = decay + self._dephasing(A, partial_omega)
 
-        return (np.abs(partial_omega * A)
-                * np.sqrt(2 * np.abs(np.log(ENV["omega_low"] * ENV["t_exp"]))))
+        return decay
 
     def dec_rate(
-            self,
-            dec_type: str,
-            states: Tuple[int, int],
-            total: bool = True
+        self,
+        dec_type: str,
+        states: Tuple[int, int],
+        total: bool = True
     ) -> float:
         """ Return the decoherence rate in [1/s] between each two eigenstates
         for different types of depolarization and dephasing.
@@ -1703,12 +2548,18 @@ def dec_rate(
                 1)``, we calculate the decoherence rate between the ground
                 state and the first excited state.
             total:
-                if False return a decoherence rate associated with a
-                transition from state m to state n for ``states=(m, n)``. if
-                True return a decoherence rate associated with both m to n
+                if ``False`` return a decoherence rate associated with a
+                transition from state m to state n for ``states=(m, n)``. If
+                ``True`` return a decoherence rate associated with both m to n
                 and n to m transitions.
 
+        Returns
+        ----------
+            Decoherence/dephasing rate between ``states`` specified by 
+            ``dec_type``.
         """
+        if len(self._efreqs) == 0:
+            raise CircuitStateError('Please diagonalize the circuit first.')
 
         omega1 = self._efreqs[states[0]]
         omega2 = self._efreqs[states[1]]
@@ -1716,21 +2567,23 @@ def dec_rate(
         state1 = self._evecs[states[0]]
         state2 = self._evecs[states[1]]
 
-        omega = np.abs(omega2 - omega1)
+        omega = sqf.abs(omega2 - omega1)
 
-        decay = 0
+        decay = sqf.zero(dtype=torch.float64, requires_grad=True)
 
-        # prevent the exponential overflow(exp(709) is the biggest number
-        # that numpy can calculate
-        if unt.hbar * omega / (unt.k_B * ENV["T"]) > 709:
+        # prevent the exponential overflow (exp(709) is the biggest number
+        # that numpy can calculate)
+        alpha = unt.hbar * omega / (unt.k_B * ENV['T'])
+        if alpha > 709:
+            logger.info('Omega=%.2e exceeded threshold; approximating '
+                        + 'spectral density function.', sqf.to_numpy(omega))
             down = 2
             up = 0
         else:
-            alpha = unt.hbar * omega / (unt.k_B * ENV["T"])
-            down = (1 + 1 / np.tanh(alpha / 2))
-            up = down * np.exp(-alpha)
+            down = (1 + 1 / sqf.tanh(alpha / 2))
+            up = down * sqf.exp(-alpha)
 
-        # for temperature dependent loss
+        # for temperature-dependent loss
         if not total:
             if states[0] > states[1]:
                 tempS = down
@@ -1739,7 +2592,7 @@ def dec_rate(
         else:
             tempS = down + up
 
-        if dec_type == "capacitive":
+        if dec_type == 'capacitive':
             for edge in self.elements.keys():
                 for el in self.elements[edge]:
                     if isinstance(el, Capacitor):
@@ -1747,54 +2600,49 @@ def dec_rate(
                     else:
                         cap = el.cap
                     if cap.Q:
-                        decay += tempS * cap.value() / cap.Q(omega) * np.abs(
+                        decay = decay + tempS * cap.get_value() / cap.Q(omega) * sqf.abs(
                             self.matrix_elements(
-                                "capacitive", edge, states)) ** 2
-
-        if dec_type == "inductive":
-            for el, _ in self._memory_ops["ind_hamil"]:
-                op = self._memory_ops["ind_hamil"][(el, _)]
-                x = 1 / el.value()
-                if el.Q:
-                    decay += tempS / el.Q(omega, ENV["T"]) * x * np.abs(
-                        (state1.dag() * op * state2).data[0, 0]) ** 2
-
-        if dec_type == "quasiparticle":
-            for el, _ in self._memory_ops['sin_half']:
-                op = (
-                    self._T_even().dag()
-                    * self._memory_ops['sin_half'][(el, _)]
-                    * self._T_odd()
-                )
-                decay += tempS * el.Y(omega, ENV["T"]) * omega * el.value() \
-                         * unt.hbar * np.abs(
-                    (state1.dag() * op * state2).data[0, 0])**2
-
-        elif dec_type == "charge":
-            # first derivative of the Hamiltonian with respect to charge noise
-            op = qt.Qobj()
-            for i in range(self.n):
-                if self._is_charge_mode(i):
-                    for j in range(self.n):
-                        op += (self.cInvTrans[i, j] * self._memory_ops["Q"][j]
-                               / np.sqrt(unt.hbar))
-                    partial_omega = np.abs((state2.dag()*op*state2 -
-                                            state1.dag()*op*state1).data[0, 0])
-                    A = (self.charge_islands[i].A * 2 * unt.e)
-                    decay += self._dephasing(A, partial_omega)
-
-        elif dec_type == "cc":
-            for el, B_idx in self._memory_ops['cos']:
-                partial_omega = self._get_partial_omega_mn(el, states=states,
-                                                           _B_idx=B_idx)
-                A = el.A * el.value(self.random)
-                decay += self._dephasing(A, partial_omega)
-
-        elif dec_type == "flux":
-            for loop in self.loops:
-                partial_omega = self._get_partial_omega_mn(loop, states=states)
-                A = loop.A
-                decay += self._dephasing(A, partial_omega)
+                                'capacitive', edge, states)) ** 2
+        elif dec_type == 'inductive':
+            for el, _ in self._memory_ops['ind_hamil']:
+                op = self._memory_ops['ind_hamil'][(el, _)]
+                Q = el.Q(omega, ENV['T'])
+                if np.isnan(sqf.to_numpy(Q)):
+                    logger.warning('Calculated Q for %s was NaN', el)
+                    decay = decay + 0
+                else:
+                    decay = decay + tempS / Q / el.get_value() * sqf.abs(
+                        sqf.operator_inner_product(state1, op, state2)) ** 2
+        elif dec_type == 'quasiparticle':
+            for el, b_id in self._memory_ops['sin_half']:
+                op = self.op('sin_half', {'el': el, 'b_id': b_id})
+                Y = el.Y(omega, ENV['T'])
+                if np.isnan(sqf.to_numpy(Y)):
+                    logger.warning('Calculated Y for %s was NaN', el)
+                    decay = decay + 0
+                else:
+                    decay = decay + (
+                        tempS * Y * omega * el.get_value()
+                        * unt.hbar
+                        * sqf.abs(sqf.operator_inner_product(state1, op, state2)) ** 2
+                    )
+        elif dec_type == 'charge':
+            if get_optim_mode():
+                decay = decay + sqtorch.dec_rate_charge_torch(self, states)
+            else:
+                decay = decay + self._dec_rate_charge_np(states)
+        elif dec_type == 'cc':
+            if get_optim_mode():
+                decay = decay + sqtorch.dec_rate_cc_torch(self, states)
+            else:
+                decay = decay + self._dec_rate_cc_np(states)
+        elif dec_type == 'flux':
+            if get_optim_mode():
+                decay = decay + sqtorch.dec_rate_flux_torch(self, states)
+            else:
+                decay = decay + self._dec_rate_flux_np(states)
+        else:
+            raise ValueError(f'The decoherence type {dec_type} is not supported.')
 
         return decay
 
@@ -1806,16 +2654,20 @@ def _get_quadratic_Q(self, A: ndarray) -> Qobj:
             A:
                 ndarray matrix that specifies the coefficient for
                 quadratic expression.
+
+        Returns
+        ----------
+            Quadratic form with charge operators using ``A``.
         """
 
-        op = qt.Qobj()
+        op = 0
 
         for i in range(self.n):
             for j in range(self.n-i):
                 if j == 0:
-                    op += 0.5 * A[i, i+j] * self._memory_ops["QQ"][i][j]
+                    op += 0.5 * A[i, i+j] * self._memory_ops['QQ'][i][j]
                 elif j > 0:
-                    op += A[i, i+j] * self._memory_ops["QQ"][i][j]
+                    op += A[i, i+j] * self._memory_ops['QQ'][i][j]
 
         return op
 
@@ -1827,17 +2679,21 @@ def _get_quadratic_phi(self, A: ndarray) -> Qobj:
             A:
                 ndarray matrix that specifies the coefficient for
                 quadratic expression.
+
+        Returns
+        ----------
+            Quadratic form with flux operators using ``A``.
         """
 
-        op = qt.Qobj()
+        op = 0
 
         # number of harmonic modes
         n_H = len(self.omega != 0)
 
         for i in range(n_H):
             for j in range(n_H):
-                phi_i = self._memory_ops["phi"][i].copy()
-                phi_j = self._memory_ops["phi"][j].copy()
+                phi_i = self._memory_ops['phi'][i].copy()
+                phi_j = self._memory_ops['phi'][j].copy()
                 if i == j:
                     op += 0.5 * A[i, i] * phi_i**2
                 elif j > i:
@@ -1846,81 +2702,84 @@ def _get_quadratic_phi(self, A: ndarray) -> Qobj:
         return op
 
     def _get_partial_H(
-            self,
-            el: Union[Capacitor, Inductor, Junction, Loop],
-            _B_idx: Optional[int] = None,
+        self,
+        el: Union[Capacitor, Inductor, Junction, Loop],
+        _b_id: Optional[int] = None,
     ) -> Qobj:
         """
-        return the gradient of the Hamiltonian with respect to elements or
+        Compute the gradient of the Hamiltonian with respect to elements or
         loop as ``qutip.Qobj`` format.
-
         Parameters
         ----------
             el:
                 Element of a circuit that can be either ``Capacitor``,
                 ``Inductor``, ``Junction``, or ``Loop``.
-            _B_idx:
-                Optional integer point to each row of B matrix (external flux
-                distribution of that element). This uses to specify that
-                gradient is calculated based on which JJ of the circuit
-                specifically (we use this option for critical current noise
-                calculation)
+            _b_id:
+                Optional integer to indicate which row of the B matrix
+                (per-element external flux distribution) to use. This specifies
+                which JJ of the circuit to consider specifically (ex. for
+                critical current noise calculation).
+
+        Returns
+        ----------
+            Partial derivative of Hamiltonian with respect to ``el``.
         """
 
-        partial_H = qt.Qobj()
+        partial_H = 0
 
         if isinstance(el, Capacitor):
-
-            cInv = np.linalg.inv(self.C)
-            A = -self.R.T @ cInv @ self.partial_C[el] @ cInv @ self.R
+            cInv = np.linalg.inv(sqf.to_numpy(self.C))
+            A = -self.R.T @ cInv @ self.partial_mats[el] @ cInv @ self.R
             partial_H += self._get_quadratic_Q(A)
 
         elif isinstance(el, Inductor):
 
-            A = -self.S.T @ self.partial_L[el]  @ self.S
+            A = -self.S.T @ self.partial_mats[el]  @ self.S
             partial_H += self._get_quadratic_phi(A)
 
-            for edge, el_ind, B_idx in self.inductor_keys:
+            for edge, el_ind, b_id in self.elem_keys[Inductor]:
                 if el == el_ind:
 
-                    phi = self._get_external_flux_at_element(B_idx)
+                    phi = self._get_external_flux_at_element(b_id)
 
-                    partial_H += -(self._memory_ops["ind_hamil"][(el, B_idx)]
-                                   / el.value()**2 / np.sqrt(unt.hbar)
+                    partial_H += -(self._memory_ops['ind_hamil'][(el, b_id)]
+                                   / np.squeeze(sqf.to_numpy(el.get_value()))**2 / np.sqrt(unt.hbar)
                                    * (unt.Phi0/2/np.pi) * phi)
 
         elif isinstance(el, Loop):
 
             loop_idx = self.loops.index(el)
 
-            for edge, el_ind, B_idx in self.inductor_keys:
-                partial_H += (self.B[B_idx, loop_idx]
-                              * self._memory_ops["ind_hamil"][(el_ind, B_idx)]
-                              / el_ind.value() * unt.Phi0 / np.sqrt(unt.hbar)
-                              / 2 / np.pi)
+            for edge, el_ind, b_id in self.elem_keys[Inductor]:
+                partial_H += (
+                    self.B[b_id, loop_idx]
+                    * self._memory_ops["ind_hamil"][(el_ind, b_id)]
+                    / sqf.to_numpy(el_ind.get_value())
+                    * unt.Phi0 / np.sqrt(unt.hbar) / 2 / np.pi
+                )
 
-            for edge, el_JJ, B_idx, W_idx in self.junction_keys:
-                partial_H += (self.B[B_idx, loop_idx] * el_JJ.value()
-                              * self._memory_ops['sin'][(el_JJ, B_idx)])
+            for edge, el_JJ, b_id, w_id in self.elem_keys[Junction]:
+                partial_H += (self.B[b_id, loop_idx] * sqf.to_numpy(el_JJ.get_value())
+                              * self._memory_ops['sin'][(el_JJ, b_id)])
 
         elif isinstance(el, Junction):
 
-            for _, el_JJ, B_idx, W_idx in self.junction_keys:
+            for _, el_JJ, b_id, w_id in self.elem_keys[Junction]:
 
-                if el == el_JJ and _B_idx is None:
-                    partial_H += -self._memory_ops['cos'][(el, B_idx)]
+                if el == el_JJ and _b_id is None:
+                    partial_H += -self._memory_ops['cos'][(el, b_id)]
 
-                elif el == el_JJ and _B_idx == B_idx:
-                    partial_H += -self._memory_ops['cos'][(el, B_idx)]
+                elif el == el_JJ and _b_id == b_id:
+                    partial_H += -self._memory_ops['cos'][(el, b_id)]
 
         return partial_H
 
-    def _get_partial_omega(
-            self,
-            el: Union[Capacitor, Inductor, Junction, Loop],
-            m: int,
-            subtract_ground: bool = True,
-            _B_idx: Optional[int] = None,
+    def get_partial_omega(
+        self,
+        el: Union[Capacitor, Inductor, Junction, Loop],
+        m: int,
+        subtract_ground: bool = True,
+        _b_id: Optional[int] = None,
     ) -> float:
         """Return the gradient of the eigen angular frequency with respect to
         elements or loop as ``qutip.Qobj`` format.
@@ -1933,38 +2792,44 @@ def _get_partial_omega(
             m:
                 Integer specifies the eigenvalue. for example ``m=0`` specifies
                 the ground state and ``m=1`` specifies the first excited state.
-            _B_idx:
-                Optional integer point to each row of B matrix (external flux
-                distribution of that element). This uses to specify that
-                gradient is calculated based on which JJ of the circuit
-                specifically (we use this option for critical current noise
-                calculation)
+            subtract_ground:
+                If ``True``, it subtracts the ground state frequency from the
+                desired frequency.
+            _b_id:
+                Optional integer to indicate which row of the B matrix
+                (per-element external flux distribution) to use. This specifies
+                which JJ of the circuit to consider specifically (ex. for
+                critical current noise calculation).
 
+        Returns
+        ----------
+            Partial derivative of eigenfrequency ``m`` with respect to ``el``,
+            in units of angular frequency.
         """
+        if len(self._evecs) == 0:
+            raise CircuitStateError('Please diagonalize the circuit first.')
 
         state_m = self._evecs[m]
-
-        partial_H = self._get_partial_H(el, _B_idx)
-
-        partial_omega_m = state_m.dag() * (partial_H*state_m)
+        partial_H = self._get_partial_H(el, _b_id)
+        partial_omega_m = sqf.operator_inner_product(
+            state_m, partial_H, state_m
+        )
 
         if subtract_ground:
-
             state_0 = self._evecs[0]
+            partial_omega_0 = sqf.operator_inner_product(
+                state_0, partial_H, state_0
+            )
 
-            partial_omega_0 = state_0.dag() * (partial_H * state_0)
-
-            return (partial_omega_m - partial_omega_0).data[0, 0].real
-
+            return sqf.real(partial_omega_m - partial_omega_0)
         else:
-
-            return partial_omega_m.data[0, 0].real
+            return sqf.real(partial_omega_m)
 
     def _get_partial_omega_mn(
-            self,
-            el: Union[Capacitor, Inductor, Junction, Loop],
-            states: Tuple[int, int],
-            _B_idx: Optional[int] = None,
+        self,
+        el: Union[Capacitor, Inductor, Junction, Loop],
+        states: Tuple[int, int],
+        _b_id: Optional[int] = None,
     ) -> float:
         """Return the gradient of the eigen angular frequency with respect to
         elements or loop as ``qutip.Qobj`` format. Note that if
@@ -1975,22 +2840,44 @@ def _get_partial_omega_mn(
             el:
                 Element of a circuit that can be either ``Capacitor``,
                 ``Inductor``, ``Junction``, or ``Loop``.
-            m:
-                Integer specifies the eigenvalue. for example ``m=0`` specifies
-                the ground state and ``m=1`` specifies the first excited state.
+            states:
+                Integers indicating indices of eigenenergies to calculate
+                the difference of.
+            _b_id:
+                Optional integer to indicate which row of the B matrix (external
+                flux distribution of that element) to use. This specifies which
+                JJ of the circuit to consider specifically (ex. for critical
+                current noise calculation).
+
+        Returns
+        ----------
+            Partial derivative of the energy difference between ``states``
+            with respect to ``el``, in units of angular frequency.
         """
 
         state_m = self._evecs[states[0]]
         state_n = self._evecs[states[1]]
 
-        partial_H = self._get_partial_H(el, _B_idx)
+        partial_H = self._get_partial_H(el, _b_id)
+
+        partial_omega_m = sqf.operator_inner_product(
+            state_m, partial_H, state_m
+        )
+        partial_omega_n = sqf.operator_inner_product(
+            state_n, partial_H, state_n
+        )
 
-        partial_omega_m = state_m.dag() * (partial_H*state_m)
-        partial_omega_n = state_n.dag() * (partial_H*state_n)
+        partial_omega_mn = partial_omega_m - partial_omega_n
+        # assert sqf.imag(partial_omega_mn)/sqf.real(partial_omega_mn) < 1e-6
 
-        return (partial_omega_m - partial_omega_n).data[0, 0].real
+        return sqf.real(partial_omega_mn)
 
-    def _get_partial_vec(self, el, m):
+    def get_partial_vec(
+        self,
+        el: Union[Element, Loop],
+        m: int,
+        epsilon=1e-12
+    ) -> Qobj:
         """Return the gradient of the eigenvectors with respect to
         elements or loop as ``qutip.Qobj`` format.
 
@@ -2002,27 +2889,33 @@ def _get_partial_vec(self, el, m):
             m:
                 Integer specifies the eigenvalue. for example ``m=0`` specifies
                 the ground state and ``m=1`` specifies the first excited state.
+
+        Returns
+        ----------
+            Partial derivative of the ``m``th eigenvector, with respect to
+            ``el``.
         """
+        if len(self._evecs) == 0:
+            raise CircuitStateError('Please diagonalize the circuit first.')
 
         state_m = self._evecs[m]
 
-        #     state_m = state_m*np.exp(-1j*np.angle(state_m[0,0]))
-
         n_eig = len(self._evecs)
 
         partial_H = self._get_partial_H(el)
-        partial_state = qt.Qobj()
+        partial_state = sqf.zeros(state_m.shape)
 
         for n in range(n_eig):
-
             if n == m:
                 continue
-
             state_n = self._evecs[n]
 
-            delta_omega = (self._efreqs[m] - self._efreqs[n])
+            delta_omega = sqf.to_numpy(self._efreqs[m] - self._efreqs[n])
 
-            partial_state += (state_n.dag()
-                              * (partial_H * state_m)) * state_n / delta_omega
+            partial_state += (
+                sqf.operator_inner_product(state_n, partial_H, state_m)
+                * state_n
+                / (delta_omega + epsilon)
+            )
 
         return partial_state
diff --git a/SQcircuit/elements.py b/SQcircuit/elements.py
old mode 100644
new mode 100755
index 737ceff..3225924
--- a/SQcircuit/elements.py
+++ b/SQcircuit/elements.py
@@ -1,183 +1,304 @@
-"""
-elements.py contains the classes for the circuit elements:
+"""elements.py contains the classes for the circuit elements:
 capacitors, inductors, and josephson junctions.
 """
 
 from typing import List, Any, Optional, Union, Callable
 
+import torch
 import numpy as np
 
-from scipy.special import kn
+from torch import Tensor
+from numpy import ndarray
 
 import SQcircuit.units as unt
+import SQcircuit.functions as sqf
 
+from SQcircuit.exceptions import (
+    raise_unit_error,
+    raise_optim_error_if_needed,
+)
+from SQcircuit.settings import get_optim_mode
 
-class Capacitor:
-    """
-    Class that contains the capacitor properties.
+###############################################################################
+# Elements
+###############################################################################
+
+class CircuitComponent:
+    """Base class for circuit components which users can optimize."""
+
+    @property
+    def internal_value(self) -> Union[float, Tensor]:
+        return self._value
+
+    @internal_value.setter
+    def internal_value(self, v: Union[float, Tensor]) -> None:
+        self._value = v
+
+    @property
+    def requires_grad(self) -> bool:
+        raise_optim_error_if_needed()
+
+        return self.internal_value.requires_grad
+
+    @requires_grad.setter
+    def requires_grad(self, f: bool) -> None:
+        raise_optim_error_if_needed()
+
+        self.internal_value.requires_grad = f
+
+    @property
+    def is_leaf(self) -> bool:
+        raise_optim_error_if_needed()
+
+        return self.internal_value.is_leaf
+
+    @property
+    def grad(self) -> Tensor:
+        raise_optim_error_if_needed()
+
+        return self.internal_value.grad
+
+    @grad.setter
+    def grad(self, new_grad: Optional[Tensor]) -> None:
+        raise_optim_error_if_needed()
+
+        self.internal_value.grad = new_grad
+
+
+class Element(CircuitComponent):
+    """Class that contains general properties of circuit elements."""
+
+    @property
+    def error(self) -> float:
+        return self._error
+
+    @error.setter
+    def error(self, e: float) -> None:
+        self._error = e
+
+    def _set_value_with_error(self, mean: float, error: float) -> None:
+        mean_th = torch.as_tensor(mean, dtype=torch.float64)
+        error_th = torch.as_tensor(error, dtype=torch.float64)
 
+        sampled_value = torch.normal(mean_th, mean_th*error_th/100)
+        self.internal_value = sampled_value
+
+        if not get_optim_mode():
+            self.internal_value = float(self.internal_value.detach().cpu().numpy())
+
+    @staticmethod
+    def _get_default_id_str(s: str, v: float, u: str) -> str:
+        """Get the default string ID for the element.
+        
+        Parameters
+        ----------
+            s:
+                The initial string of the id_string.
+            v:
+                The value of the element as float number.
+            u:
+                The unit of the element as string.
+        """
+        if not isinstance(s, str):
+            raise ValueError('The input must be a string.')
+
+        return (s + '_{}_{}').format(v, u)
+
+
+class Capacitor(Element):
+    """Class that contains the capacitor properties.
+    
     Parameters
     ----------
-    value:
-        The value of the capacitor.
-    unit:
-        The unit of input value. If ``unit`` is "THz", "GHz", and ,etc.,
-        the value specifies the charging energy of the capacitor. If ``unit``
-        is "fF", "pF", and ,etc., the value specifies the capacitance in
-        farad. If ``unit`` is ``None``, the default unit of capacitor is "GHz".
-    Q:
-        Quality factor of the dielectric of the capacitor which is one over
-        tangent loss. It can be either a float number or a Python function of
-        angular frequency.
-    error:
-        The error in fabrication as a percentage.
-    id_str:
-        ID string for the capacitor.
+        value:
+            The value of the capacitor.
+        unit:
+            The unit of input value. If ``unit`` is "THz", "GHz", and, etc.,
+            the value specifies the charging energy of the capacitor. If
+            ``unit`` is "fF", "pF", and, etc., the value specifies the
+            capacitance in farad. If ``unit`` is ``None``, the default unit of
+            capacitor is "GHz".
+        requires_grad:
+            A boolean variable that specifies whether autograd should record
+            operations on this element. This feature is specific to the
+            ``PyTorch`` engine.
+        Q:
+            Quality factor of the dielectric of the capacitor which is one
+            over tangent loss. It can be either a float number or a Python
+            function of angular frequency.
+        error:
+            The error of fabrication in percentage.
+        id_str:
+            ID string for the capacitor.
     """
+    value_unit = 'F'
 
     def __init__(
         self,
         value: float,
         unit: Optional[str] = None,
-        Q: Union[Any, Callable[[float], float]] = "default",
+        requires_grad: bool = False,
+        Q: Union[Any, Callable[[float], float]] = 'default',
         error: float = 0,
         id_str: Optional[str] = None,
     ) -> None:
 
-        if (unit not in unt.freq_list and
-                unit not in unt.farad_list and
-                unit is not None):
-            error = "The input unit for the capacitor is not correct. " \
-                    "Look at the documentation for the correct input format."
-            raise ValueError(error)
+        if unit is None:
+            unit = unt.get_unit_cap()
+        else:
+            self._check_unit_format(unit)
 
-        self.cValue = value
-        self.error = error
+        self.set_value(value, unit, error)
         self.type = type(self)
 
-        if unit is None:
-            self.unit = unt.get_unit_cap()
-        else:
-            self.unit = unit
+        if requires_grad:
+            self.requires_grad = requires_grad
 
-        if Q == "default":
-            self.Q = lambda omega: 1e6 * (
-                    2 * np.pi * 6e9 / np.abs(omega)) ** 0.7
+        if Q == 'default':
+            self.Q = self._default_q_cap
         elif isinstance(Q, float) or isinstance(Q, int):
             self.Q = lambda omega: Q
         else:
             self.Q = Q
 
         if id_str is None:
-            self.id_str = "C_{}_{}".format(value, self.unit)
+            self.id_str = self._get_default_id_str('C', value, unit)
         else:
             self.id_str = id_str
 
-    def value(self, random: bool = False) -> float:
-        """
-        Return the value of the capacitor in farad units. If `random` is
-        `True`, it samples from a normal distribution with variance defined
-        by the fabrication error.
+    @staticmethod
+    def _check_unit_format(u: str):
+        """Check if the unit input has the correct format."""
+
+        if (u not in unt.freq_list and
+                u not in unt.farad_list and
+                u is not None):
+            raise_unit_error()
 
+    def set_value(self, v: float, u: str = 'F', e: float = 0.0) -> None:
+        """Set the value for the capacitor.
+        
         Parameters
         ----------
-            random:
-                A boolean flag which specifies whether the output is
-                deterministic or random.
+            v:
+                The value of the element.
+            u:
+                The unit of input value.
+            e:
+                The fabrication error in percentage.
         """
-        if self.unit in unt.farad_list:
-            cMean = self.cValue * unt.farad_list[self.unit]
-        else:
-            E_c = self.cValue * unt.freq_list[self.unit] * (
-                    2 * np.pi * unt.hbar)
-            cMean = unt.e ** 2 / 2 / E_c
+        self._check_unit_format(u)
+        self.error = e
 
-        if not random:
-            return cMean
+        if u in unt.farad_list:
+            mean = v * unt.farad_list[u]
         else:
-            return np.random.normal(cMean, cMean * self.error / 100, 1)[0]
+            energy = v * unt.freq_list[u] * (2*np.pi*unt.hbar)
+            mean = unt.e ** 2 / 2 / energy
 
-    def energy(self) -> float:
-        """
-        Return the charging energy of the capacitor in frequency unit of
-        SQcircuit (gigahertz by default).
+        self._set_value_with_error(mean, e)
+
+    def get_value(self, u: str = 'F') -> Union[float, Tensor]:
+        """Return the value of the element in specified unit.
+        
+        Parameters
+        ----------
+            u:
+                The unit of input value. The default is "F".
         """
-        if self.unit in unt.freq_list:
-            return self.cValue * unt.freq_list[
-                self.unit] / unt.get_unit_freq()
+
+        if u in unt.farad_list:
+            return self.internal_value / unt.farad_list[u]
+
+        elif u in unt.freq_list:
+            E_c = (
+                unt.e**2 / 2 / self.internal_value / (2*np.pi*unt.hbar)
+                / unt.freq_list[u]
+            )
+            return E_c
+
         else:
-            c = self.cValue * unt.farad_list[self.unit]
-            return unt.e ** 2 / 2 / c / (
-                    2 * np.pi * unt.hbar) / unt.get_unit_freq()
+            raise_unit_error()
+
+    @staticmethod
+    def partial_mat(edge_mat):
+
+        return edge_mat
+
+    @staticmethod
+    def _default_q_cap(omega):
+        """Default function for capacitor quality factor."""
+
+        return 1e6 * (2 * np.pi * 6e9 / sqf.abs(omega))**0.7
 
 
 class VerySmallCap(Capacitor):
 
     def __init__(self):
-        super().__init__(1e-20, "F", Q=None)
+        super().__init__(1e-20, 'F', Q=None)
 
 
 class VeryLargeCap(Capacitor):
 
     def __init__(self):
-        super().__init__(1e20, "F", Q=None)
+        super().__init__(1e20, 'F', Q=None)
 
 
-class Inductor:
-    """
-    Class that contains the inductor properties.
-
+class Inductor(Element):
+    """Class that contains the inductor properties.
+    
     Parameters
     ----------
-    value:
-        The value of the inductor.
-    unit:
-        The unit of input value. If ``unit`` is "THz", "GHz", and ,etc.,
-        the value specifies the inductive energy of the inductor. If ``unit``
-        is "fH", "pH", and ,etc., the value specifies the inductance in henry.
-        If ``unit`` is ``None``, the default unit of inductor is "GHz".
-    loops:
-        List of loops in which the inductor resides.
-    cap:
-        Capacitor associated to the inductor, necessary for correct
-        time-dependent external fluxes scheme.
-    Q:
-        Quality factor of the inductor needed for inductive loss calculation.
-        It can be either a float number or a Python function of angular
-        frequency and temperature.
-    error:
-        The error in fabrication as a percentage.
-    id_str:
-        ID string for the inductor.
+        value:
+            The value of the inductor.
+        unit:
+            The unit of input value. If ``unit`` is "THz", "GHz", and ,etc.,
+            the value specifies the inductive energy of the inductor. If
+            ``unit`` is "fH", "pH", and ,etc., the value specifies the
+            inductance in henry. If ``unit`` is ``None``, the default unit of
+            inductor is "GHz".
+        requires_grad:
+            A boolean variable that specifies whether autograd should record
+            operations on this element. This feature is specific to the
+            ``PyTorch`` engine.
+        loops:
+            List of loops in which the inductor resides.
+        cap:
+            Capacitor associated to the inductor, necessary for correct
+            time-dependent external fluxes scheme.
+        Q:
+            Quality factor of the inductor needed for inductive loss
+            calculation. It can be either a float number or a Python function of
+            angular frequency and temperature.
+        error:
+            The error in fabrication as a percentage.
+        id_str:
+            ID string for the inductor.
     """
+    value_unit = 'H'
 
     def __init__(
             self,
             value: float,
             unit: str = None,
-            cap: Optional["Capacitor"] = None,
-            Q: Union[Any, Callable[[float, float], float]] = "default",
+            requires_grad: bool = False,
+            cap: Optional['Capacitor'] = None,
+            Q: Union[Any, Callable[[float, float], float]] = 'default',
             error: float = 0,
-            loops: Optional[List["Loop"]] = None,
+            loops: Optional[List['Loop']] = None,
             id_str: Optional[str] = None
     ) -> None:
 
-        if (unit not in unt.freq_list and
-                unit not in unt.henry_list and
-                unit is not None):
-            error = "The input unit for the inductor is not correct. " \
-                    "Look at the documentation for the correct input format."
-            raise ValueError(error)
+        if unit is None:
+            unit = unt.get_unit_ind()
+        else:
+            self._check_unit_format(unit)
 
-        self.lValue = value
-        self.error = error
+        self.set_value(value, unit, error)
         self.type = type(self)
-        self.id_str = id_str
 
-        if unit is None:
-            self.unit = unt.get_unit_ind()
-        else:
-            self.unit = unit
+        if requires_grad:
+            self.requires_grad = requires_grad
 
         if cap is None:
             self.cap = VerySmallCap()
@@ -189,125 +310,181 @@ def __init__(
         else:
             self.loops = loops
 
-        def qInd(omega, T):
-            alpha = unt.hbar * 2 * np.pi * 0.5e9 / (2 * unt.k_B * T)
-            beta = unt.hbar * omega / (2 * unt.k_B * T)
-
-            return 500e6 * (kn(0, alpha) * np.sinh(alpha)) / (
-                    kn(0, beta) * np.sinh(beta))
-
-        if Q == "default":
-            self.Q = qInd
+        if Q == 'default':
+            self.Q = self._default_q_ind
         elif isinstance(Q, float) or isinstance(Q, int):
             self.Q = lambda omega, T: Q
         else:
             self.Q = Q
 
         if id_str is None:
-            self.id_str = "L_{}_{}".format(value, self.unit)
+            self.id_str = self._get_default_id_str('L', value, unit)
         else:
             self.id_str = id_str
 
-    def value(self, random: bool = False) -> float:
-        """
-        Return the value of the inductor in henry units. If `random` is
-        `True`, it samples from a normal distribution with variance defined
-        by the fabrication error.
+    @staticmethod
+    def _check_unit_format(u):
+        """Check if the unit input has the correct format."""
+
+        if (
+            u not in unt.freq_list and
+            u not in unt.henry_list and
+            u is not None
+        ):
+            raise_unit_error()
 
+    def set_value(self, v: float, u: str = 'H', e: float = 0.0) -> None:
+        """Set the value for the element.
+        
         Parameters
         ----------
-            random:
-                A boolean flag which specifies whether the output is
-                deterministic or random.
+            v:
+                The value of the element.
+            u:
+                The unit of input value.
+            e:
+                The fabrication error in percentage.
         """
-        if self.unit in unt.henry_list:
-            lMean = self.lValue * unt.henry_list[self.unit]
+        self._check_unit_format(u)
+        self.error = e
+
+        if u in unt.henry_list:
+            mean = v * unt.henry_list[u]
         else:
-            E_l = self.lValue * unt.freq_list[self.unit] * (
-                    2 * np.pi * unt.hbar)
-            lMean = (unt.Phi0 / 2 / np.pi) ** 2 / E_l
+            energy = v * unt.freq_list[u] * (2*np.pi*unt.hbar)
+            mean = (unt.Phi0/2/np.pi)**2 / energy
+
+        self._set_value_with_error(mean, e)
+
+    def get_value(self, u: str = 'H') -> Union[float, Tensor]:
+        """Return the value of the element in specified unit.
+        
+        Parameters
+        ----------
+            u:
+                The unit of input value. The default is "H".
+        """
+
+        if u in unt.henry_list:
+            return self.internal_value / unt.henry_list[u]
+
+        elif u in unt.freq_list:
+            l = self.internal_value
+            E_l = (unt.Phi0/2/np.pi)**2/l/(2*np.pi*unt.hbar)/unt.freq_list[u]
+            return E_l
 
-        if not random:
-            return lMean
         else:
-            return np.random.normal(lMean, lMean * self.error / 100, 1)[0]
+            raise_unit_error()
+
+    @staticmethod
+    def _default_q_ind(omega, T):
+        """Default function for inductor quality factor."""
+
+        alpha = unt.hbar * 2 * np.pi * 0.5e9 / (2 * unt.k_B * T)
+        beta = unt.hbar * omega / (2 * unt.k_B * T)
+
+        return 500e6 * sqf.exp(
+            sqf.log_k0(alpha)
+            + sqf.log_sinh(alpha)
+            - sqf.log_k0(beta)
+            - sqf.log_sinh(beta)
+        )
+
+    def get_key(self, edge, B_idx, *_):
+        """Return the inductor key.
 
-    def energy(self) -> float:
+        Parameters
+        ----------
+            edge:
+                Edge that element is part of.
+            B_idx:
+                The inductive element index
         """
-        Return the inductive energy of the capacitor in frequency unit of
-        SQcircuit (gigahertz by default).
+
+        return edge, self, B_idx
+
+    def get_cap_for_flux_dist(self, flux_dist):
+
+        if flux_dist == 'all':
+            return self.cap.get_value()
+        elif flux_dist == 'junctions':
+            return VeryLargeCap().get_value()
+        elif flux_dist == 'inductors':
+            return VerySmallCap().get_value()
+
+    def partial_mat(self, edge_mat: ndarray) -> ndarray:
+        """Get the partial_mat based on input edge_mat.
+
+        Parameters
+        ----------
+            edge_mat:
+                Matrix representation of the edge that element is part of.
         """
-        if self.unit in unt.freq_list:
-            return self.lValue * unt.freq_list[
-                self.unit] / unt.get_unit_freq()
-        else:
-            l = self.lValue * unt.henry_list[self.unit]
-            return (unt.Phi0 / 2 / np.pi) ** 2 / l / (
-                    2 * np.pi * unt.hbar) / unt.get_unit_freq()
 
+        return edge_mat / sqf.to_numpy(self.get_value()**2)
 
-class Junction:
-    """
-    Class that contains the Josephson junction properties.
 
+class Junction(Element):
+    """Class that contains the Josephson junction properties.
+    
     Parameters
     -----------
-    value:
-        The value of the Josephson junction.
-    unit: str
-        The unit of input value. The ``unit`` can be "THz", "GHz", and ,etc.,
-        that specifies the junction energy of the inductor. If ``unit`` is
-        ``None``, the default unit of junction is "GHz".
-    loops:
-        List of loops in which the Josephson junction reside.
-    cap:
-        Capacitor associated to the josephson junction, necessary for the
-        correct time-dependent external fluxes scheme.
-    A:
-        Normalized noise amplitude related to critical current noise.
-    x:
-        Quasiparticle density
-    delta:
-        Superconducting gap
-    Y:
-        Real part of admittance.
-    error:
-        The error in fabrication as a percentage.
-    id_str:
-        ID string for the junction.
+        value:
+            The value of the Josephson junction.
+        unit: str
+            The unit of input value. The ``unit`` can be "THz", "GHz", and
+            ,etc., that specifies the junction energy of the inductor. If
+            ``unit`` is ``None``, the default unit of junction is "GHz".
+        requires_grad:
+            A boolean variable that specifies whether autograd should record
+            operations on this element. This feature is specific to the
+            ``PyTorch`` engine.
+        loops:
+            List of loops in which the Josephson junction reside.
+        cap:
+            Capacitor associated to the josephson junction, necessary for the
+            correct time-dependent external fluxes scheme.
+        A:
+            Normalized noise amplitude related to critical current noise.
+        x:
+            Quasiparticle density
+        delta:
+            Superconducting gap
+        Y:
+            Real part of admittance.
+        error:
+            The error in fabrication as a percentage.
+        id_str:
+            ID string for the junction.
     """
+    value_unit = 'Hz'
 
     def __init__(
         self,
         value: float,
         unit: Optional[str] = None,
-        cap: Optional[str] = None,
+        requires_grad: bool = False,
+        cap: Optional[Capacitor] = None,
         A: float = 1e-7,
         x: float = 3e-06,
         delta: float = 3.4e-4,
-        Y: Union[Any, Callable[[float, float], float]] = "default",
+        Y: Union[Any, Callable[[float, float], float]] = 'default',
         error: float = 0,
-        loops: Optional[List["Loop"]] = None,
+        loops: Optional[List['Loop']] = None,
         id_str: Optional[str] = None,
     ) -> None:
 
-        if (unit not in unt.freq_list and
-                unit is not None):
-            error = "The input unit for the Josephson Junction is not " \
-                    "correct. Look at the documentation for the correct " \
-                    "input format."
-            raise ValueError(error)
+        if unit is None:
+            unit = unt.get_unit_JJ()
+        else:
+            self._check_unit_format(unit)
 
-        self.jValue = value
-        self.error = error
+        self.set_value(value, unit, error)
         self.type = type(self)
         self.A = A
-        self.id_str = id_str
 
-        if unit is None:
-            self.unit = unt.get_unit_JJ()
-        else:
-            self.unit = unit
+        if requires_grad:
+            self.requires_grad = requires_grad
 
         if cap is None:
             self.cap = VerySmallCap()
@@ -319,53 +496,118 @@ def __init__(
         else:
             self.loops = loops
 
-        def yQP(omega, T):
-            alpha = unt.hbar * omega / (2 * unt.k_B * T)
-            y = np.sqrt(2 / np.pi) * (8 / (delta * 1.6e-19) / (
-                    unt.hbar * 2 * np.pi / unt.e ** 2)) \
-                * (2 * (delta * 1.6e-19) / unt.hbar / omega) ** 1.5 \
-                * x * np.sqrt(alpha) * kn(0, alpha) * np.sinh(alpha)
-            return y
-
-        if Y == "default":
-            self.Y = yQP
+        if Y == 'default':
+            self.Y = self._get_default_y_func(delta, x)
         else:
             self.Y = Y
 
         if id_str is None:
-            self.id_str = "JJ_{}_{}".format(value, self.unit)
+            self.id_str = self._get_default_id_str('JJ', value, unit)
         else:
             self.id_str = id_str
 
-    def value(self, random: bool = False) -> float:
+    @staticmethod
+    def _check_unit_format(u):
+        """Check if the unit input has the correct format."""
+
+        if u not in unt.freq_list and u is not None:
+            raise_unit_error()
+
+    def set_value(self, v: float, u: str = 'Hz', e: float = 0.0) -> None:
+        """Set the value for the element.
+        
+        Parameters
+        ----------
+            v:
+                The value of the element.
+            u:
+                The unit of input value.
+            e:
+                The fabrication error in percentage.
         """
-        Return the value of the Josephson Junction in angular frequency.
-        If `random` is `True`, it samples from a normal distribution with
-        variance defined by the fabrication error.
+        self._check_unit_format(u)
+        self.error = e
+
+        mean = v * unt.freq_list[u] * 2 * np.pi
+
+        self._set_value_with_error(mean, e)
 
+    def get_value(self, u: str = 'Hz') -> Union[float, Tensor]:
+        """Return the value of the element in specified unit.
+        
         Parameters
         ----------
-            random:
-                A boolean flag which specifies whether the output
-                is deterministic or random.
+            u:
+                The unit of input value. The default is "Hz".
         """
-        jMean = self.jValue * unt.freq_list[self.unit] * 2 * np.pi
 
-        if not random:
-            return jMean
+        if u in unt.freq_list:
+            return self.internal_value / unt.freq_list[u]
         else:
-            return np.random.normal(jMean, jMean * self.error / 100, 1)[0]
+            raise_unit_error()
 
+    def get_key(self, edge, B_idx, W_idx, *_):
+        """Return the junction key.
 
-class Loop:
-    """
-    Class that contains the inductive loop properties, closed path of
-    inductive elements.
+        Parameters
+        ----------
+            edge:
+                Edge that element is part of.
+            B_idx:
+                The inductive element index
+            W_idx:
+                The JJ index
+        """
+
+        return edge, self, B_idx, W_idx
+
+    def get_cap_for_flux_dist(self, flux_dist):
+
+        if flux_dist == 'all':
+            return self.cap.get_value()
+        elif flux_dist == 'junctions':
+            return VerySmallCap().get_value()
+        elif flux_dist == 'inductors':
+            return VeryLargeCap().get_value()
+
+    @staticmethod
+    def _get_default_y_func(
+        delta: float,
+        x: float
+    ) -> Callable[[Union[float, Tensor]], float]:
+
+        def _default_y_junc(
+            omega: Union[float, Tensor],
+            T: float
+        ) -> Union[float, Tensor]:
+            """Default function for junction admittance."""
+
+            alpha = unt.hbar * omega / (2 * unt.k_B * T)
+            y = (
+                np.sqrt(2 / np.pi)
+                * (8 / (delta * 1.6e-19) / (unt.hbar * 2 * np.pi / unt.e ** 2))
+                * (2 * (delta * 1.6e-19) / unt.hbar / omega) ** 1.5
+                * x * sqf.sqrt(alpha) * sqf.k0e(alpha)
+                * (1 - sqf.exp(-2 * alpha)) / 2
+            )
+
+            return y
 
+        return _default_y_junc
+
+
+class Loop(CircuitComponent):
+    """Class that contains the inductive loop properties, closed path of
+    inductive elements.
+    
     Parameters
     ----------
         value:
-            Value of the external flux at the loop.
+            Value of the external flux in the loop.
+        requires_grad:
+            A boolean variable that specifies whether autograd should record
+            operations on this loop. This feature is specific to the ``PyTorch``
+            engine.
         A:
             Normalized noise amplitude related to flux noise.
         id_str:
@@ -376,31 +618,35 @@ def __init__(
         self,
         value: float = 0,
         A: float = 1e-6,
+        requires_grad: bool = False,
         id_str: Optional[str] = None
     ) -> None:
 
-        self.lpValue = value * 2 * np.pi
+        self.set_flux(value)
+
+        if requires_grad:
+            self.requires_grad = requires_grad
+
         self.A = A * 2 * np.pi
         # indices of inductive elements.
         self.indices = []
-        # k1 matrix related to this specific loop
-        self.K1 = []
+        # k matrix related to this specific loop
+        self.k_mat = []
 
         if id_str is None:
-            self.id_str = "loop"
+            self.id_str = 'loop'
         else:
             self.id_str = id_str
 
     def reset(self) -> None:
-        self.K1 = []
+        self.k_mat = []
         self.indices = []
 
     def value(self, random: bool = False) -> float:
-        """
-        Return the value of the external flux. If `random` is `True`, it
+        """Return the value of the external flux. If ``random`` is `True``, it
         samples from a normal distribution with variance defined by the flux
         noise amplitude.
-
+        
         Parameters
         ----------
             random:
@@ -408,70 +654,73 @@ def value(self, random: bool = False) -> float:
                 deterministic or random.
         """
         if not random:
-            return self.lpValue
+            return self.internal_value
         else:
-            return np.random.normal(self.lpValue, self.A, 1)[0]
+            return np.random.normal(self.internal_value, self.A, 1)[0]
 
     def set_flux(self, value: float) -> None:
-        """
-        Set the external flux associated to the loop.
-
+        """Set the external flux associated to the loop.
+        
         Parameters
         ----------
             value:
                 The external flux value
         """
-        self.lpValue = value * 2 * np.pi
+        if get_optim_mode():
+            value = torch.as_tensor(value, dtype=torch.float64)
+
+        self.internal_value = value * 2 * np.pi
+
+    def set_noise(self, A: float) -> None:
+        self.A = A
 
     def add_index(self, index):
         self.indices.append(index)
 
-    def addK1(self, w):
-        self.K1.append(w)
-
-    def getP(self):
-        K1 = np.array(self.K1)
-        a = np.zeros_like(K1)
-        select = np.sum(K1 != a, axis=0) != 0
-        # eliminate the zero columns
-        K1 = K1[:, select]
-        if K1.shape[0] == K1.shape[1]:
-            K1 = K1[:, 0:-1]
-        b = np.zeros((1, K1.shape[0]))
+    def add_to_k_mat(self, w):
+        self.k_mat.append(w)
+
+    def get_g(self):
+
+        k_mat = sqf.remove_dependent_columns(np.array(self.k_mat))
+
+        b = np.zeros((1, k_mat.shape[0]))
         b[0, 0] = 1
-        p = np.linalg.inv(np.concatenate((b, K1.T), axis=0)) @ b.T
-        return p.T
+        g = np.linalg.inv(np.concatenate((b, k_mat.T), axis=0)) @ b.T
+        return g.T
 
 
 class Charge:
-    """
-    class that contains the charge island properties.
-    """
+    """Class that contains the charge island properties.
 
+    Parameters
+    ----------
+        value:
+            The value of the charge offset.
+        noise:
+            The amplitude of the charge noise.
+    """
     def __init__(self, value: float = 0, A: float = 1e-4) -> None:
-        """
-       inputs:
-            -- value: The value of the offset.
-            -- noise: The amplitude of the charge noise.
-        """
         self.chValue = value
         self.A = A
 
     def value(self, random: bool = False) -> float:
-        """
-        returns the value of charge bias. If random flag is true, it samples
-        from a normal distribution.
-        inputs:
-            -- random: A flag which specifies whether the output is picked
+        """Returns the value of charge bias. If ``random`` flag is true, it
+        samples from a normal distribution.
+
+        Parameters
+        ----------
+            random:
+                A flag which specifies whether the output is picked 
                 deterministically or randomly.
         """
         if not random:
             return self.chValue
         else:
-            return np.random.normal(self.chValue, self.noise, 1)[0]
+            return np.random.normal(self.chValue, self.A)
 
-    def setOffset(self, value: float) -> None:
+    def set_offset(self, value: float) -> None:
         self.chValue = value
 
-    def setNoise(self, A: float) -> None:
+    def set_noise(self, A: float) -> None:
         self.A = A
diff --git a/SQcircuit/exceptions.py b/SQcircuit/exceptions.py
new file mode 100755
index 0000000..8438ee6
--- /dev/null
+++ b/SQcircuit/exceptions.py
@@ -0,0 +1,41 @@
+"""exceptions.py contains the error and warning messages for SQcircuit"""
+from typing import List, Union
+
+from SQcircuit.settings import (
+    get_optim_mode,
+    get_engine
+)
+
+
+UNIT_ERROR = ('The input unit is not valid. Look at the API documentation for '
+              'the allowed unit formats.')
+
+
+def raise_unit_error():
+    raise ValueError(UNIT_ERROR)
+
+
+def raise_optim_error_if_needed():
+    if get_optim_mode() is False:
+        raise ModeError('PyTorch')
+
+
+class ModeError(Exception):
+    """Exception raised when trying to use a feature of SQcircuit which
+    is not supported by the currently activated engine."""
+    def __init__(self, correct_engine: Union[str, List[str]]):
+        self.correct_engine = correct_engine
+    def __str__(self):
+        if isinstance(self.correct_engine, list):
+            correct_eng_str = ' or '.join(
+                [f"'{eng}'" for eng in self.correct_engine]
+            )
+        else:
+            correct_eng_str = f"'{self.correct_engine}'"
+        return (f'The current operation is not supported by the \'{get_engine()}\' '
+                f'engine. Please set the engine to {correct_eng_str}.')
+
+
+class CircuitStateError(Exception):
+    """Exception raised when the circuit state does not permit a method to be
+    run."""
diff --git a/SQcircuit/functions.py b/SQcircuit/functions.py
new file mode 100755
index 0000000..e5d6258
--- /dev/null
+++ b/SQcircuit/functions.py
@@ -0,0 +1,417 @@
+"""Utility module with functions implemented in both PyTorch and numpy,
+depending on optimization mode."""
+from typing import Union
+
+import numpy as np
+from numpy import ndarray
+import qutip as qt
+from qutip import Qobj
+from qutip.core.data import Dia, CSR, Dense
+import scipy
+from scipy.special import kve
+import torch
+from torch import Tensor
+from torch.autograd.function import once_differentiable
+
+from SQcircuit.settings import get_optim_mode
+
+###############################################################################
+# Mathematical functions
+###############################################################################
+
+def abs(x):
+    if isinstance(x, Tensor):
+        return torch.abs(x)
+    return np.abs(x)
+
+
+def cosh(x):
+    if isinstance(x, Tensor):
+        return torch.cosh(x)
+    return np.cosh(x)
+
+
+def exp(x):
+    if isinstance(x, Tensor):
+        return torch.exp(x)
+    return np.exp(x)
+
+
+def imag(x):
+    if isinstance(x, Tensor):
+        return torch.imag(x)
+    return np.imag(x)
+
+
+def log(x):
+    if isinstance(x, Tensor):
+        return torch.log(x)
+    return np.log(x)
+
+
+def log_sinh(x):
+    if isinstance(x, Tensor):
+        return x + torch.log(1 - torch.exp(-2 * x)) - torch.log(torch.tensor(2))
+    else:
+        return x + np.log(1 - np.exp(-2 * x)) - np.log(2)
+
+
+def maximum(a, b):
+    if isinstance(a, Tensor):
+        if a > b:
+            return 0 * b + a
+        else:
+            return b
+    return np.maximum(a, b)
+
+
+def minimum(a, b):
+    if isinstance(a, Tensor):
+        if a > b:
+            return 0 * a + b
+        else:
+            return a
+    return np.minimum(a, b)
+
+
+def normal(mean, var):
+    if isinstance(mean, Tensor):
+        return torch.normal(mean, var)
+    return np.random.normal(mean, var)
+
+
+def pow(x, a):
+    if isinstance(x, Tensor):
+        return torch.pow(x, a)
+    return x ** a
+
+
+def real(x):
+    if isinstance(x, Tensor):
+        return torch.real(x)
+    return np.real(x)
+
+
+def round(x, a=3):
+    if isinstance(x, Tensor):
+        return torch.round(x * 10**a) / (10**a)
+    return np.round(x, a)
+
+
+def sqrt(x):
+    if isinstance(x, Tensor):
+        return torch.sqrt(x)
+    return np.sqrt(x)
+
+
+def tanh(x):
+    if isinstance(x, Tensor):
+        return torch.tanh(x)
+    return np.tanh(x)
+
+
+def sum(a):
+    if isinstance(a, Tensor):
+        return torch.sum(a)
+    return np.sum(a)
+
+
+def sinh(x):
+    if isinstance(x, Tensor):
+        return torch.sinh(x)
+    return np.sinh(x)
+
+
+###############################################################################
+# Special functions
+###############################################################################
+
+def k0e(x):
+    if isinstance(x, Tensor):
+        return K0eSolver.apply(x)
+    return kve(0, x)
+
+
+def log_k0(x):
+    if isinstance(x, Tensor):
+        return LogK0Solver.apply(x)
+    return log(k0e(x)) - x
+
+
+class K0eSolver(torch.autograd.Function):
+    """Computes ``torch.special.scaled_modified_bessel_k0`` with gradient.
+    """
+    @staticmethod
+    def forward(ctx, x):
+        ctx.save_for_backward(x)
+        return torch.special.scaled_modified_bessel_k0(x)
+
+    @staticmethod
+    @once_differentiable
+    def backward(ctx, grad_output):
+        x, = ctx.saved_tensors
+        x = to_numpy(x)
+
+        return grad_output * torch.tensor(kve(0, x) - kve(1, x),
+                                          dtype=torch.float64)
+
+
+class LogK0Solver(torch.autograd.Function):
+    """Computes the logarithm of the modified Bessel function of the second
+    kind for n = 0.
+    """
+    @staticmethod
+    def forward(ctx, x):
+        ctx.save_for_backward(x)
+        x = to_numpy(x)
+        return torch.as_tensor(LogK0Solver.log_kv(0, x), dtype=torch.float64)
+
+    @staticmethod
+    def log_kv(n, x):
+        return np.log(kve(n, x)) - x
+
+    @staticmethod
+    @once_differentiable
+    def backward(ctx, grad_output):
+        x, = ctx.saved_tensors
+        x = to_numpy(x)
+        # K0'(z) = -K1(z); see Eq. 10.29.3 of DLMF
+        return grad_output * -1 * torch.exp(
+            torch.tensor(LogK0Solver.log_kv(1, x) - LogK0Solver.log_kv(0, x),
+                         dtype=torch.float64)
+        )
+
+
+###############################################################################
+# Creation routines
+###############################################################################
+
+def eye(N: int) -> Union[Qobj, Tensor]:
+    """Return identity operator in qutip or torch.
+
+    parameters
+    -----------
+        N:
+            Size of the operator.
+    """
+    if get_optim_mode():
+        return torch.eye(N)
+    return qt.qeye(N)
+
+
+def zero(dtype=torch.complex128, requires_grad=False):
+    if get_optim_mode():
+        return torch.tensor(0, dtype=dtype, requires_grad=requires_grad)
+    return 0
+
+
+def zeros(shape, dtype=torch.complex128):
+    if get_optim_mode():
+        return torch.zeros(shape, dtype=dtype)
+    return np.zeros(shape)
+
+
+###############################################################################
+# Matrix operations
+###############################################################################
+
+def block_diag(*args: Union[ndarray, Tensor]) -> Union[ndarray, Tensor]:
+    if get_optim_mode():
+        return torch.block_diag(*args)
+
+    return scipy.linalg.block_diag(*args)
+
+
+def dag(state):
+    assert len(state.shape) <= 2
+
+    if isinstance(state, np.ndarray) and state.ndim == 1:
+        return np.conj(state)
+    elif isinstance(state, torch.Tensor):
+        if state.dim() == 1:
+            return torch.conj(state)
+        else:
+            return torch.conj(torch.transpose(state, 0, 1))
+    elif isinstance(state, Qobj):
+        return state.dag()
+    else:
+        raise ValueError(f'Object type {type(state)} is not supported.')
+
+
+def diag(v):
+    if get_optim_mode():
+        return torch.diag(v)
+
+    return np.diag(v)
+
+
+def tensor_product(*args: Union[Qobj, Tensor]) -> Union[Qobj, Tensor]:
+    if get_optim_mode():
+        return tensor_product_torch(*args)
+
+    return qt.tensor(*args)
+
+
+def tensor_product_torch(*args: Tensor) -> Tensor:
+    """ Pytorch function similar to ``qutip.tensor``."""
+    op_list = args
+
+    for n, op in enumerate(op_list):
+        if n == 0:
+            out = op
+        else:
+            out = torch.kron(out, op)
+
+    return out
+
+
+def mat_inv(A):
+    if isinstance(A, Tensor):
+        return torch.linalg.inv(A)
+    return np.linalg.inv(A)
+
+
+def mat_mul(A, B):
+    if get_optim_mode():
+        if isinstance(A, Tensor) and isinstance(B, Tensor):
+            if A.is_sparse and B.is_sparse:
+                return torch.sparse.mm(A, B)
+            if (A.is_sparse and not B.is_sparse) or (not A.is_sparse and B.is_sparse):
+                return torch.sparse.mm(A, B)
+            return torch.mm(A, B)
+        elif isinstance(A, Tensor) and isinstance(B, Qobj):
+            B = qobj_to_tensor(B)
+            # TODO: Add additional check on input dimensions, to make sure this works
+            return torch.transpose(torch.sparse.mm(B, A), 0, 1)
+        elif isinstance(A, Qobj) and isinstance(B, Tensor):
+            A = qobj_to_tensor(A)
+            return torch.sparse.mm(A, B)
+        A = dense(A)
+        B = dense(B)
+        return torch.matmul(torch.as_tensor(A, dtype=torch.complex128),
+                            torch.as_tensor(B, dtype=torch.complex128))
+    if isinstance(A, Qobj) and isinstance(B, Qobj):
+        return A * B
+    return A @ B
+
+
+def operator_inner_product(state1, op, state2):
+    if get_optim_mode():
+        state1 = torch.unsqueeze(state1, 1)
+        state2 = torch.unsqueeze(state2, 1)
+
+    # torch.sparse.mm requires sparse in first arg, dense in second
+    A = mat_mul(op, state2)
+    B = mat_mul(dag(state1), A)
+    return unwrap(B)
+
+
+def remove_zero_columns(matrix):
+    """Removes columns from the matrix that contain only zero values."""
+
+    # Identify columns that are not all zeros
+    non_zero_columns = np.any(matrix != 0, axis=0)
+
+    # Filter the matrix to keep only non-zero columns
+    filtered_matrix = matrix[:, non_zero_columns]
+
+    return filtered_matrix
+
+
+def remove_dependent_columns(matrix, tol=1e-5):
+    """Remove dependent columns from the given matrix."""
+
+    # first remove the zero columns from the matrix
+    no_zero_col_matrix = remove_zero_columns(matrix)
+
+    # Perform SVD
+    _, s, _ = np.linalg.svd(no_zero_col_matrix)
+
+    # Identify columns with singular values above tolerance
+    independent_columns = np.where(np.abs(s) > tol)[0]
+
+    # Reconstruct matrix with only independent columns
+    reduced_matrix = no_zero_col_matrix[:, independent_columns]
+
+    return reduced_matrix
+
+
+def sort(a):
+    if get_optim_mode():
+        return torch.sort(a)
+    return np.sort(a)
+
+
+###############################################################################
+# Type-casting
+###############################################################################
+
+def array(object, dtype=torch.float64):
+    if get_optim_mode():
+        return torch.as_tensor(object, dtype=dtype)
+    return np.array(object)
+
+
+def copy(x):
+    if get_optim_mode():
+        if isinstance(x, Qobj):
+            return x.copy()
+        return x.clone()
+    return x.copy()
+
+
+def dense(obj):
+    if isinstance(obj, Qobj):
+        return obj.full()
+    if isinstance(obj, torch.Tensor) and obj.layout != torch.strided:
+        return obj.to_dense()
+    return obj
+
+
+def mat_to_op(A: Union[ndarray, Tensor]):
+    if isinstance(A, Tensor):
+        return A
+
+    return Qobj(A).to('csr')
+
+
+def to_numpy(A: Union[ndarray, float, Tensor]):
+    if isinstance(A, Tensor):
+        return A.detach().numpy()
+    elif isinstance(A, Qobj):
+        return A.full()
+    return A
+
+
+def qobj_to_tensor(qobj, dtype=torch.complex128):
+    """Convert ``qobj`` to tensor (sparse if ``qobj`` is not dense). """
+
+    if qobj.dtype == CSR:
+        # Convert to coo, as there does not seem to be a way to convert directly
+        # from csr format to PyTorch sparse tensor
+        coo_scipy = qobj.data_as('csr_matrix').tocoo()
+    elif qobj.dtype == Dia:
+        coo_scipy = qobj.data_as('dia_matrix').tocoo()
+    elif qobj.dtype == Dense:
+        return torch.as_tensor(qobj.full(), dtype=dtype)
+    else:
+        raise ValueError(f'The datatype {qobj.dtype} is not supported.')
+
+    # Now deal with COO
+    indices = np.vstack((coo_scipy.row, coo_scipy.col))
+    values = coo_scipy.data
+    shape = coo_scipy.shape
+
+    indices_tensor = torch.LongTensor(indices)
+    values_tensor = torch.tensor(values, dtype=dtype)
+    coo_tensor = torch.sparse_coo_tensor(
+        indices_tensor, values_tensor, torch.Size(shape), dtype=dtype
+    )
+
+    return coo_tensor
+
+
+def unwrap(x):
+    if get_optim_mode():
+        return x[0, 0]
+    return x
diff --git a/SQcircuit/noise.py b/SQcircuit/noise.py
old mode 100644
new mode 100755
index ff3fc78..55ffe72
--- a/SQcircuit/noise.py
+++ b/SQcircuit/noise.py
@@ -8,10 +8,10 @@
 
 
 ENV = {
-    "T":  0.015,    # experiment time
-    "omega_low": 2 * np.pi,    # low-frequency cut off
-    "omega_high": 2 * np.pi * 3 * 1e9,    # high-frequency cut off
-    "t_exp": 10e-6,    # experiment time
+    'T':  0.015,    # temperature
+    'omega_low': 2 * np.pi,    # low-frequency cut off
+    'omega_high': 2 * np.pi * 3 * 1e9,    # high-frequency cut off
+    't_exp': 10e-6,    # experiment time
 }
 
 
@@ -27,7 +27,7 @@ def set_temp(T: float) -> None:
 
     global ENV
 
-    ENV["T"] = T
+    ENV['T'] = T
 
 
 def set_low_freq(value: float, unit: str) -> None:
@@ -45,7 +45,7 @@ def set_low_freq(value: float, unit: str) -> None:
 
     global ENV
 
-    ENV["omega_low"] = 2 * np.pi * value * unt.freq_list[unit]
+    ENV['omega_low'] = 2 * np.pi * value * unt.freq_list[unit]
 
 
 def set_high_freq(value: float, unit: str) -> None:
@@ -63,7 +63,7 @@ def set_high_freq(value: float, unit: str) -> None:
 
     global ENV
 
-    ENV["omega_high"] = 2 * np.pi * value * unt.freq_list[unit]
+    ENV['omega_high'] = 2 * np.pi * value * unt.freq_list[unit]
 
 
 def set_t_exp(value: float, unit: str) -> None:
@@ -81,7 +81,7 @@ def set_t_exp(value: float, unit: str) -> None:
 
     global ENV
 
-    ENV["t_exp"] = value * unt.time_list[unit]
+    ENV['t_exp'] = value * unt.time_list[unit]
 
 
 def reset_to_default() -> None:
diff --git a/SQcircuit/settings.py b/SQcircuit/settings.py
old mode 100644
new mode 100755
index d623c2d..d9fc918
--- a/SQcircuit/settings.py
+++ b/SQcircuit/settings.py
@@ -2,7 +2,81 @@
 
 # accuracy settings related to computation in SQcircuit algorithm
 ACC = {
-    "sing_mode_detect": 1e-11,  # singular mode detection accuracy
-    "Gram–Schmidt": 1e-7,  # Gram–Schmidt process accuracy
-    "har_mode_elim": 1e-11,  # harmonic mode elimination accuracy
+    'sing_mode_detect': 1e-11,  # singular mode detection accuracy
+    'Gram-Schmidt': 1e-6,  # Gram–Schmidt process accuracy
+    'har_mode_elim': 1e-11,  # harmonic mode elimination accuracy
 }
+
+# Global variable holding current engine
+_ENG = 'numpy'
+
+SUPPORTED_ENGINES = ['NumPy', 'PyTorch']
+SUPPORTED_ENGINES_LOWER = [e.lower() for e in SUPPORTED_ENGINES]
+SUPPORTED_OPTIM_ENGINES = ['PyTorch']
+
+def set_engine(eng: str) -> None:
+    """Choose the internal numerical package for ``SQcircuit` to use.  This
+    sets the type of objects used for element values, computed eigenfrequencies
+    and eigenvectors, etc.
+
+    - ``'NumPy'``
+    - ``'PyTorch'`` (allows gradient-based optimization)
+    
+    Parameters
+    ----------
+    eng:
+        Numerical engine for SQcircuit to use.
+    """
+    global _ENG
+
+    eng = eng.lower()
+
+    if eng not in SUPPORTED_ENGINES_LOWER:
+        raise ValueError(
+            'Invalid engine passed. Currently SQcircuit supports '
+            + ' and '.join([f"'{eng}'" for eng in SUPPORTED_ENGINES])
+            + '.'
+        )
+
+    _ENG = eng
+
+def get_engine() -> str:
+    """Get the internal numerical engine which ``SQcircuit`` is using.
+
+    Returns
+    ----------
+        The numerical engine currently in use.
+    """
+    return _ENG.lower()
+
+
+def set_optim_mode(s: bool) -> None:
+    """Utility function for setting engine of ``SQcircuit``. If the optimization
+    mode is ``True``, then ``SQcircuit`` uses ``PyTorch``; if it is false
+    it uses ``NumPy``.
+
+    Parameters
+    ----------
+    s:
+        State of the optimization mode as boolean variable.
+    """
+
+    if s:
+        eng = 'PyTorch'
+    else:
+        eng = 'NumPy'
+
+    set_engine(eng)
+
+
+def get_optim_mode() -> bool:
+    """Utility function for getting the current engine of ``SQcircuit``. If
+    the engine is ``NumPy``, the optimization mode is ``False``; if it is
+    ``PyTorch``, the optimization mode is ``True``.
+    
+    Returns
+    ----------
+        If the numerical engine is NumPy (``False``) or PyTorch (``True``)
+    """
+
+    return get_engine() == 'pytorch'
diff --git a/SQcircuit/storage.py b/SQcircuit/storage.py
old mode 100644
new mode 100755
index 29abe12..421236d
--- a/SQcircuit/storage.py
+++ b/SQcircuit/storage.py
@@ -33,8 +33,8 @@ def load(fromFile: str):
             return pickle.load(inp)
 
     def plot(self, typeP):
-        assert len(self.efreq.shape) <= 2, \
-            "SQcircuit can only plot 1D grid data for now"
+        if len(self.efreq.shape) > 2:
+            raise NotImplementedError('SQcircuit can only plot 1D grid data for now.')
 
         plt.figure(figsize=(6.5, 5), linewidth=1)
         plt.tight_layout()
@@ -42,9 +42,9 @@ def plot(self, typeP):
         for i in range(self.numEig):
             plt.plot(self.grid[0], self.efreq[i, :] - self.efreq[0, :],
                      linewidth=2.2)
-        if typeP == "flux":
-            plt.xlabel(r"$\Phi_{ext}/\Phi_0$", fontsize=18)
-        elif typeP == "charge":
-            plt.xlabel(r"$n_g$", fontsize=18)
-        plt.ylabel(r"$f_i-f_0$[GHz]", fontsize=18)
+        if typeP == 'flux':
+            plt.xlabel(r'$\Phi_{ext}/\Phi_0$', fontsize=18)
+        elif typeP == 'charge':
+            plt.xlabel(r'$n_g$', fontsize=18)
+        plt.ylabel(r'$f_i-f_0$[GHz]', fontsize=18)
         plt.show()
diff --git a/SQcircuit/sweep.py b/SQcircuit/sweep.py
old mode 100644
new mode 100755
index f41dfb3..753812d
--- a/SQcircuit/sweep.py
+++ b/SQcircuit/sweep.py
@@ -3,10 +3,13 @@
 """
 
 from itertools import product
+import logging
 
 from SQcircuit.circuit import *
 from SQcircuit.storage import *
 
+logger = logging.getLogger(__name__)
+
 
 class Sweep(SQdata):
     """
@@ -25,7 +28,7 @@ def __init__(self, cr: Circuit, numEig: int, properties=None):
         self.numEig = numEig
 
         if properties is None:
-            properties = ["efreq"]
+            properties = ['efreq']
 
         self.properties = properties
         # type of the data
@@ -39,14 +42,14 @@ def __init__(self, cr: Circuit, numEig: int, properties=None):
         # eigenfrequencies of the circuit
         self.efreq = None
         # the decayRates
-        if "loss" in properties:
+        if 'loss' in properties:
             self.dec = {
-                "capacitive": None,
-                "inductive": None,
-                "quasiparticle": None,
-                "charge": None,
-                "cc": None,
-                "flux": None
+                'capacitive': None,
+                'inductive': None,
+                'quasiparticle': None,
+                'charge': None,
+                'cc': None,
+                'flux': None
             }
         else:
             self.dec = None
@@ -68,14 +71,14 @@ def _gridIndex(grid):
     def sweepFlux(self, loops: list, grid: list,
                   toFile: str = None, plotF: bool = False):
 
-        self.type = "sweepFlux"
+        self.type = 'sweepFlux'
         self.params = loops
         self.grid = grid
 
         # table of eigenfrequencies that we want to calculate
         self.efreq = np.zeros((self.numEig, *self._gridDims(grid)))
 
-        if "loss" in self.properties:
+        if 'loss' in self.properties:
             # dictionary that contains the decoherence rate for each loss
             # mechanism
             self.dec = {key: np.zeros(self._gridDims(grid))
@@ -89,15 +92,15 @@ def sweepFlux(self, loops: list, grid: list,
 
             evec, _ = self.cr.diag(self.numEig)
             self.efreq[:, indices] = evec.reshape(self.efreq[:, indices].shape)
-            if "loss" in self.properties:
+            if 'loss' in self.properties:
                 for dec_type in self.dec.keys():
                     self.dec[dec_type][indices] = \
                         self.cr.dec_rate(dec_type=dec_type, states=(1, 0))
 
-        print("Sweeping process is finished!")
+        logger.info('Sweeping process is finished!')
 
         if plotF:
-            self.plot("flux")
+            self.plot('flux')
 
         if toFile:
             self.save(toFile)
@@ -108,14 +111,14 @@ def sweepFlux(self, loops: list, grid: list,
     def sweepCharge(self, modes: list, grid: list,
                     toFile: str = None, plotF: bool = False):
 
-        self.type = "sweepCharge"
+        self.type = 'sweepCharge'
         self.params = modes
         self.grid = grid
 
         # table of eigenfrequencies that we want to calculate
         self.efreq = np.zeros((self.numEig, *self._gridDims(grid)))
 
-        if "loss" in self.properties:
+        if 'loss' in self.properties:
             # dictionary that contains the decoherence rate for each
             # loss mechanism
             self.dec = {key: np.zeros(self._gridDims(grid))
@@ -129,18 +132,17 @@ def sweepCharge(self, modes: list, grid: list,
 
             evec, _ = self.cr.diag(self.numEig)
             self.efreq[:, indices] = evec.reshape(self.efreq[:, indices].shape)
-            if "loss" in self.properties:
+            if 'loss' in self.properties:
                 for dec_type in self.dec.keys():
                     self.dec[dec_type][indices] = \
                         self.cr.dec_rate(dec_type=dec_type, states=(1, 0))
 
-        print("Sweeping process is finished!")
+        logger.info('Sweeping process is finished!')
 
         if plotF:
-            self.plot("charge")
+            self.plot('charge')
 
         if toFile:
             self.save(toFile)
-            pass
         else:
             return self.efreq, self.dec
diff --git a/SQcircuit/symbolic.py b/SQcircuit/symbolic.py
new file mode 100755
index 0000000..fde80de
--- /dev/null
+++ b/SQcircuit/symbolic.py
@@ -0,0 +1,178 @@
+"""symbolic.py"""
+import numpy as np
+import sympy as sm
+from sympy import Expr, Symbol
+from sympy.physics.quantum import Operator
+
+from SQcircuit.elements import Inductor, Junction
+
+
+class ExplicitSymbol(Symbol):
+    def __new__(cls, plainname, latexname):
+        self = super().__new__(cls, plainname)
+        self.lsymbol = Symbol(latexname)
+        return self
+
+    def _latex(self, printer):
+        return printer.doprint(self.lsymbol)
+
+
+class qOperator(Operator):
+    """
+    Class extending sympy physics operator, mostly for custom printing purposes.
+    Created with a base ``name` and a `subscript``. When printing in LaTeX, a
+    hat will be put over the name, but not in plaintext.
+    """
+    def __new__(cls, name, subscript=None):
+        if subscript:
+            self = super().__new__(cls,f'{name}_{subscript}')
+        else:
+            self = super().__new__(cls, name)
+        self.opname = name
+        self.sub = subscript
+        return self
+
+    def _latex(self, printer):
+        if self.opname[-1] == 'd':
+            tex = fr'\hat{{{self.opname[:-1]}}}^\dagger'
+        else:
+            tex = fr'\hat{{{self.opname}}}'
+        if self.sub:
+            tex  += fr'_{{{self.sub}}}'
+        return printer.doprint(Symbol(tex))
+
+
+def phi_op(i):
+    return ExplicitSymbol(f'phi_{i}',
+                          fr'\hat{{\varphi}}_{{{i}}}')
+
+
+def a(i: int) -> qOperator:
+    return qOperator('a', i)
+
+
+def ad(i: int) -> qOperator:
+    return qOperator('ad', i)
+
+
+def n(i: int) -> qOperator:
+    return qOperator('n', i)
+
+
+def phi_ext(i: int) -> ExplicitSymbol:
+    return ExplicitSymbol(f'phi_e{i}',
+                          r'\varphi_{\text{ext}_{' + str(i) + '}}')
+
+
+def phi_zp(i: int) -> ExplicitSymbol:
+    return ExplicitSymbol(f'zp_{i}',
+                          r'\varphi_{\text{zp}_{' + str(i) + '}}')
+
+
+def omega(i: int) -> Symbol:
+    return Symbol(rf'omega_{i}')
+
+
+def ng(i: int) -> ExplicitSymbol:
+    return ExplicitSymbol(f'ng_{i}',
+                          fr'n_{{g_{{{i}}}}}')
+
+
+def EC(i: int, j: int) -> ExplicitSymbol:
+    return ExplicitSymbol(f'EC_{i}{j}',
+                          fr'E_{{C_{{{i}{j}}}}}')
+
+
+def EL(i: int) -> ExplicitSymbol:
+    return ExplicitSymbol(f'EL_{i}',
+                          fr'E_{{L_{{{i}}}}}')
+
+
+def EJ(i: int) -> ExplicitSymbol:
+    return ExplicitSymbol(f'EJ_{i}',
+                          fr'E_{{J_{{{i}}}}}')
+
+
+H = qOperator('H')
+
+
+def har_mode_hamil(coeff_dict, do_sum=True) -> Expr:
+    terms =  [sm.Mul(omega(i+1), ad(i+1), a(i+1), evaluate=False)
+              for i in range(coeff_dict['har_dim'])]
+    if do_sum:
+        return sm.Add(*terms, evaluate=False)
+
+    return terms
+
+
+def charge_mode_hamil(coeff_dict, do_sum=True) -> Expr:
+    hamil = 0
+    for i in range(coeff_dict['har_dim'], coeff_dict['n_modes']):
+        for j in range(i, coeff_dict['n_modes']):
+            hamil += EC(i+1,j+1) * (n(i+1) - ng(i+1)) * (n(i+1) - ng(j+1))
+
+    if do_sum:
+        return hamil
+
+    return sm.Add.make_args(hamil)
+
+
+def inductive_hamil(elem_keys, coeff_dict, do_sum=True) -> Expr:
+    S = coeff_dict['S']
+    B = coeff_dict['B']
+
+    terms = []
+    for i, (edge, _, b_id) in enumerate(elem_keys[Inductor]):
+        if np.sum(np.abs(B[b_id, :])) == 0:
+            continue
+        if 0 in edge:
+            w = S[edge[0] + edge[1] - 1, :]
+        else:
+            w = S[edge[0] - 1, :] - S[edge[1] - 1, :]
+        w = np.round(w[:coeff_dict['har_dim']], 3)
+
+        w_sym = sm.Matrix(w)
+        B_sym = sm.Matrix(B[b_id, :])
+
+        phis = [phi_op(i+1) for i in range(len(w_sym))]
+        phi_exts = [phi_ext(i+1) for i in range(len(B_sym))]
+
+        terms.append(sm.Mul(EL(i+1), sm.nsimplify(w_sym.dot(phis)),
+                            sm.nsimplify(B_sym.dot(phi_exts)), evaluate=False))
+    if do_sum:
+        return sum(terms)
+
+    return terms
+
+
+def jj_hamil(elem_keys, coeff_dict, do_sum=True) -> Expr:
+    terms = []
+    for i, (_, _, b_id, w_id) in enumerate(elem_keys[Junction]):
+        W_sym = sm.Matrix(coeff_dict['W'][w_id, :])
+        phis = [phi_op(i+1) for i in range(len(W_sym))]
+
+        B_sym = sm.Matrix(coeff_dict['B'][b_id, :])
+        phi_exts = [phi_ext(i+1) for i in range(len(B_sym))]
+
+        terms.append(- EJ(i+1) * sm.cos(sm.Add(sm.nsimplify(W_sym.dot(phis)),
+                                             sm.nsimplify(B_sym.dot(phi_exts))
+                                             ))
+        )
+    if do_sum:
+        return sum(terms)
+
+    return terms
+
+
+def construct_hamiltonian(cr) -> Expr:
+    har_terms = har_mode_hamil(cr.descrip_vars, do_sum=False)
+    charge_terms = charge_mode_hamil(cr.descrip_vars, do_sum=False)
+    ind_terms = inductive_hamil(cr.elem_keys, cr.descrip_vars, do_sum=False)
+    JJ_terms = jj_hamil(cr.elem_keys, cr.descrip_vars, do_sum=False)
+
+    all_terms = []
+    for terms in [har_terms, charge_terms, ind_terms, JJ_terms]:
+        if sum(terms) != 0:
+            all_terms += terms
+
+    return sm.Add(*all_terms, evaluate=False)
diff --git a/SQcircuit/systems.py b/SQcircuit/systems.py
new file mode 100755
index 0000000..1321f17
--- /dev/null
+++ b/SQcircuit/systems.py
@@ -0,0 +1,327 @@
+"""systems.py contains the classes for the interaction and connection of
+superconducting circuits.
+"""
+
+from typing import List, Tuple, Union
+
+import numpy as np
+import qutip as qt
+
+from numpy import ndarray
+from qutip import Qobj
+
+import SQcircuit.units as unt
+import SQcircuit.functions as sqf
+
+from SQcircuit.circuit import Circuit
+from SQcircuit.elements import Capacitor, Inductor, Junction
+
+
+class Couple:
+    """Class  that stores the coupling between two different circuits.
+
+    Parameters
+    ----------
+        cr1:
+            Circuit of the system one.
+        attr1:
+            node or edge related to system one.
+        cr2:
+            Circuit of the system two.
+        attr2:
+            node or edge related to system two.
+        el:
+            The element of the coupling. (For now only capacitors)
+    """
+    def __init__(
+            self,
+            cr1: Circuit,
+            attr1: Union[int, Tuple[int, int]],
+            cr2: Tuple[Circuit, int],
+            attr2: Union[int, Tuple[int, int]],
+            el: Union[Capacitor, Inductor, Junction]
+    ) -> None:
+
+        # information of the circuits
+        self.circuits = [cr1, cr2]
+
+        # attribute of each circuit
+        self.attrs = [attr1, attr2]
+
+        # element of coupling
+        self.el = el
+
+
+class System:
+    """Class that contains method for calculating the spectrum and properties
+    of a system of coupled circuits.
+
+    Parameters
+    ----------
+        couplings:
+            List of couplings that describes the interaction between
+            different part of the system.
+    """
+
+    def __init__(self, couplings: List['Couple']) -> None:
+
+        self.couplings = couplings
+
+        # list of circuits
+        self.circuits = self._get_all_circuits()
+
+        # number of sub-circuits (subsystems).
+        self.n_sub = len(self.circuits)
+
+        # overall number of nodes
+        self.n_N = sum([circ.n for circ in self.circuits])
+
+        # truncation numbers for each sub-circuit which is the number of
+        # eigenvalue of each sub-circuit.
+        self.trunc_nums = self._get_all_circuits_num_eig()
+
+    def _get_all_circuits(self) -> List[Circuit]:
+        """Return all the circuits described in ``System.couplings`` as a
+        list."""
+
+        # list of circuits
+        circuits = []
+
+        for couple in self.couplings:
+            for circuit in couple.circuits:
+
+                if circuit not in circuits:
+                    circuits.append(circuit)
+
+        return circuits
+
+    def _get_all_circuits_num_eig(self) -> List[int]:
+        """Return the number of eigenvalues for all circuit as a list.
+        """
+
+        num_eig_list = []
+
+        for circ in self.circuits:
+            num_eig_list.append(len(circ.efreqs))
+
+        return num_eig_list
+
+    def _node_idx_in_sys(self, cr: Circuit, i: int) -> int:
+        """Return node index in the general system.
+
+        Parameters
+        ----------
+            cr:
+                Circuit of interest as a subsystem.
+            i:
+                Integer specifies the node index in the sub-circuit.
+        """
+
+        # number of nodes up to the cr circuit.
+        N = 0
+        for circ in self.circuits:
+            if circ == cr:
+                break
+            N += circ.n
+
+        return N+i-1
+
+    def _node_idx_in_sub_sys(self, i: int) -> Tuple[int, int]:
+        """Return circuit and node index related to node index in the overall
+        system.
+
+        Parameters
+        ----------
+            i:
+                node index in the overall system.
+        """
+
+        # number of nodes
+        N = 0
+
+        # node in sub-circuit
+        node_in_subsys: int = 0
+
+        # index of the sub-circuit in overall system.
+        sub_idx: int = 0
+
+        for circ in self.circuits:
+            if i < N+circ.n:
+                # node in sub-circuit
+                node_in_subsys = i-N+1
+
+                # index of the circuit in overall system.
+                sub_idx = self.circuits.index(circ)
+                break
+            N += circ.n
+
+        return node_in_subsys, sub_idx
+
+    def _bare_cap_matrix(self) -> ndarray:
+        """Return the capacitance matrix of the entire system as``ndarray``
+        without considering the coupling capacitors."""
+
+        # list of capacitance matrix for each circuit
+        cap_matrices = [circ.C for circ in self.circuits]
+
+        return sqf.block_diag(*cap_matrices)
+
+    def cap_matrix(self) -> ndarray:
+        """Return the capacitance matrix of the entire system as``ndarray``."""
+
+        C = self._bare_cap_matrix()
+
+        for couple in self.couplings:
+
+            if isinstance(couple.el, Capacitor):
+
+                i = self._node_idx_in_sys(couple.circuits[0], couple.attrs[0])
+
+                j = self._node_idx_in_sys(couple.circuits[1], couple.attrs[1])
+
+                C[i, i] += couple.el.get_value()
+                C[j, j] += couple.el.get_value()
+                C[i, j] -= couple.el.get_value()
+                C[j, i] -= couple.el.get_value()
+
+        return C
+
+    def _op_in_sys(self, sub_op: Qobj, sub_idx: int) -> Qobj:
+        """Return the subsystem operator in the tensor product space of the
+        overall system as ``Qutip.Qobj`` format.
+
+        Parameters
+        ----------
+            sub_op:
+                Subsystem operator with format of ``Qutip.Qobj``.
+            sub_idx:
+                Integer that specifies the index of the subsystem in the
+                overall system.
+        """
+
+        op_list: List[Qobj] = []
+
+        for i in range(self.n_sub):
+
+            if i == sub_idx:
+                op_list.append(sub_op)
+            else:
+                op_list.append(sqf.eye(self.trunc_nums[i]))
+
+        return sqf.tensor_product(*op_list)
+
+    def _op_times_op_in_sys(
+            self,
+            sub_op_1: Qobj,
+            sub_idx_1: int,
+            sub_op_2: Qobj,
+            sub_idx_2: int
+    ) -> Qobj:
+        """Return the multiplication of two subsystem operators in the
+        tensor product space of the overall system as ``Qutip.Qobj`` format.
+
+        Parameters
+        ----------
+            sub_op_1:
+                First subsystem operator with format of ``Qutip.Qobj``.
+            sub_idx_1:
+                Integer that specifies the first index of the subsystem in the
+                overall system.
+                        sub_op_1:
+                Second subsystem operator with format of ``Qutip.Qobj``.
+            sub_idx_1:
+                Integer that specifies the second index of the subsystem in the
+                overall system.
+        """
+
+        # If both operators are in the same subsystems.
+        if sub_idx_1 == sub_idx_2:
+            return self._op_in_sys(sqf.mat_mul(sub_op_1, sub_op_2), sub_idx_1)
+
+        op_list: List[Qobj] = []
+
+        for i in range(self.n_sub):
+
+            if i == sub_idx_1:
+                op_list.append(sub_op_1)
+            elif i == sub_idx_2:
+                op_list.append(sub_op_2)
+            else:
+                op_list.append(sqf.eye(self.trunc_nums[i]))
+
+        return sqf.tensor_product(*op_list)
+
+    def _QQ_op(self, m: int, n: int) -> Qobj:
+        """Return Q_m*Q_n operator.
+
+        Parameters
+        ----------
+            m:
+                Integer that specifies the index of the first charge operator
+                in the overall system.
+            n:
+                Integer that specifies the index of the second charge operator
+                in the overall system.
+        """
+
+        node_in_subsys, m_sub_idx = self._node_idx_in_sub_sys(m)
+        Q_m = self.circuits[m_sub_idx].charge_op(node_in_subsys, basis='eig')
+
+        node_in_subsys, n_sub_idx = self._node_idx_in_sub_sys(n)
+        Q_n = self.circuits[n_sub_idx].charge_op(node_in_subsys, basis='eig')
+
+        return self._op_times_op_in_sys(Q_m, m_sub_idx, Q_n, n_sub_idx)
+
+    def _quadratic_Q(self, A: ndarray) -> Qobj:
+        """Return quadratic form of 1/2 * Q^T * A * Q
+
+        Parameters
+        ----------
+            A:
+                ndarray matrix that specifies the coefficient for
+                quadratic expression.
+        """
+
+        op = 0
+
+        for i in range(self.n_N):
+            for j in range(self.n_N):
+                if i == j:
+                    op += 0.5 * A[i, i] * self._QQ_op(i, j)
+                elif j > i:
+                    op += A[i, j] * self._QQ_op(i, j)
+
+        return op
+
+    def _H_local(self) -> Qobj:
+        """Return summation of local Hamiltonian in default frequency unit
+        of SQcircuit as ``Qutip.Qobj`` format.
+        """
+
+        op = 0
+
+        for i, circ in enumerate(self.circuits):
+
+            op += self._op_in_sys(sqf.mat_to_op(sqf.diag(circ.efreqs)), i)
+
+        return op
+
+    def _H_int(self) -> Qobj:
+        """Return interaction Hamiltonian in default frequency unit
+        of SQcircuit as ``Qutip.Qobj`` format.
+        """
+
+        delta_C_inv = (sqf.mat_inv(self.cap_matrix())
+                       - sqf.mat_inv(self._bare_cap_matrix()))
+
+        R = sqf.block_diag(*[sqf.array(circ.R) for circ in self.circuits])
+
+        return self._quadratic_Q(R.T @ delta_C_inv @ R) / (
+                2*np.pi*unt.get_unit_freq())
+
+    def hamiltonian(self) -> Qobj:
+        """Return Hamiltonian of the overall system in default frequency
+        unit of SQcircuit as ``Qutip.Qobj`` format.
+        """
+
+        return self._H_local() + self._H_int()
diff --git a/SQcircuit/texts.py b/SQcircuit/texts.py
old mode 100644
new mode 100755
index 4a2c584..b2d4da9
--- a/SQcircuit/texts.py
+++ b/SQcircuit/texts.py
@@ -1,13 +1,25 @@
-""" Module for text and latex output of SQcircuit"""
+"""Module for text and LaTeX output of SQcircuit"""
 
 import numpy as np
-from IPython.display import display, Latex
+from IPython.display import display, Math
+import sympy as sm
+from sympy.printing.latex import LatexPrinter
+from sympy.printing.pretty.pretty import PrettyPrinter
+from sympy.printing.pretty.stringpict import prettyForm
 
+from SQcircuit.elements import Capacitor, Inductor, Junction
+import SQcircuit.symbolic as sym
+
+
+ELEMENT_NAMES = {
+    Capacitor: 'C',
+    Inductor: 'L',
+    Junction: 'JJ'
+}
 
 def is_notebook():
-    """
-    The function that checks whether we are working in notebook environment or
-    Python terminal.
+    """Checks whether we are working in notebook environment or the Python
+    terminal.
     """
     try:
         shell = get_ipython().__class__.__name__
@@ -21,203 +33,187 @@ def is_notebook():
         return False  # Probably standard Python interpreter
 
 
+def newDiv(self, den):
+    """prettyForm division to always be inline x/y
+    """
+    return prettyForm(binding=prettyForm.DIV, s=self.s + '/' + den.s)
+
+
 class HamilTxt:
     """
     Class that contains the methods for printing the Hamiltonian in text or
     latex format.
     """
 
-    def __init__(self, tp='ltx'):
+    def __init__(self, tp='ltx', _test=False):
         self.tp = tp
         self.line = 60 * "-" + "\n"
-
-    def Ec(self, i, j):
-        if self.tp == 'ltx':
-            return "E_{C_{" + f"{i}{j}" + "}}"
-        elif self.tp == 'txt':
-            return f"EC_{i}{j}"
-
-    def Ej(self, i):
-        if self.tp == 'ltx':
-            return "E_{J_{" + f"{i}" + "}}"
-        elif self.tp == 'txt':
-            return f"EJ_{i}"
-
-    def El(self, i):
-        if self.tp == 'ltx':
-            return "E_{L_{" + f"{i}" + "}}"
-        elif self.tp == 'txt':
-            return f"EL_{i}"
-
-    def phi(self, i):
-        if self.tp == 'ltx':
-            return "\hat{\\varphi}_" + f"{i}"
-        elif self.tp == 'txt':
-            return f"\u03C6_{i}"
-
-    def phiExt(self, i):
-        if self.tp == 'ltx':
-            return "\\varphi_{\\text{ext}_{" + f"{i}" + "}}"
-        elif self.tp == 'txt':
-            return f"\u03C6_e{i}"
-
-    def zp(self, i):
-        if self.tp == 'ltx':
-            return "\\varphi_{zp_{" + f"{i}" + "}}"
-        elif self.tp == 'txt':
-            return f"zp_{i}"
-
-    def omega(self, i, F=True):
-        if self.tp == 'ltx':
-            if F:
-                return f"\omega_{i}"
-            else:
-                return f"\omega_{i}" + "/2\pi"
-        elif self.tp == 'txt':
-            if F:
-                return f"\u03C9_{i}"
+        if tp == 'ltx':
+            self.printer = LatexPrinter({'order': 'none', 'fold_short_frac': True})
+        elif tp == 'txt':
+            prettyForm.__truediv__ = newDiv
+            if _test:
+                self.printer = PrettyPrinter({
+                    'use_unicode': False,
+                    'order': 'none',
+                    'wrap_line': True,
+                    'num_columns': 80
+                })
             else:
-                return f"\u03C9_{i}" + "/2\u03C0"
+                self.printer = PrettyPrinter({
+                    'use_unicode': False,
+                    'order': 'none'
+                })
+        else:
+            raise ValueError('Permitted values for `tp` are \'ltx\' and '
+                             '\'txt\'.')
 
-    def tPi(self):
+    def tab(self) -> str:
         if self.tp == 'ltx':
-            return "/2\pi"
+            return 11*"~"
         elif self.tp == 'txt':
-            return "/2\u03C0"
+            return 7*" "
 
-    def a(self, i):
+    def plaintxt(self, text) -> str:
         if self.tp == 'ltx':
-            return f"\hat a_{i}"
-        elif self.tp == 'txt':
-            return f"a_{i}"
+            return rf'\text{{{text}}}'
+        else:
+            return text
 
-    def ad(self, i):
-        if self.tp == 'ltx':
-            return f"\hat a^\dagger_{i}"
-        elif self.tp == 'txt':
-            return f"ad_{i}"
+    def ham_txt(self, coeff_dict) -> str:
+        sym_ham = coeff_dict['H']
+        return self.printer.doprint(sm.Eq(sym.H, sym_ham)) + '\n'
 
-    def n(self, i, j):
-        if self.tp == 'ltx':
-            if i == j:
-                return "(\hat{n}_" + f"{i}" + "-n_{g_{" + f"{i}" + "}})^2"
-            else:
-                return "(\hat{n}_" + f"{i}" + "-n_{g_{" + f"{i}" + "}})" +\
-                       "(\hat{n}_" + f"{j}" + "-n_{g_{" + f"{j}" + "}})"
-        elif self.tp == 'txt':
-            if i == j:
-                return f"(n_{i}-ng_{i})^2"
+    def mode_txt(self, coeff_dict) -> str:
+        txt = ''
+        for i in range(coeff_dict['n_modes']):
+            if i < coeff_dict['har_dim']:
+                kind = 'harmonic'
+                omega_val  = np.round(coeff_dict['omega'][i], 5)
+                zp_val = coeff_dict['phi_zp'][i]
+                info = [
+                    sm.Eq(sym.phi_op(i+1),
+                          sym.phi_zp(i+1)*(sym.a(i+1) + sym.ad(i+1))),
+                    sm.Eq(sym.omega(i+1)/(2 * sm.pi), float(f'{omega_val:.5e}')),
+                    sm.Eq(sym.phi_zp(i+1), float(f'{zp_val:.2e}'))
+                ]
             else:
-                return f"(n_{i}-ng_{i})(n_{j}-ng_{j})"
+                kind = 'charge'
+                ng_val = np.round(coeff_dict['ng'][i - coeff_dict['har_dim']], 3)
+                info = [sm.Eq(sym.ng(i+1), ng_val)]
+            txt += (
+                self.plaintxt(f'mode {i+1}:') + self.tab()
+                + self.plaintxt(kind) + self.tab()
+            )
+            txt += self.tab().join([self.printer.doprint(e) for e in info]) + '\n'
+        return txt
 
-    def ng(self, i):
-        if self.tp == 'ltx':
-            return "n_{g_{" + f"{i}" + "}}"
+    def param_txt(self, coeff_dict) -> str:
+        params = []
+        # For the LC part, only print out if not completely absorbed into the
+        # omegas of the harmonic modes.
+        ## Only capacitve energies on charge modes
+        for i in range(coeff_dict['har_dim'], coeff_dict['n_modes']):
+            for j in range(i, coeff_dict['n_modes']):
+                params.append(sm.Eq(sym.EC(i+1,j+1),
+                                    np.round(coeff_dict['EC'][i, j], 3)))
+        ## Only inductive energy if it has external flux assigned to it
+        for i, EL_val in enumerate(coeff_dict['EL']):
+            if coeff_dict['EL_has_ext_flux'][i]:
+                params.append(sm.Eq(sym.EL(i+1),
+                                    np.round(EL_val, 3)))
+        # Need to print all junctions
+        for i, EJ_val in enumerate(coeff_dict['EJ']):
+            params.append(sm.Eq(sym.EJ(i+1),
+                                np.round(EJ_val, 3)))
+        txt = (
+            self.plaintxt('parameters:') + self.tab()
+            + self.tab().join([self.printer.doprint(p) for p in params])
+        )
+        return txt + '\n'
+
+    def loop_txt(self, coeff_dict) -> str:
+        info = [sm.Eq(sym.phi_ext(i+1)/(2*sm.pi),
+                      np.round(coeff_dict['loops'][i], 2))
+                for i in range(coeff_dict['n_loops'])]
+        txt = (
+            self.plaintxt('loops:') + self.tab()
+            + self.tab().join([self.printer.doprint(l) for l in info])
+        )
+        return txt
 
-        elif self.tp == 'txt':
-            return f"ng_{i}"
+    def print_circuit_description(self, coeff_dict) -> str:
+        finalTxt = (
+            self.ham_txt(coeff_dict) + self.line
+            + self.mode_txt(coeff_dict) + self.line 
+            + self.param_txt(coeff_dict)
+            + self.loop_txt(coeff_dict)
+        )
 
-    def mode(self, i):
-        if self.tp == 'ltx':
-            return "\\text{mode}" + f"~{i}:"
-        elif self.tp == 'txt':
-            return f"mode {i}:"
+        self.display(finalTxt)
+        return finalTxt
 
-    def param(self):
-        if self.tp == 'ltx':
-            return "\\text{parameters}:"
-        elif self.tp == 'txt':
-            return f"parameters:"
+    @staticmethod
+    def print_loop_description(cr) -> str:
+        # maximum length of element ID strings
+        nr = max(
+            [len(el.id_str) for _, el, _, _ in cr.elem_keys[Junction]]
+            + [len(el.id_str) for _, el, _ in cr.elem_keys[Inductor]]
+        )
 
-    def har(self):
-        if self.tp == 'ltx':
-            return "\\text{harmonic}"
-        elif self.tp == 'txt':
-            return "harmonic"
+        # maximum length of loop ID strings
+        nh = max([len(lp.id_str) for lp in cr.loops])
 
-    def ch(self):
-        if self.tp == 'ltx':
-            return "\\text{charge}~~~~~"
-        elif self.tp == 'txt':
-            return "charge  "
+        # number of loops
+        nl = len(cr.loops)
 
-    def loops(self):
-        if self.tp == 'ltx':
-            return "\\text{loops}:~~~~~~~~~"
-        elif self.tp == 'txt':
-            return f"loops:     "
+        # space between elements in rows
+        ns = 5
 
-    def H(self):
-        if self.tp == 'ltx':
-            return "\hat{H} =~"
-        elif self.tp == 'txt':
-            return "H = "
+        # header with names of loops
+        header = (nr + ns + len(', b1:')) * " "
+        header += (' ' * 10).join([f'{lp.id_str:{nh}}' for lp in cr.loops])
 
-    def cos(self):
-        if self.tp == 'ltx':
-            return "\cos"
-        elif self.tp == 'txt':
-            return "cos"
+        loop_description_txt = header + '\n'
 
-    def eq(self):
-        if self.tp == 'ltx':
-            return "~=~"
-        elif self.tp == 'txt':
-            return " = "
+        # add line under header
+        loop_description_txt += "-" * len(header) + '\n'
 
-    def tab(self):
-        if self.tp == 'ltx':
-            return 11*"~"
-        elif self.tp == 'txt':
-            return 7*" "
 
-    def p(self):
-        if self.tp == 'ltx':
-            return "~+~"
-        elif self.tp == 'txt':
-            return " + "
+        for i in range(cr.B.shape[0]):
+            el = None
+            for _, el_ind, b_id in cr.elem_keys[Inductor]:
+                if i == b_id:
+                    el = el_ind
+            for _, el_ind, b_id, _ in cr.elem_keys[Junction]:
+                if i == b_id:
+                    el = el_ind
 
-    def neg(self):
-        if self.tp == 'ltx':
-            return "~-~"
-        elif self.tp == 'txt':
-            return " - "
+            id_str = el.id_str
+            row = f'{id_str:{nr}}' + f'{{:{ns + len(", b1:")}}}'.format(', b' + str(i+1) + ':')
+            row += ('').join([f'{np.round(np.abs(cr.B[i, j]), 2):<{nh + 10}}'
+                              for j in range(nl)])
+            loop_description_txt += row + '\n'
 
-    @staticmethod
-    def ltx(txt):
-        return f"${txt}$"
+        print(loop_description_txt)
+        return loop_description_txt
 
     @staticmethod
-    def linear(method, w, st=True):
-
+    def print_el_description(elements, precision=3) -> str:
         txt = ''
-
-        for j in range(len(w)):
-
-            if w[j] == 1:
-                if np.sum(np.abs(w[:j])) == 0 and st is True:
-                    txt += method(j+1)
-                else:
-                    txt += '+' + method(j+1)
-            elif w[j] == -1:
-                txt += '-' + method(j+1)
-            elif w[j] == 0:
-                continue
-            else:
-                if w[j] > 0:
-                    if np.sum(np.abs(w[:j])) == 0 and st is True:
-                        txt += str(w[j]) + method(j+1)
-                    else:
-                        txt += '+' + str(w[j]) + method(j+1)
-                else:
-                    txt += '-' + str(-w[j]) + method(j + 1)
-
+        for edge, el_list in elements.items():
+            txt += f'Edge {edge}:\n'
+            for el in el_list:
+                txt += (f'\t{ELEMENT_NAMES[type(el)]} = '
+                        f'{el.get_value(el.value_unit):.{precision}e} '
+                        f'{el.value_unit}\n')
+
+        print(txt, end=None)
         return txt
 
-    def display(self, text):
-
+    def display(self, text) -> None:
         if self.tp == 'ltx':
             for line in text.split('\n'):
-                display(Latex(self.ltx(line)))
+                display(Math(line))
         elif self.tp == 'txt':
             print(text)
diff --git a/SQcircuit/torch_extensions.py b/SQcircuit/torch_extensions.py
new file mode 100755
index 0000000..4c918cd
--- /dev/null
+++ b/SQcircuit/torch_extensions.py
@@ -0,0 +1,1016 @@
+from typing import List, Optional, Tuple, Union
+
+import numpy as np
+import qutip as qt
+import torch
+from torch import Tensor
+from torch.autograd import Function
+from torch.autograd.function import once_differentiable
+
+from SQcircuit import Capacitor, Element, Inductor, Junction, Loop
+from SQcircuit.noise import ENV
+from SQcircuit import functions as sqf
+from SQcircuit import units as unt
+
+SupportedGradElements = Union[Capacitor, Inductor, Junction, Loop]
+EIGENVECTOR_MAX_GRAD = None
+
+###############################################################################
+# Interfaces to custom Torch functions
+###############################################################################
+
+
+def eigencircuit(circuit: 'Circuit', n_eig: int) -> Tensor:
+    """Given a circuit, computes the concatenated tensor including both 
+    eigenvalues and eigenvectors of a circuit using the ``EigenSolver`` class.
+
+    Parameters
+    ----------
+        circuit:
+            A circuit for which the eigensystem will be solved.
+        n_eig:
+            Number of eigenvalues to output. The lower `n_eig`, the
+                faster `SQcircuit` finds the eigenvalues.
+
+    Returns
+    ----------
+        A concatenated tensor of the eigenfrequencies and eigenvectors
+    """
+    return EigenSolver.apply(
+        torch.stack(circuit.parameters) if circuit.parameters else torch.tensor([]),
+        circuit,
+        n_eig,
+        EIGENVECTOR_MAX_GRAD
+    )
+
+
+def set_max_eigenvector_grad(n: Optional[int]) -> None:
+    """Sets the maximum number of eigenvectors to compute the gradient for when
+    calling ``diag``. Setting this to ``None`` will compute the gradient for
+    all eigenvectors. 
+    
+    Parameters
+    ----------
+        n:
+            The maximum number of eigenvectors to compute the gradient for, or
+            ``None``.
+    """
+    global EIGENVECTOR_MAX_GRAD
+    EIGENVECTOR_MAX_GRAD = n
+
+
+def dec_rate_cc_torch(circuit: 'Circuit', states: Tuple[int, int]) -> Tensor:
+    """Given a circuit, computes critical current dephasing rate using the
+    ``DecRateCC`` class.
+
+    Parameters
+    ----------
+        circuit:
+            A circuit to compute the dephasing rate of.
+        states:
+            A tuple of states to compute the dephasing between.
+
+    Returns
+    ----------
+        The critical current dephasing rate of ``circuit`` between ``states``.
+    """
+    return DecRateCC.apply(
+        torch.stack(circuit.parameters) if circuit.parameters else torch.tensor([]),
+        circuit,
+        states
+    )
+
+
+def dec_rate_charge_torch(
+        circuit: 'Circuit',
+        states: Tuple[int, int]
+) -> Tensor:
+    """Given a circuit, computes charge dephasing rate using the
+    ``DecRateCharge`` class.
+
+    Parameters
+    ----------
+        circuit:
+            A circuit to compute the dephasing rate of.
+        states:
+            A tuple of states to compute the dephasing between.
+
+    Returns
+    ----------
+        The charge dephasing rate of ``circuit`` between ``states``.
+    """
+    return DecRateCharge.apply(
+        torch.stack(circuit.parameters) if circuit.parameters else torch.tensor([]),
+        circuit,
+        states
+    )
+
+
+def dec_rate_flux_torch(circuit: 'Circuit', states: Tuple[int, int]) -> Tensor:
+    """Given a circuit, computes charge dephasing rate using the
+    ``DecRateFlux`` class.
+
+    Parameters
+    ----------
+        circuit:
+            A circuit to compute the dephasing rate of.
+        states:
+            A tuple of states to compute the dephasing between.
+
+    Returns
+    ----------
+        The flux rate of ``circuit`` between ``states``.
+    """
+    return DecRateFlux.apply(
+        torch.stack(circuit.parameters) if circuit.parameters else torch.tensor([]),
+        circuit,
+        states
+    )
+
+
+###############################################################################
+# Eigensolver
+###############################################################################
+
+class EigenSolver(Function):
+    """
+    Subclass of ``torch.Function`` which implements (once-differentiable)
+    forward and backwards pass for diagonalizing a ``SQcircuit.Circuit.``
+    object.
+    """
+
+    @staticmethod
+    def forward(
+        ctx,
+        element_tensors: Tensor,
+        circuit: 'Circuit',
+        n_eig: int,
+        eigenvector_max_grad: Optional[int]
+    ) -> Tensor:
+        """
+        Forward pass for diagonalizing a `Circuit` object.
+
+        Parameters
+        ----------
+            ctx:
+                The Torch context argument
+            element_tensors:
+                The elements of ``circuit`` which require gradient, as a tensor
+                given by ``circuit.parameters``. This argument is not strictly
+                used by the forward pass, but the gradient is computed for it,
+                it must be an input by convention.
+            circuit:
+                The ``SQcircuit`` circuit to diagonalize.
+            n_eig:
+                The number of eigenvalues/vectors to compute.
+            eigenvector_max_grad:
+                The maximum number of eigenvectors to compute the gradient
+                for.
+
+        Returns
+        ----------
+            A concatenated tensor of the eigenvalues and eigenvectors.
+        """
+
+        # Compute forward pass for eigenvalues/vectors
+        eigenvalues, eigenvectors = circuit._diag_np(n_eig=n_eig)
+
+        # Construct eigenvalues tensor
+        eigenvalues = [eigenvalue * 2 * np.pi * unt.get_unit_freq() for
+                       eigenvalue in eigenvalues]
+        eigenvalue_tensors = [torch.as_tensor(eigenvalue, dtype=torch.float64)
+                              for eigenvalue in eigenvalues]
+        eigenvalue_tensor = torch.stack(eigenvalue_tensors)
+        eigenvalue_tensor = torch.unsqueeze(eigenvalue_tensor, dim=-1)
+        # Construct eigenvectors tensor
+        eigenvector_tensors = [torch.as_tensor(eigenvector.full(),
+                                               dtype=torch.complex128)
+                                for eigenvector in eigenvectors]
+        eigenvector_tensor = torch.squeeze(torch.stack(eigenvector_tensors))
+
+        # Setup context -- needs to be done after diagonalization so that
+        # memory ops are filled
+        eigenvalues = torch.real(eigenvalue_tensor)
+        ctx.circuit = circuit.safecopy()
+
+        # Save eigenvalues, vectors into `ctx` circuit
+        ctx.circuit._efreqs = eigenvalues
+        ctx.circuit._evecs = eigenvector_tensor
+
+        # Number of eigenvalues
+        ctx.n_eig = n_eig
+        # Number of eigenvectors to use in computation of partial_omega
+        if eigenvector_max_grad is None:
+            ctx.eigenvector_max_grad = ctx.n_eig
+        else:
+            ctx.eigenvector_max_grad = eigenvector_max_grad
+        # Output shape
+        ctx.out_shape = element_tensors.shape
+
+        # Return concatenated Tensor
+        return torch.cat([eigenvalue_tensor, eigenvector_tensor], dim=-1)
+
+    @staticmethod
+    @once_differentiable
+    def backward(ctx, grad_output) -> Tuple[Tensor]:
+        """
+        Backward pass for diagonalizing a `Circuit` object.
+
+        Parameters
+        ----------
+            ctx:
+                The Torch context argument
+            grad_output:
+                The Torch grad output
+
+        Returns
+        ----------
+            The gradient for the ``element_tensors`` input for the forward pass,
+            and ``None`` for all other inputs.         
+        """
+
+        # Extract key parts of `ctx`
+        elements = ctx.circuit.parameters_elems
+        m, n, l = (
+            len(ctx.circuit.parameters), # number of parameters
+            ctx.n_eig,                   # number of eigenvalues
+            (grad_output.shape[1] - 1),  # length of eigenvectors
+        )
+        # Break grad_output into eigenvalue sub-tensor and eigenvector
+        # sub-tensor
+        grad_output_eigenvalue = grad_output[:, 0]
+        grad_output_eigenvector = grad_output[:, 1:]
+
+        partial_omega = torch.zeros([m, n], dtype=torch.float64)
+        partial_eigenvec = torch.zeros([m, n, l], dtype=torch.complex128)
+        for el_idx in range(m):
+            for eigen_idx in range(n):
+                # Compute backward pass for eigenvalues
+                partial_omega[el_idx, eigen_idx] = ctx.circuit.get_partial_omega(
+                        el=elements[el_idx],
+                        m=eigen_idx,
+                        subtract_ground=False
+                )
+                # Compute backwards pass for only _some_ of the eigenvectors.
+                # This computation is expensive, and the gradient of higher
+                # eigenvectors is infrequently used. (However, _computing_)
+                # many eigenvectors is necessary for accurate first-order
+                # gradients of _any_ eigenvector.)
+                if eigen_idx < ctx.eigenvector_max_grad:
+                    partial_tensor = torch.squeeze(torch.as_tensor(
+                        ctx.circuit.get_partial_vec(
+                            el=elements[el_idx],
+                            m=eigen_idx
+                        ),
+                        dtype=torch.complex128
+                    ))
+                    partial_eigenvec[el_idx, eigen_idx, :] = partial_tensor
+        eigenvalue_grad = torch.sum(
+            partial_omega * torch.conj(grad_output_eigenvalue), axis=-1)
+        eigenvector_grad = torch.sum(
+            partial_eigenvec * torch.conj(grad_output_eigenvector),
+            axis=(-1, -2))
+
+        return (
+            torch.real(eigenvalue_grad + eigenvector_grad).view(ctx.out_shape),
+            None,
+            None,
+            None
+        )
+
+###############################################################################
+# Decoherence rate helper functions
+###############################################################################
+
+
+def partial_squared_omega(
+    cr: 'Circuit',
+    grad_el: SupportedGradElements,
+    partial_H: qt.Qobj,
+    partial_H_squared: qt.Qobj,
+    states: Tuple[int, int]
+) -> float:
+    """ Calculates the second derivative of the difference between the ``m``th 
+    and ``n``th eigenfrequencies with respect to an arbitrary parameter ``x``
+    and ``grad_el``.
+
+    Parameters
+    ----------
+        cr:
+            The ``Circuit`` object to differentiate the eigenfrequencies of
+        grad_el:
+            The element to take the gradient of the frequencies with respect to
+        partial_H:
+            The derivative of the circuit Hamiltonian with respect to the
+            arbitrary parameter.
+        partial_H_squared:
+            The second derivative of the circuit Hamiltonian with respect to 
+            the arbitrary parameter and ``grad_el``.
+        states:
+            The indices ``(m, n)`` of the eigenfrequencies to differentiate.
+    
+    Returns
+    ----------
+        The second derivative of the eigenfrequency difference.         
+    """
+
+    m, n = states
+
+    state_m = cr.evecs[m]
+    partial_state_m = cr.get_partial_vec(grad_el, m)
+    state_n = cr.evecs[n]
+    partial_state_n = cr.get_partial_vec(grad_el, n)
+
+    # Compute the first term of the second-order derivative, from
+    # differentiating the state
+    if partial_H == 0:
+        # operator_inner_product behaves badly if passed a float
+        p2_omega_1 = 0
+    else:
+        p2_omega_1 = 2 * np.real(
+            sqf.operator_inner_product(partial_state_m, partial_H, state_m)
+            - sqf.operator_inner_product(partial_state_n, partial_H, state_n)
+        )
+
+    # Compute the second term of the second-order derivative, from
+    # differentating partial_H
+    if partial_H_squared == 0:
+        p2_omega_2 = 0
+    else:
+        p2_omega_2 = (
+            sqf.operator_inner_product(state_m, partial_H_squared, state_m)
+            - sqf.operator_inner_product(state_n, partial_H_squared, state_n)
+        )
+
+    # Return sum
+    p2_omega = p2_omega_1 + p2_omega_2
+    # The eigenfrequencies (and hence derivatives) should be real since H
+    # is Hermitian, but numerical imprecision can result in small complex
+    # components.
+    return sqf.to_numpy(p2_omega.real)
+
+
+def partial_dephasing_rate(
+    A: float,
+    partial_A: float,
+    partial_omega_mn: float,
+    partial_squared_omega_mn: float
+) -> float:
+    """Calculate the derivative of the dephasing rate with noise amplitude
+    ``A`` that depends on ``partial_omega_mn``.
+
+    Parameters
+    ----------
+        A:
+            The noise amplitude of the dephasing
+        partial_A:
+            The derivative of ``A`` with respect to an external parameter
+        partial_omega_mn:
+            The derivative of the difference between eigenfrequencies 
+            used to calculate the dephasing rate
+        partial_squared_omega_mn:
+            The derivative of ``partial_omega_mn`` with respect to an external
+            parameter
+
+    Returns
+    ----------
+        The derivative of dephasing rate based on the input parameters.
+    """
+    return (
+        np.sign(partial_omega_mn)
+        * np.sqrt(2 * np.abs(np.log(ENV["omega_low"] * ENV["t_exp"])))
+        * (partial_A * partial_omega_mn + A * partial_squared_omega_mn)
+    )
+
+
+def get_B_indices(
+    cr: 'Circuit',
+    el: Union[Junction, Inductor]
+) -> List[int]:
+    """
+    Return the list of ``b_id``'s with the element ``el`` (the same element
+    could be placed at multiple branches).
+
+    Parameters
+    ----------
+        cr:
+            The ``Circuit`` object.
+        el:
+            The element to find in ``cr``.
+
+    Returns
+    ----------
+        A list of rows of the ``B`` matrix the element is associated with.
+    """
+    B_indices = []
+    if isinstance(el, Junction):
+        for _, el_JJ, b_id, _ in cr.elem_keys[Junction]:
+            if el_JJ is el:
+                B_indices.append(b_id)
+    elif isinstance(el, Inductor):
+        for _, el_ind, b_id in cr.elem_keys[Inductor]:
+            if el_ind is el:
+                B_indices.append(b_id)
+
+    return B_indices
+
+
+###############################################################################
+# Charge noise
+###############################################################################
+
+def partial_H_ng(
+    cr: 'Circuit',
+    charge_idx: int
+) -> qt.Qobj:
+    """Calculates the derivative of the Hamiltonian of ``cr`` with
+    respect to the gate charge on the ``charge_idx`` charge mode.
+
+    Parameters
+    ----------
+        cr:
+            The ``Circuit`` object to differentiate the Hamiltonian of
+        charge_idx:
+            The charge mode whose gate charge to differentiate with respect to
+
+    Returns
+    ----------
+        The partial derivative of ``cr``'s Hamiltonian with respect to gate
+        charge.
+    """
+    if not cr._is_charge_mode(charge_idx):
+        raise ValueError('The mode index passed is not a charge mode!')
+
+    op = 0
+    for j in range(cr.n):
+        op += (
+            cr.cInvTrans[charge_idx, j]
+            * cr._memory_ops["Q"][j]
+            / np.sqrt(unt.hbar)
+        )
+    return op
+
+
+def partial_squared_H_ng(
+    cr: 'Circuit',
+    charge_idx: int,
+    grad_el: SupportedGradElements
+) -> qt.Qobj:
+    """Calculates the second derivative of the Hamiltonian of ``cr`` with 
+    respect to the gate charge on the ``charge_idx`` charge mode and
+    ``grad_el``.
+
+    Parameters
+    ----------
+        cr:
+            The ``Circuit`` object to differentiate the Hamiltonian of
+        charge_idx:
+            The charge mode whose gate charge to differentiate with respect to
+        grad_el:
+            The circuit element to differentiate with respect to
+
+    Returns
+    ----------
+        The second derivative of the Hamiltonian with respect to gate charge and
+        ``grad_el``.
+    """
+    if not cr._is_charge_mode(charge_idx):
+        raise ValueError('The mode index passed is not a charge mode!')
+
+    # The charge operators only multiply the capacitance matrix
+    if not isinstance(grad_el, Capacitor):
+        return 0
+
+    cInv = np.linalg.inv(sqf.to_numpy(cr.C))
+    A = cInv @ cr.partial_mats[grad_el] @ cInv
+
+    op = 0
+    for j in range(cr.n):
+        op += A[charge_idx, j] * cr._memory_ops["Q"][j] / np.sqrt(unt.hbar)
+    return -op
+
+
+def partial_omega_ng(
+    cr: 'Circuit',
+    charge_idx: int,
+    states: Tuple[int, int]
+) -> float:
+    """Calculates the derivative of the difference between the ``m``th
+    and ``n``th eigenfrequencies of ``cr`` with respect to the gate charge of
+    the ``charge_idx`` charge mode.
+
+    Parameters
+    ----------
+        cr:
+            The `Circuit`` object to differentiate the eigenfrequencies of
+        charge_idx:
+            The charge mode whose gate charge to differentiate with respect to
+        states:
+            The numbers ``(m, n)`` of the eigenfrequencies to differentiate.
+
+    Returns
+    ----------
+        The first derivative of the eigenfrequency difference with respect to
+        gate charge.   
+    """
+    if not cr._is_charge_mode(charge_idx):
+        raise ValueError('The mode index passed is not a charge mode!')
+
+    state_m = cr.evecs[states[0]]
+    state_n = cr.evecs[states[1]]
+    op = partial_H_ng(cr, charge_idx)
+
+    partial_omega_mn = (
+        sqf.operator_inner_product(state_m, op, state_m)
+        - sqf.operator_inner_product(state_n, op, state_n)
+    )
+
+    # The eigenfrequencies (and hence derivatives) should be real since H
+    # is Hermitian, but numerical imprecision can result in small complex
+    # components.
+    return sqf.to_numpy(partial_omega_mn.real)
+
+
+def partial_squared_omega_mn_ng(
+    cr: 'Circuit',
+    charge_idx: int,
+    grad_el: SupportedGradElements,
+    states: Tuple[int, int]
+) -> float:
+    """Calculates the second derivative of the difference between the ``m``th 
+    and ``n``th eigenfrequencies with respect to the gate charge on the
+    ```charge_idx`` mode and ``grad_el``.
+
+    Parameters
+    ----------
+        cr:
+            The ``Circuit`` object to differentiate the eigenfrequencies of
+        charge_idx:
+            The charge mode whose gate charge to differentiate with respect to
+        grad_el:
+            The circuit element to differentiate with respect to
+        states:
+            The numbers ``(m, n)`` of the eigenfrequencies to differentiate.
+
+    Returns
+    ----------
+        The second derivative of the eigenfrequency difference with respect to
+        gate charge and ``grad_el``.  
+    """
+    if not cr._is_charge_mode(charge_idx):
+        raise ValueError('The mode index passed is not a charge mode!')
+
+    partial_H = partial_H_ng(cr, charge_idx)
+    partial_H_squared = partial_squared_H_ng(cr, charge_idx, grad_el)
+
+    return partial_squared_omega(
+        cr,
+        grad_el,
+        partial_H,
+        partial_H_squared,
+        states
+    )
+
+
+def partial_charge_dec(
+    cr: 'Circuit',
+    grad_el: SupportedGradElements,
+    states: Tuple[int, int]
+) -> float:
+    """
+    Calculate the derivative of the charge dephasing rate between the states
+    given in ``states`` with respect to ``grad_el``.
+
+    Parameters
+    ----------
+        cr:
+            The ``Circuit`` object.
+        grad_el:
+            The circuit element to differentiate with respect to
+        states:
+            A tuple ``(m, n)`` of states to consider charge decoherence between.
+    
+    Returns
+    ----------
+        The first partial derivative of the charge dephasing rate.
+    """
+    dec_rate_grad = 0
+    for i in range(cr.n):
+        if cr._is_charge_mode(i):
+            partial_omega_mn = partial_omega_ng(cr, i, states)
+            partial_squared_omega_mn = partial_squared_omega_mn_ng(
+                cr=cr,
+                charge_idx=i,
+                grad_el=grad_el,
+                states=states
+            )
+            A = cr.charge_islands[i].A * 2 * unt.e
+            partial_A = 0
+
+            dec_rate_grad += partial_dephasing_rate(
+                A,
+                partial_A,
+                partial_omega_mn,
+                partial_squared_omega_mn
+            )
+
+    return dec_rate_grad
+
+
+class DecRateCharge(Function):
+    """
+    Torch ``Function`` wrapper to compute forward and backwards pass for the
+    charge dephasing rate. 
+
+    The forward pass is computed using an internal method of the ``Circuit``
+    class, and the gradient is computed using helper functions in the
+    ``torch_extensions.py`` module.
+    """
+    @staticmethod
+    def forward(
+        circuit_parameters: Tensor,
+        circuit: 'Circuit',
+        states: Tuple[int, int]
+    ) -> Tensor:
+        return torch.as_tensor(circuit._dec_rate_charge_np(states),
+                               dtype=torch.float64)
+
+    @staticmethod
+    def setup_context(ctx, inputs, output):
+        circuit_parameters, circuit, states = inputs
+
+        ctx.circuit = circuit.safecopy(save_eigs=True)
+        ctx.states = states
+
+    @staticmethod
+    @once_differentiable
+    def backward(ctx, grad_output) -> Tuple[Tensor, None, None]:
+        output_grad = torch.zeros(len(ctx.circuit._parameters))
+
+        for idx, elem in enumerate(ctx.circuit._parameters.keys()):
+            output_grad[idx] = grad_output * partial_charge_dec(
+                ctx.circuit,
+                elem,
+                ctx.states
+            )
+
+        return output_grad, None, None
+
+
+###############################################################################
+# Critical current noise
+###############################################################################
+
+def partial_squared_H_EJ(
+    cr: 'Circuit',
+    EJ_el: Junction,
+    b_id: int,
+    grad_el: SupportedGradElements
+) -> qt.Qobj:
+    """ Calculates the second derivative of the Hamiltonian of ``cr`` with 
+    respect to the Josephson energy of ``EJ_el`` charge mode and ``grad_el``.
+
+    Parameters
+    ----------
+        cr:
+            The ``Circuit`` object to differentiate the Hamiltonian of.
+        EJ_el:
+            The Josephson junction to differentiate with respect to.
+        b_id:
+            The index of the ``cr.B`` matrix identifying which branch the 
+            ``EJ_el`` is on.
+        grad_el:
+            The circuit element to differentiate with respect to.
+
+    Returns
+    ----------
+        The second deritative of the Hamiltonain with respec to ``EJ_el`` and
+        ``grad_el``.
+    """
+    if not isinstance(grad_el, Loop):
+        return 0
+
+    loop_idx = cr.loops.index(grad_el)
+    return cr.B[b_id, loop_idx] * cr._memory_ops['sin'][(EJ_el, b_id)]
+
+
+def partial_squared_omega_mn_EJ(
+    cr: 'Circuit',
+    EJ_el: Junction,
+    b_id: int,
+    grad_el: SupportedGradElements,
+    states: Tuple[int, int]
+) -> float:
+    """ Calculates the second derivative of the difference between the ``m``th 
+    and ``n``th eigenfrequencies with respect to ``EJ_el`` and ``grad_el``.
+
+    Parameters
+    ----------
+        cr:
+            The ``Circuit`` object to differentiate the eigenfrequencies of.
+        EJ_el:
+            A Josephson junction to differentiate with respect to.
+        b_id:
+            A number
+        grad_el:
+            A circuit element to differentiate with respect to.
+        states:
+            The numbers ``(m, n)`` of the eigenfrequencies to differentiate.
+
+    Returns
+    ----------
+        The second derivative of the eigenfrequency difference with respect to
+        ``EJ_el`` and ``grad_el``.
+    """
+    partial_H = cr._get_partial_H(EJ_el, _b_id = b_id)
+    partial_H_squared = partial_squared_H_EJ(cr, EJ_el, b_id, grad_el)
+
+    return partial_squared_omega(
+        cr,
+        grad_el,
+        partial_H,
+        partial_H_squared,
+        states
+    )
+
+
+def partial_cc_dec(
+    cr: 'Circuit',
+    grad_el: SupportedGradElements,
+    states: Tuple[int, int]
+) -> float:
+    """Calculates the derivative of the critical current dephasing rate between
+    the states given  in ``states`` with respect to ``grad_el``.
+
+    Parameters
+    ----------
+        cr:
+            The ``Circuit`` object.
+        states:
+            A tuple ``(m, n)`` of states to consider charge decoherence between.
+        grad_el:
+            The circuit element to differentiate with respect to.
+    
+    Returns
+    ----------
+        The derivative of the critical current dephasing rate between ``states``
+        with respect to ``grad_el``.
+    """
+
+    dec_rate_grad = 0
+    # Sum over all cosine operators because each Josephson junction is
+    # associated with exactly one.
+    for EJ_el, b_id in cr._memory_ops['cos']:
+        partial_omega_mn = sqf.to_numpy(cr._get_partial_omega_mn(
+            EJ_el,
+            states=states,
+            _b_id=b_id
+        ))
+        partial_squared_omega_mn = partial_squared_omega_mn_EJ(
+            cr,
+            EJ_el,
+            b_id,
+            grad_el,
+            states
+        )
+        A = sqf.to_numpy(EJ_el.A * EJ_el.get_value())
+        partial_A = EJ_el.A if grad_el is EJ_el else 0
+
+        dec_rate_grad += partial_dephasing_rate(
+            A,
+            partial_A,
+            partial_omega_mn,
+            partial_squared_omega_mn
+        )
+
+    return dec_rate_grad
+
+
+class DecRateCC(Function):
+    """
+    Torch ``Function`` wrapper to compute forward and backwards pass for the
+    critical current dephasing rate. 
+
+    The forward pass is computed using an internal method of the ``Circuit``
+    class, and the gradient is computed using helper functions in the
+    ``torch_extensions.py`` module.
+    """
+    @staticmethod
+    def forward(
+        element_tensors: Tensor,
+        circuit: 'Circuit',
+        states: Tuple[int, int]
+    ) -> Tensor:
+
+        return torch.as_tensor(circuit._dec_rate_cc_np(states),
+                               dtype=torch.float64)
+
+    @staticmethod
+    def setup_context(ctx, inputs, output):
+        circuit_parameters, circuit, states = inputs
+
+        ctx.circuit = circuit.safecopy(save_eigs=True)
+        ctx.states = states
+
+    @staticmethod
+    @once_differentiable
+    def backward(ctx, grad_output) -> Tuple[Tensor, None, None]:
+        output_grad = torch.zeros(len(ctx.circuit._parameters))
+
+        for idx, elem in enumerate(ctx.circuit._parameters.keys()):
+            output_grad[idx] = grad_output * partial_cc_dec(
+                ctx.circuit,
+                elem,
+                ctx.states
+            )
+
+        return output_grad, None, None
+
+
+###############################################################################
+# Flux noise
+###############################################################################
+
+def partial_squared_H_phi(
+    cr: 'Circuit',
+    loop: Loop,
+    grad_el: SupportedGradElements
+) -> qt.Qobj:
+    """Calculates the second derivative of the Hamiltonian of ``cr`` with 
+    respect to the external flux through ``loop`` and ``grad_el``.
+
+    Parameters
+    ----------
+        cr:
+            The ``Circuit`` object to differentiate the Hamiltonian of.
+        loop:
+            The loop with external flux to differentiate with respect to.
+        grad_el:
+            The circuit element to differentiate with respect to.
+    
+    Returns
+    ----------
+        The second derivative of the Hamiltonian with respect to external flux
+        and ``grad_el``.
+    """
+    # Only inductors and junctions (and loops) are associated with external
+    # flux.
+    if isinstance(grad_el, Capacitor):
+        return 0
+
+    loop_idx = cr.loops.index(loop)
+    B_indices = get_B_indices(cr, grad_el)
+
+    H_squared = 0
+    if isinstance(grad_el, Junction):
+        for b_id in B_indices:
+            H_squared += (
+                    cr.B[b_id, loop_idx]
+                    * cr._memory_ops['sin'][(grad_el, b_id)]
+            )
+        return H_squared
+    elif isinstance(grad_el, Inductor):
+        for b_id in B_indices:
+            H_squared += (
+                cr.B[b_id, loop_idx]
+                / -sqf.to_numpy(grad_el.get_value()**2)
+                * unt.Phi0 / np.sqrt(unt.hbar) / 2 / np.pi
+                * cr._memory_ops["ind_hamil"][(grad_el, b_id)]
+            )
+        return H_squared
+    elif isinstance(grad_el, Loop):
+        loop_idx_1 = cr.loops.index(loop)
+        loop_idx_2 = cr.loops.index(grad_el)
+        for edge, el_JJ, b_id, W_idx in cr.elem_keys[Junction]:
+            H_squared += (
+                sqf.to_numpy(el_JJ.get_value())
+                * cr.B[b_id, loop_idx_2]
+                * cr.B[b_id, loop_idx_1]
+                * cr._memory_ops['cos'][(el_JJ, b_id)]
+            )
+        return H_squared
+    else:
+        raise NotImplementedError
+
+
+def partial_squared_omega_mn_phi(
+    cr: 'Circuit',
+    loop: Loop,
+    grad_el: SupportedGradElements,
+    states: Tuple[int, int]
+) -> float:
+    """Calculate the second derivative of the difference between the ``m``th 
+    and `n`th eigenfrequencies with respect to ``loop`` and ``grad_el``.
+
+    Parameters
+    ----------
+        cr:
+            The ``Circuit`` object to differentiate the eigenfrequencies of.
+        loop:
+            A loop in ``cr`` to differentiate with respect to.
+        grad_el:
+            A circuit element to differentiate with respect to.
+        states:
+            The numbers ``(m, n)`` of the eigenfrequencies to differentiate.
+
+    Returns
+    ----------
+        The gradient of the eigenfrequency difference with respect to
+        ``grad_el`` and the flux through ``loop``.
+    """
+    partial_H = cr._get_partial_H(loop)
+    partial_H_squared = partial_squared_H_phi(cr, loop, grad_el)
+
+    return partial_squared_omega(
+        cr,
+        grad_el,
+        partial_H,
+        partial_H_squared,
+        states
+    )
+
+
+def partial_flux_dec(
+    cr: 'Circuit',
+    grad_el: SupportedGradElements,
+    states: Tuple[int, int]
+) -> float:
+    """Calculate the derivative of the flux dephasing rate between the states 
+    given in `states` with respect to `grad_el`.
+
+    Parameters
+    ----------
+        cr:
+            The ``Circuit`` object.
+        states:
+            A tuple ``(m, n)`` of states to consider charge decoherence between.
+        grad_el:
+            The circuit element to differentiate with respect to.
+
+    Returns
+    ----------
+        The derivative of the flux dephasing rate with respect to ``grad_el``.
+    """
+    dec_rate_grad = 0
+    for loop in cr.loops:
+        partial_omega_mn = sqf.to_numpy(cr._get_partial_omega_mn(
+            loop,
+            states=states
+        ))
+        partial_squared_omega_mn = partial_squared_omega_mn_phi(
+            cr,
+            loop,
+            grad_el,
+            states
+        )
+
+        A = loop.A
+        partial_A = 0
+
+        dec_rate_grad += partial_dephasing_rate(
+            A,
+            partial_A,
+            partial_omega_mn,
+            partial_squared_omega_mn
+        )
+    return dec_rate_grad
+
+
+class DecRateFlux(Function):
+    """
+    Torch ``Function`` wrapper to compute forward and backwards pass for the
+    flux dephasing rate. 
+
+    The forward pass is computed using an internal method of the ``Circuit``
+    class, and the gradient is computed using helper functions in the
+    ``torch_extensions.py`` module.
+    """
+    @staticmethod
+    def forward(
+        circuit_parameters: Tensor,
+        circuit: 'Circuit',
+        states: Tuple[int, int]
+    ) -> Tensor:
+        return torch.as_tensor(circuit._dec_rate_flux_np(states),
+                               dtype=torch.float64)
+
+    @staticmethod
+    def setup_context(ctx, inputs, output):
+        circuit_parameters, circuit, states = inputs
+
+        ctx.circuit = circuit.safecopy(save_eigs=True)
+        ctx.states = states
+
+    @staticmethod
+    @once_differentiable
+    def backward(ctx, grad_output) -> Tuple[Tensor, None, None]:
+        output_grad = torch.zeros(len(ctx.circuit._parameters))
+
+        for idx, elem in enumerate(ctx.circuit._parameters.keys()):
+            output_grad[idx] = grad_output * partial_flux_dec(
+                ctx.circuit,
+                elem,
+                ctx.states
+            )
+
+        return output_grad, None, None
diff --git a/SQcircuit/units.py b/SQcircuit/units.py
old mode 100644
new mode 100755
index d8d59f7..eabe49a
--- a/SQcircuit/units.py
+++ b/SQcircuit/units.py
@@ -1,5 +1,5 @@
 """units.py module contain the physical constants and units"""
-
+from SQcircuit.exceptions import raise_unit_error
 
 henry_list = {'H': 1.0, 'mH': 1.0e-3, 'uH': 1.0e-6,
               'nH': 1.0e-9, 'pH': 1.0e-12, 'fH': 1.0e-15}
@@ -27,28 +27,28 @@
 k_B = 1.38e-23
 
 # main frequency unit of the SQcircuit
-_unit_freq = freq_list["GHz"]
+_unit_freq = freq_list['GHz']
 
 # default unit of capacitors
-_unit_cap = "GHz"
+_unit_cap = 'GHz'
 
 # default unit of inductors
-_unit_ind = "GHz"
+_unit_ind = 'GHz'
 
 # default unit of JJs
-_unit_JJ = "GHz"
+_unit_JJ = 'GHz'
 
 
 def set_unit_freq(unit: str) -> None:
-    """
-    Change the main frequency unit of the SQcircuit.
+    """Change the main frequency unit of the SQcircuit.
 
     Parameters
     ----------
         unit:
             The desired frequency unit, which can be "THz", "GHz", and ,etc.
     """
-    assert unit in freq_list, "The input format is not correct."
+    if unit not in freq_list:
+        raise_unit_error()
 
     global _unit_freq
 
@@ -56,8 +56,7 @@ def set_unit_freq(unit: str) -> None:
 
 
 def get_unit_freq() -> float:
-    """
-    get current frequency unit of the SQcircuit.
+    """Get current frequency unit of the SQcircuit.
 
     Returns
     ----------
@@ -69,8 +68,7 @@ def get_unit_freq() -> float:
 
 
 def set_unit_cap(unit: str) -> None:
-    """
-    Change the default unit for capacitors
+    """Change the default unit for capacitors
 
     Parameters
     ----------
@@ -79,9 +77,7 @@ def set_unit_cap(unit: str) -> None:
             etc., or "fF", "pF", and ,etc.
     """
     if unit not in freq_list and unit not in farad_list:
-        error = ("The input unit is not correct. Look at the documentation "
-                 "for the correct input format.")
-        raise ValueError(error)
+        raise_unit_error()
 
     global _unit_cap
 
@@ -89,8 +85,7 @@ def set_unit_cap(unit: str) -> None:
 
 
 def get_unit_cap() -> str:
-    """
-    Get current unit of capacitor
+    """Get current unit of capacitor
 
     Returns
     ----------
@@ -102,8 +97,7 @@ def get_unit_cap() -> str:
 
 
 def set_unit_ind(unit: str) -> None:
-    """
-    Change the default unit for inductors
+    """Change the default unit for inductors
 
     Parameters
     ----------
@@ -112,9 +106,7 @@ def set_unit_ind(unit: str) -> None:
              etc., or "fH", "pH", and ,etc.
     """
     if unit not in freq_list and unit not in henry_list:
-        error = ("The input unit is not correct. Look at the documentation "
-                 "for the correct input format.")
-        raise ValueError(error)
+        raise_unit_error()
 
     global _unit_ind
 
@@ -122,8 +114,7 @@ def set_unit_ind(unit: str) -> None:
 
 
 def get_unit_ind() -> str:
-    """
-    Get current unit of inductor
+    """Get current unit of inductor
 
     Returns
     ----------
@@ -135,8 +126,7 @@ def get_unit_ind() -> str:
 
 
 def set_unit_JJ(unit: str) -> None:
-    """
-    Change the default unit for Josephson junctions.
+    """Change the default unit for Josephson junctions.
 
     Parameters
     ----------
@@ -144,7 +134,8 @@ def set_unit_JJ(unit: str) -> None:
             The desired Josephson junction default unit, which can be "THz",
             "GHz", and ,etc.
     """
-    assert unit in freq_list, "The input format is not correct."
+    if unit not in freq_list:
+        raise_unit_error()
 
     global _unit_JJ
 
@@ -152,8 +143,7 @@ def set_unit_JJ(unit: str) -> None:
 
 
 def get_unit_JJ() -> str:
-    """
-    Get current unit of Josephson junction unit
+    """Get current unit of Josephson junction unit
 
     Returns
     ----------
diff --git a/pics/README_logo.png b/pics/README_logo.png
deleted file mode 100644
index 7745cf8..0000000
Binary files a/pics/README_logo.png and /dev/null differ
diff --git a/pics/README_zeroPi.png b/pics/README_zeroPi.png
deleted file mode 100644
index bf35470..0000000
Binary files a/pics/README_zeroPi.png and /dev/null differ
diff --git a/pics/dark_logo_sq.png b/pics/dark_logo_sq.png
new file mode 100644
index 0000000..94d499e
Binary files /dev/null and b/pics/dark_logo_sq.png differ
diff --git a/pics/light_logo_sq.png b/pics/light_logo_sq.png
new file mode 100644
index 0000000..4622b55
Binary files /dev/null and b/pics/light_logo_sq.png differ
diff --git a/requirements.txt b/requirements.txt
old mode 100644
new mode 100755
index 226593c..2ddb27f
--- a/requirements.txt
+++ b/requirements.txt
@@ -1,6 +1,9 @@
 numpy>=1.14.2
 scipy>=1.1.0
 matplotlib>=3.0.0
-qutip>=4.3.1
+qutip>=5.0.0
 dill>=0.3.4
-IPython>=7.0.0
\ No newline at end of file
+IPython>=7.0.0
+torch>=2.0
+mpmath>=1.2.1
+sympy>=1.13.0
\ No newline at end of file
diff --git a/setup.py b/setup.py
old mode 100644
new mode 100755
index 276df0d..24df143
--- a/setup.py
+++ b/setup.py
@@ -24,9 +24,9 @@
 """
 
 # version of the SQcircuit
-MAJOR = 0
+MAJOR = 1
 MINOR = 0
-PATCH = 17
+PATCH = 0
 
 VERSION = "%d.%d.%d" % (MAJOR, MINOR, PATCH)
 


Test patch:
diff --git a/.github/workflows/pytest.yml b/.github/workflows/pytest.yml
old mode 100644
new mode 100755
index 717ba55..4087c68
--- a/.github/workflows/pytest.yml
+++ b/.github/workflows/pytest.yml
@@ -10,9 +10,7 @@ jobs:
       matrix:
         python-version: [3.9]
     steps:
-    - uses: actions/checkout@v1
-      with:
-        ref: main
+    - uses: actions/checkout@v4
     - name: Set up Python ${{ matrix.python-version }}
       uses: actions/setup-python@v1
       with:
@@ -38,4 +36,4 @@ jobs:
         name: codecov-umbrella
         fail_ci_if_error: true
         path_to_write_report: ./codecov_report.txt
-        verbose: true
\ No newline at end of file
+        verbose: true
diff --git a/SQcircuit/tests/__init__.py b/SQcircuit/tests/__init__.py
old mode 100644
new mode 100755
diff --git a/SQcircuit/tests/conftest.py b/SQcircuit/tests/conftest.py
old mode 100644
new mode 100755
index d93f0ce..bc0408d
--- a/SQcircuit/tests/conftest.py
+++ b/SQcircuit/tests/conftest.py
@@ -3,11 +3,11 @@
 """
 import os
 
-import numpy as np
-
-from SQcircuit.sweep import *
-from SQcircuit.storage import SQdata
 from SQcircuit.circuit import Circuit
+from SQcircuit.storage import SQdata
+from SQcircuit.sweep import *
+from SQcircuit.settings import set_optim_mode
+
 
 TESTDIR = os.path.dirname(os.path.abspath(__file__))
 DATADIR = os.path.join(TESTDIR, "data/qubits")
@@ -24,6 +24,7 @@ def setup_class(cls):
 
     def test_transform_process(self):
         # load the data
+        set_optim_mode(False)
         data = SQdata.load(DATADIR + "/" + self.fileName)
 
         # build the new circuit based on data circuit parameters
@@ -77,3 +78,138 @@ def test_data(self):
             for decType in data.dec.keys():
                 assert np.allclose(dec[decType], data.dec[decType]),\
                     "The \"{}\" loss has issue".format(decType)
+                
+
+def create_transmon_numpy(trunc_num):
+    cap_value, ind_value, Q = 7.746, 5, 1e6
+    cap_unit, ind_unit = 'fF', 'GHz'
+
+    set_optim_mode(False)
+    C_numpy = Capacitor(cap_value, cap_unit, Q=Q)
+    J_numpy = Junction(ind_value, ind_unit)
+    circuit_numpy = Circuit({(0, 1): [C_numpy, J_numpy], })
+    circuit_numpy.set_trunc_nums([trunc_num, ])
+    return circuit_numpy
+
+
+def create_transmon_torch(trunc_num):
+    cap_value, ind_value, Q = 7.746, 5, 1e6
+    cap_unit, ind_unit = 'fF', 'GHz'
+
+    set_optim_mode(True)
+    C_torch = Capacitor(cap_value, cap_unit, Q=Q, requires_grad=True)
+    J_torch = Junction(ind_value, ind_unit, requires_grad=True)
+    circuit_torch = Circuit({(0, 1): [C_torch, J_torch]})
+    circuit_torch.set_trunc_nums([trunc_num, ])
+    return circuit_torch
+
+
+def create_flux_transmon_numpy(trunc_num, phi_ext):
+    cap_value, ind_value, Q = 7.746, 5, 1e6
+    cap_unit, ind_unit = 'fF', 'GHz'
+    disorder = 1.1
+
+    set_optim_mode(False)
+    loop = Loop(phi_ext)
+    C_numpy = Capacitor(cap_value, cap_unit, Q=Q)
+    J1_numpy = Junction(ind_value, ind_unit, loops=[loop])
+    J2_numpy = Junction(ind_value * disorder, ind_unit, loops=[loop])
+    circuit_numpy = Circuit({(0, 1): [C_numpy, J1_numpy, J2_numpy], })
+    circuit_numpy.set_trunc_nums([trunc_num, ])
+    return circuit_numpy
+
+
+def create_flux_transmon_torch(trunc_num, phi_ext):
+    cap_value, ind_value, Q = 7.746, 5, 1e6
+    cap_unit, ind_unit = 'fF', 'GHz'
+    disorder = 1.1
+
+    set_optim_mode(True)
+    loop_torch = Loop(phi_ext, requires_grad=True)
+    C_torch = Capacitor(cap_value, cap_unit, Q=Q, requires_grad=True)
+    J1_torch = Junction(ind_value, ind_unit, loops=[loop_torch], requires_grad=True)
+    J2_torch = Junction(ind_value * disorder, ind_unit, loops=[loop_torch], requires_grad=True)
+    circuit_torch = Circuit({(0, 1): [C_torch, J1_torch, J2_torch], })
+    circuit_torch.set_trunc_nums([trunc_num, ])
+    return circuit_torch
+
+
+def create_fluxonium_numpy(trunc_num, phi_ext=0):
+    set_optim_mode(False)
+    loop = Loop(phi_ext)
+    C_numpy = Capacitor(3.6, 'GHz', Q=1e6, requires_grad=False)
+    L_numpy = Inductor(0.46, 'GHz', Q=500e6, loops=[loop], requires_grad=False)
+    JJ_numpy = Junction(10.2, 'GHz', A=1e-7, x=3e-06, loops=[loop], requires_grad=False)
+    circuit_numpy = Circuit({(0, 1): [C_numpy, L_numpy, JJ_numpy], }, flux_dist='junctions')
+    circuit_numpy.set_trunc_nums([trunc_num, ])
+    return circuit_numpy
+
+
+def create_fluxonium_torch(trunc_num, phi_ext=0):
+    set_optim_mode(True)
+    loop = Loop(phi_ext)
+    C_torch = Capacitor(3.6, 'GHz', Q=1e6, requires_grad=True)
+    L_torch = Inductor(0.46, 'GHz', Q=500e6, loops=[loop], requires_grad=True)
+    JJ_torch = Junction(10.2, 'GHz', A=1e-7, x=3e-06, loops=[loop], requires_grad=True)
+    circuit_torch = Circuit({(0, 1): [C_torch, L_torch, JJ_torch], }, flux_dist='junctions')
+    circuit_torch.set_trunc_nums([trunc_num, ])
+    return circuit_torch
+
+
+def create_fluxonium_torch_flux(trunc_num, phi_ext=0):
+    set_optim_mode(True)
+    loop = Loop(phi_ext, requires_grad=True)
+    C_torch = Capacitor(3.6, 'GHz', Q=1e6, requires_grad=True)
+    L_torch = Inductor(0.46, 'GHz', Q=500e6, loops=[loop], requires_grad=True)
+    JJ_torch = Junction(10.2, 'GHz', A=1e-7, x=3e-06, loops=[loop], requires_grad=True)
+    circuit_torch = Circuit({(0, 1): [C_torch, L_torch, JJ_torch], }, flux_dist='junctions')
+    circuit_torch.set_trunc_nums([trunc_num, ])
+    return circuit_torch
+
+
+def create_JJL_numpy(trunc_num, phi_ext):
+    set_optim_mode(False)
+    Cs = [3.6, 1.2, 2.0]
+    Js = [10.2, 6.8]
+    Ls = [0.46]
+    loop = Loop(phi_ext)
+    C1 = Capacitor(Cs[0], 'GHz', requires_grad=False)
+    C2 = Capacitor(Cs[1], 'GHz', requires_grad=False)
+    C3 = Capacitor(Cs[2], 'GHz', requires_grad=False)
+    L = Inductor(Ls[0], 'GHz', loops=[loop], requires_grad=False)
+    J1 = Junction(Js[0], 'GHz',loops=[loop], requires_grad=False)
+    J2 = Junction(Js[1], 'GHz',loops=[loop], requires_grad=False)
+
+    circuit_numpy = Circuit(
+        {(0, 1): [C1, J1],
+         (1, 2): [C2, J2],
+         (2, 0): [L, C3]
+         },
+        flux_dist='junctions'
+    )
+    circuit_numpy.set_trunc_nums([trunc_num, trunc_num])
+    return circuit_numpy
+
+
+def create_JJL_torch(trunc_num, phi_ext):
+    set_optim_mode(True)
+    Cs = [3.6, 1.2, 2.0]
+    Js = [10.2, 6.8]
+    Ls = [0.46]
+    loop = Loop(phi_ext, requires_grad=False)
+    C1 = Capacitor(Cs[0], 'GHz', requires_grad=True)
+    C2 = Capacitor(Cs[1], 'GHz', requires_grad=True)
+    C3 = Capacitor(Cs[2], 'GHz', requires_grad=True)
+    L = Inductor(Ls[0], 'GHz', loops=[loop], requires_grad=True)
+    J1 = Junction(Js[0], 'GHz',loops=[loop], requires_grad=True)
+    J2 = Junction(Js[1], 'GHz',loops=[loop], requires_grad=True)
+
+    circuit_torch = Circuit(
+        {(0, 1): [C1, J1],
+         (1, 2): [C2, J2],
+         (2, 0): [L, C3]
+         },
+        flux_dist='junctions'
+    )
+    circuit_torch.set_trunc_nums([trunc_num, trunc_num])
+    return circuit_torch
diff --git a/SQcircuit/tests/data/descriptions/elements.txt b/SQcircuit/tests/data/descriptions/elements.txt
new file mode 100644
index 0000000..33ee9cc
--- /dev/null
+++ b/SQcircuit/tests/data/descriptions/elements.txt
@@ -0,0 +1,14 @@
+Edge (0, 1):
+	C = 1.937e-15 F
+	JJ = 3.142e+10 Hz
+Edge (0, 2):
+	L = 1.257e-06 H
+Edge (3, 0):
+	C = 1.291e-13 F
+Edge (1, 2):
+	C = 1.291e-13 F
+Edge (1, 3):
+	L = 1.257e-06 H
+Edge (2, 3):
+	C = 1.614e-15 F
+	JJ = 4.398e+10 Hz
diff --git a/SQcircuit/tests/data/descriptions/flux_dist_all.txt b/SQcircuit/tests/data/descriptions/flux_dist_all.txt
old mode 100644
new mode 100755
index 3a65bf2..8152c56
--- a/SQcircuit/tests/data/descriptions/flux_dist_all.txt
+++ b/SQcircuit/tests/data/descriptions/flux_dist_all.txt
@@ -1,1 +1,5 @@
-             loop1          \n----------------------------\nJJ1, b1:     0.33           \nJJ2, b2:     0.33           \nind, b3:     0.33           \n
\ No newline at end of file
+             loop1
+------------------
+JJ1, b1:     0.33           
+JJ2, b2:     0.33           
+ind, b3:     0.33           
diff --git a/SQcircuit/tests/data/descriptions/flux_dist_inductors.txt b/SQcircuit/tests/data/descriptions/flux_dist_inductors.txt
old mode 100644
new mode 100755
index dd7600f..87fd3d9
--- a/SQcircuit/tests/data/descriptions/flux_dist_inductors.txt
+++ b/SQcircuit/tests/data/descriptions/flux_dist_inductors.txt
@@ -1,1 +1,5 @@
-             loop1          \n----------------------------\nJJ1, b1:     0.0            \nJJ2, b2:     0.0            \nind, b3:     1.0            \n
\ No newline at end of file
+             loop1
+------------------
+JJ1, b1:     0.0            
+JJ2, b2:     0.0            
+ind, b3:     1.0            
diff --git a/SQcircuit/tests/data/descriptions/flux_dist_junctions.txt b/SQcircuit/tests/data/descriptions/flux_dist_junctions.txt
old mode 100644
new mode 100755
index 7f97c68..c07356d
--- a/SQcircuit/tests/data/descriptions/flux_dist_junctions.txt
+++ b/SQcircuit/tests/data/descriptions/flux_dist_junctions.txt
@@ -1,1 +1,5 @@
-             loop1          \n----------------------------\nJJ1, b1:     0.5            \nJJ2, b2:     0.5            \nind, b3:     0.0            \n
\ No newline at end of file
+             loop1
+------------------
+JJ1, b1:     0.5            
+JJ2, b2:     0.5            
+ind, b3:     0.0            
diff --git a/SQcircuit/tests/data/descriptions/zero_pi_inductors_ltx-a.txt b/SQcircuit/tests/data/descriptions/zero_pi_inductors_ltx-a.txt
new file mode 100755
index 0000000..ab5678c
--- /dev/null
+++ b/SQcircuit/tests/data/descriptions/zero_pi_inductors_ltx-a.txt
@@ -0,0 +1,8 @@
+\hat{H} = \omega_{1} \hat{a}^\dagger_{1} \hat{a}_{1} + \omega_{2} \hat{a}^\dagger_{2} \hat{a}_{2} + E_{C_{33}} \left(- n_{g_{3}} + \hat{n}_{3}\right)^{2} + E_{L_{1}} \left(- \hat{\varphi}_{1} - \hat{\varphi}_{2}\right) \left(- \varphi_{\text{ext}_{1}} / 2\right) + E_{L_{2}} \left(- \hat{\varphi}_{1} + \hat{\varphi}_{2}\right) \left(- \varphi_{\text{ext}_{1}} / 2\right) - E_{J_{1}} \cos{\left(\hat{\varphi}_{1} + \hat{\varphi}_{3} \right)} - E_{J_{2}} \cos{\left(- \hat{\varphi}_{3} + \hat{\varphi}_{1} \right)}
+------------------------------------------------------------
+\text{mode 1:}~~~~~~~~~~~\text{harmonic}~~~~~~~~~~~\hat{\varphi}_{1} = \varphi_{\text{zp}_{1}} \left(\hat{a}_{1} + \hat{a}^\dagger_{1}\right)~~~~~~~~~~~\omega_{1} / 2 \pi = 3.22489~~~~~~~~~~~\varphi_{\text{zp}_{1}} = 2.49
+\text{mode 2:}~~~~~~~~~~~\text{harmonic}~~~~~~~~~~~\hat{\varphi}_{2} = \varphi_{\text{zp}_{2}} \left(\hat{a}_{2} + \hat{a}^\dagger_{2}\right)~~~~~~~~~~~\omega_{2} / 2 \pi = 0.39497~~~~~~~~~~~\varphi_{\text{zp}_{2}} = 0.872
+\text{mode 3:}~~~~~~~~~~~\text{charge}~~~~~~~~~~~n_{g_{3}} = 0
+------------------------------------------------------------
+\text{parameters:}~~~~~~~~~~~E_{C_{33}} = 0.296~~~~~~~~~~~E_{L_{1}} = 0.13~~~~~~~~~~~E_{L_{2}} = 0.13~~~~~~~~~~~E_{J_{1}} = 5.0~~~~~~~~~~~E_{J_{2}} = 5.0
+\text{loops:}~~~~~~~~~~~\varphi_{\text{ext}_{1}} / 2 \pi = 0.5
\ No newline at end of file
diff --git a/SQcircuit/tests/data/descriptions/zero_pi_inductors_ltx-b.txt b/SQcircuit/tests/data/descriptions/zero_pi_inductors_ltx-b.txt
new file mode 100755
index 0000000..ad8f54c
--- /dev/null
+++ b/SQcircuit/tests/data/descriptions/zero_pi_inductors_ltx-b.txt
@@ -0,0 +1,8 @@
+\hat{H} = \omega_{1} \hat{a}^\dagger_{1} \hat{a}_{1} + \omega_{2} \hat{a}^\dagger_{2} \hat{a}_{2} + E_{C_{33}} \left(- n_{g_{3}} + \hat{n}_{3}\right)^{2} + E_{L_{1}} \left(- \hat{\varphi}_{1} - \hat{\varphi}_{2}\right) \left(- \varphi_{\text{ext}_{1}} / 2\right) + E_{L_{2}} \left(- \hat{\varphi}_{1} + \hat{\varphi}_{2}\right) \left(- \varphi_{\text{ext}_{1}} / 2\right) - E_{J_{1}} \cos{\left(- \hat{\varphi}_{3} + \hat{\varphi}_{1} \right)} - E_{J_{2}} \cos{\left(\hat{\varphi}_{1} + \hat{\varphi}_{3} \right)}
+------------------------------------------------------------
+\text{mode 1:}~~~~~~~~~~~\text{harmonic}~~~~~~~~~~~\hat{\varphi}_{1} = \varphi_{\text{zp}_{1}} \left(\hat{a}_{1} + \hat{a}^\dagger_{1}\right)~~~~~~~~~~~\omega_{1} / 2 \pi = 3.22489~~~~~~~~~~~\varphi_{\text{zp}_{1}} = 2.49
+\text{mode 2:}~~~~~~~~~~~\text{harmonic}~~~~~~~~~~~\hat{\varphi}_{2} = \varphi_{\text{zp}_{2}} \left(\hat{a}_{2} + \hat{a}^\dagger_{2}\right)~~~~~~~~~~~\omega_{2} / 2 \pi = 0.39497~~~~~~~~~~~\varphi_{\text{zp}_{2}} = 0.872
+\text{mode 3:}~~~~~~~~~~~\text{charge}~~~~~~~~~~~n_{g_{3}} = 0
+------------------------------------------------------------
+\text{parameters:}~~~~~~~~~~~E_{C_{33}} = 0.296~~~~~~~~~~~E_{L_{1}} = 0.13~~~~~~~~~~~E_{L_{2}} = 0.13~~~~~~~~~~~E_{J_{1}} = 5.0~~~~~~~~~~~E_{J_{2}} = 5.0
+\text{loops:}~~~~~~~~~~~\varphi_{\text{ext}_{1}} / 2 \pi = 0.5
\ No newline at end of file
diff --git a/SQcircuit/tests/data/descriptions/zero_pi_inductors_ltx.txt b/SQcircuit/tests/data/descriptions/zero_pi_inductors_ltx.txt
deleted file mode 100644
index d59dbdf..0000000
--- a/SQcircuit/tests/data/descriptions/zero_pi_inductors_ltx.txt
+++ /dev/null
@@ -1,1 +0,0 @@
-\\hat{H} =~\\omega_1\\hat a^\\dagger_1\\hat a_1~+~E_{C_{22}}(\\hat{n}_2-n_{g_{2}})^2~~+~E_{L_{1}}(-\\hat{\\varphi}_1)(-0.5\\varphi_{\\text{ext}_{1}})~+~E_{L_{2}}(-\\hat{\\varphi}_1)(-0.5\\varphi_{\\text{ext}_{1}})~-~E_{J_{1}}\\cos(-\\hat{\\varphi}_1+\\hat{\\varphi}_2)~-~E_{J_{2}}\\cos(-\\hat{\\varphi}_1-\\hat{\\varphi}_2)\n------------------------------------------------------------\n\\text{mode}~1:~~~~~~~~~~~\\text{harmonic}~~~~~~~~~~~\\hat{\\varphi}_1~=~\\varphi_{zp_{1}}(\\hat a_1+\\hat a^\\dagger_1)~~~~~~~~~~~\\omega_1/2\\pi~=~3.22489~~~~~~~~~~~\\varphi_{zp_{1}}~=~2.49e+00\n\\text{mode}~2:~~~~~~~~~~~\\text{charge}~~~~~~~~~~~~~~~~n_{g_{2}}~=~0\n------------------------------------------------------------\n\\text{parameters}:~~~~~~~~~~~E_{C_{22}}~=~0.296~~~~~~~~~~~E_{L_{1}}~=~0.13~~~~~~~~~~~E_{L_{2}}~=~0.13~~~~~~~~~~~E_{J_{1}}~=~5.0~~~~~~~~~~~E_{J_{2}}~=~5.0~~~~~~~~~~~\n\\text{loops}:~~~~~~~~~~~~~~~~~~~~\\varphi_{\\text{ext}_{1}}/2\\pi~=~0.5~~~~~~~~~~~
\ No newline at end of file
diff --git a/SQcircuit/tests/data/descriptions/zero_pi_inductors_txt-a.txt b/SQcircuit/tests/data/descriptions/zero_pi_inductors_txt-a.txt
new file mode 100755
index 0000000..a9a2df8
--- /dev/null
+++ b/SQcircuit/tests/data/descriptions/zero_pi_inductors_txt-a.txt
@@ -0,0 +1,12 @@
+                                            2                                  ↪
+H = ω₁⋅ad₁⋅a₁ + ω₂⋅ad₂⋅a₂ + EC₃₃⋅(-ng₃ + n₃)  + EL₁⋅(-φ₁ - φ₂)⋅-φₑ₁/2 + EL₂⋅(- ↪
+
+↪                                                       
+↪ φ₁ + φ₂)⋅-φₑ₁/2 - EJ₁⋅cos(φ₁ + φ₃) - EJ₂⋅cos(-φ₃ + φ₁)
+------------------------------------------------------------
+mode 1:       harmonic       φ₁ = zp₁⋅(a₁ + ad₁)       ω₁/2⋅π = 3.22489       zp₁ = 2.49
+mode 2:       harmonic       φ₂ = zp₂⋅(a₂ + ad₂)       ω₂/2⋅π = 0.39497       zp₂ = 0.872
+mode 3:       charge       ng₃ = 0
+------------------------------------------------------------
+parameters:       EC₃₃ = 0.296       EL₁ = 0.13       EL₂ = 0.13       EJ₁ = 5.0       EJ₂ = 5.0
+loops:       φₑ₁/2⋅π = 0.5
\ No newline at end of file
diff --git a/SQcircuit/tests/data/descriptions/zero_pi_inductors_txt-b.txt b/SQcircuit/tests/data/descriptions/zero_pi_inductors_txt-b.txt
new file mode 100755
index 0000000..a6cb6f5
--- /dev/null
+++ b/SQcircuit/tests/data/descriptions/zero_pi_inductors_txt-b.txt
@@ -0,0 +1,12 @@
+                                            2                                  ↪
+H = ω₁⋅ad₁⋅a₁ + ω₂⋅ad₂⋅a₂ + EC₃₃⋅(-ng₃ + n₃)  + EL₁⋅(-φ₁ - φ₂)⋅-φₑ₁/2 + EL₂⋅(- ↪
+
+↪                                                       
+↪ φ₁ + φ₂)⋅-φₑ₁/2 - EJ₁⋅cos(-φ₃ + φ₁) - EJ₂⋅cos(φ₁ + φ₃)
+------------------------------------------------------------
+mode 1:       harmonic       φ₁ = zp₁⋅(a₁ + ad₁)       ω₁/2⋅π = 3.22489       zp₁ = 2.49
+mode 2:       harmonic       φ₂ = zp₂⋅(a₂ + ad₂)       ω₂/2⋅π = 0.39497       zp₂ = 0.872
+mode 3:       charge       ng₃ = 0
+------------------------------------------------------------
+parameters:       EC₃₃ = 0.296       EL₁ = 0.13       EL₂ = 0.13       EJ₁ = 5.0       EJ₂ = 5.0
+loops:       φₑ₁/2⋅π = 0.5
\ No newline at end of file
diff --git a/SQcircuit/tests/data/descriptions/zero_pi_inductors_txt.txt b/SQcircuit/tests/data/descriptions/zero_pi_inductors_txt.txt
deleted file mode 100644
index f38d3c1..0000000
--- a/SQcircuit/tests/data/descriptions/zero_pi_inductors_txt.txt
+++ /dev/null
@@ -1,1 +0,0 @@
-H = ω_1ad_1a_1 + EC_22(n_2-ng_2)^2  + EL_1(-φ_1)(-0.5φ_e1) + EL_2(-φ_1)(-0.5φ_e1) - EJ_1cos(-φ_1+φ_2) - EJ_2cos(-φ_1-φ_2)\n------------------------------------------------------------\nmode 1:       harmonic       φ_1 = zp_1(a_1+ad_1)       ω_1/2π = 3.22489       zp_1 = 2.49e+00\nmode 2:       charge         ng_2 = 0\n------------------------------------------------------------\nparameters:       EC_22 = 0.296       EL_1 = 0.13       EL_2 = 0.13       EJ_1 = 5.0       EJ_2 = 5.0       \nloops:            φ_e1/2π = 0.5       
\ No newline at end of file
diff --git a/SQcircuit/tests/data/descriptions/zero_pi_junctions_ltx-a.txt b/SQcircuit/tests/data/descriptions/zero_pi_junctions_ltx-a.txt
new file mode 100755
index 0000000..8a1e750
--- /dev/null
+++ b/SQcircuit/tests/data/descriptions/zero_pi_junctions_ltx-a.txt
@@ -0,0 +1,8 @@
+\hat{H} = \omega_{1} \hat{a}^\dagger_{1} \hat{a}_{1} + \omega_{2} \hat{a}^\dagger_{2} \hat{a}_{2} + E_{C_{33}} \left(- n_{g_{3}} + \hat{n}_{3}\right)^{2} - E_{J_{1}} \cos{\left(- \varphi_{\text{ext}_{1}} / 2 + \hat{\varphi}_{1} + \hat{\varphi}_{3} \right)} - E_{J_{2}} \cos{\left(\varphi_{\text{ext}_{1}} / 2 - \hat{\varphi}_{1} + \hat{\varphi}_{3} \right)}
+------------------------------------------------------------
+\text{mode 1:}~~~~~~~~~~~\text{harmonic}~~~~~~~~~~~\hat{\varphi}_{1} = \varphi_{\text{zp}_{1}} \left(\hat{a}_{1} + \hat{a}^\dagger_{1}\right)~~~~~~~~~~~\omega_{1} / 2 \pi = 3.22489~~~~~~~~~~~\varphi_{\text{zp}_{1}} = 2.49
+\text{mode 2:}~~~~~~~~~~~\text{harmonic}~~~~~~~~~~~\hat{\varphi}_{2} = \varphi_{\text{zp}_{2}} \left(\hat{a}_{2} + \hat{a}^\dagger_{2}\right)~~~~~~~~~~~\omega_{2} / 2 \pi = 0.39497~~~~~~~~~~~\varphi_{\text{zp}_{2}} = 0.872
+\text{mode 3:}~~~~~~~~~~~\text{charge}~~~~~~~~~~~n_{g_{3}} = 0
+------------------------------------------------------------
+\text{parameters:}~~~~~~~~~~~E_{C_{33}} = 0.296~~~~~~~~~~~E_{J_{1}} = 5.0~~~~~~~~~~~E_{J_{2}} = 5.0
+\text{loops:}~~~~~~~~~~~\varphi_{\text{ext}_{1}} / 2 \pi = 0.5
\ No newline at end of file
diff --git a/SQcircuit/tests/data/descriptions/zero_pi_junctions_ltx-b.txt b/SQcircuit/tests/data/descriptions/zero_pi_junctions_ltx-b.txt
new file mode 100755
index 0000000..525b161
--- /dev/null
+++ b/SQcircuit/tests/data/descriptions/zero_pi_junctions_ltx-b.txt
@@ -0,0 +1,8 @@
+\hat{H} = \omega_{1} \hat{a}^\dagger_{1} \hat{a}_{1} + \omega_{2} \hat{a}^\dagger_{2} \hat{a}_{2} + E_{C_{33}} \left(- n_{g_{3}} + \hat{n}_{3}\right)^{2} - E_{J_{1}} \cos{\left(\varphi_{\text{ext}_{1}} / 2 - \hat{\varphi}_{1} + \hat{\varphi}_{3} \right)} - E_{J_{2}} \cos{\left(- \varphi_{\text{ext}_{1}} / 2 + \hat{\varphi}_{1} + \hat{\varphi}_{3} \right)}
+------------------------------------------------------------
+\text{mode 1:}~~~~~~~~~~~\text{harmonic}~~~~~~~~~~~\hat{\varphi}_{1} = \varphi_{\text{zp}_{1}} \left(\hat{a}_{1} + \hat{a}^\dagger_{1}\right)~~~~~~~~~~~\omega_{1} / 2 \pi = 3.22489~~~~~~~~~~~\varphi_{\text{zp}_{1}} = 2.49
+\text{mode 2:}~~~~~~~~~~~\text{harmonic}~~~~~~~~~~~\hat{\varphi}_{2} = \varphi_{\text{zp}_{2}} \left(\hat{a}_{2} + \hat{a}^\dagger_{2}\right)~~~~~~~~~~~\omega_{2} / 2 \pi = 0.39497~~~~~~~~~~~\varphi_{\text{zp}_{2}} = 0.872
+\text{mode 3:}~~~~~~~~~~~\text{charge}~~~~~~~~~~~n_{g_{3}} = 0
+------------------------------------------------------------
+\text{parameters:}~~~~~~~~~~~E_{C_{33}} = 0.296~~~~~~~~~~~E_{J_{1}} = 5.0~~~~~~~~~~~E_{J_{2}} = 5.0
+\text{loops:}~~~~~~~~~~~\varphi_{\text{ext}_{1}} / 2 \pi = 0.5
\ No newline at end of file
diff --git a/SQcircuit/tests/data/descriptions/zero_pi_junctions_ltx.txt b/SQcircuit/tests/data/descriptions/zero_pi_junctions_ltx.txt
deleted file mode 100644
index f597182..0000000
--- a/SQcircuit/tests/data/descriptions/zero_pi_junctions_ltx.txt
+++ /dev/null
@@ -1,1 +0,0 @@
-\\hat{H} =~\\omega_1\\hat a^\\dagger_1\\hat a_1~+~E_{C_{22}}(\\hat{n}_2-n_{g_{2}})^2~~-~E_{J_{1}}\\cos(-\\hat{\\varphi}_1+\\hat{\\varphi}_2+0.5\\varphi_{\\text{ext}_{1}})~-~E_{J_{2}}\\cos(-\\hat{\\varphi}_1-\\hat{\\varphi}_2+0.5\\varphi_{\\text{ext}_{1}})\n------------------------------------------------------------\n\\text{mode}~1:~~~~~~~~~~~\\text{harmonic}~~~~~~~~~~~\\hat{\\varphi}_1~=~\\varphi_{zp_{1}}(\\hat a_1+\\hat a^\\dagger_1)~~~~~~~~~~~\\omega_1/2\\pi~=~3.22489~~~~~~~~~~~\\varphi_{zp_{1}}~=~2.49e+00\n\\text{mode}~2:~~~~~~~~~~~\\text{charge}~~~~~~~~~~~~~~~~n_{g_{2}}~=~0\n------------------------------------------------------------\n\\text{parameters}:~~~~~~~~~~~E_{C_{22}}~=~0.296~~~~~~~~~~~E_{J_{1}}~=~5.0~~~~~~~~~~~E_{J_{2}}~=~5.0~~~~~~~~~~~\n\\text{loops}:~~~~~~~~~~~~~~~~~~~~\\varphi_{\\text{ext}_{1}}/2\\pi~=~0.5~~~~~~~~~~~
\ No newline at end of file
diff --git a/SQcircuit/tests/data/descriptions/zero_pi_junctions_txt-a.txt b/SQcircuit/tests/data/descriptions/zero_pi_junctions_txt-a.txt
new file mode 100755
index 0000000..67d4069
--- /dev/null
+++ b/SQcircuit/tests/data/descriptions/zero_pi_junctions_txt-a.txt
@@ -0,0 +1,12 @@
+                                            2                                  ↪
+H = ω₁⋅ad₁⋅a₁ + ω₂⋅ad₂⋅a₂ - EC₃₃⋅(-ng₃ + n₃)  - EJ₁⋅cos(-φₑ₁/2 + φ₁ + φ₃) - EJ ↪
+
+↪                       
+↪ ₂⋅cos(φₑ₁/2 - φ₁ + φ₃)
+------------------------------------------------------------
+mode 1:       harmonic       φ₁ = zp₁⋅(a₁ + ad₁)       ω₁/2⋅π = 3.22489       zp₁ = 2.49
+mode 2:       harmonic       φ₂ = zp₂⋅(a₂ + ad₂)       ω₂/2⋅π = 0.39497       zp₂ = 0.872
+mode 3:       charge       ng₃ = 0
+------------------------------------------------------------
+parameters:       EC₃₃ = 0.296       EJ₁ = 5.0       EJ₂ = 5.0
+loops:       φₑ₁/2⋅π = 0.5
\ No newline at end of file
diff --git a/SQcircuit/tests/data/descriptions/zero_pi_junctions_txt-b.txt b/SQcircuit/tests/data/descriptions/zero_pi_junctions_txt-b.txt
new file mode 100755
index 0000000..9a47c5c
--- /dev/null
+++ b/SQcircuit/tests/data/descriptions/zero_pi_junctions_txt-b.txt
@@ -0,0 +1,12 @@
+                                            2                                  ↪
+H = ω₁⋅ad₁⋅a₁ + ω₂⋅ad₂⋅a₂ + EC₃₃⋅(-ng₃ + n₃)  - EJ₁⋅cos(φₑ₁/2 - φ₁ + φ₃) - EJ₂ ↪
+
+↪                       
+↪ ⋅cos(-φₑ₁/2 + φ₁ + φ₃)
+------------------------------------------------------------
+mode 1:       harmonic       φ₁ = zp₁⋅(a₁ + ad₁)       ω₁/2⋅π = 3.22489       zp₁ = 2.49
+mode 2:       harmonic       φ₂ = zp₂⋅(a₂ + ad₂)       ω₂/2⋅π = 0.39497       zp₂ = 0.872
+mode 3:       charge       ng₃ = 0
+------------------------------------------------------------
+parameters:       EC₃₃ = 0.296       EJ₁ = 5.0       EJ₂ = 5.0
+loops:       φₑ₁/2⋅π = 0.5
\ No newline at end of file
diff --git a/SQcircuit/tests/data/descriptions/zero_pi_junctions_txt.txt b/SQcircuit/tests/data/descriptions/zero_pi_junctions_txt.txt
deleted file mode 100644
index 321f804..0000000
--- a/SQcircuit/tests/data/descriptions/zero_pi_junctions_txt.txt
+++ /dev/null
@@ -1,1 +0,0 @@
-H = ω_1ad_1a_1 + EC_22(n_2-ng_2)^2  - EJ_1cos(-φ_1+φ_2+0.5φ_e1) - EJ_2cos(-φ_1-φ_2+0.5φ_e1)\n------------------------------------------------------------\nmode 1:       harmonic       φ_1 = zp_1(a_1+ad_1)       ω_1/2π = 3.22489       zp_1 = 2.49e+00\nmode 2:       charge         ng_2 = 0\n------------------------------------------------------------\nparameters:       EC_22 = 0.296       EJ_1 = 5.0       EJ_2 = 5.0       \nloops:            φ_e1/2π = 0.5       
\ No newline at end of file
diff --git a/SQcircuit/tests/data/phase_coord/zeropi_0 b/SQcircuit/tests/data/phase_coord/zeropi_0
old mode 100644
new mode 100755
diff --git a/SQcircuit/tests/data/phase_coord/zeropi_1 b/SQcircuit/tests/data/phase_coord/zeropi_1
old mode 100644
new mode 100755
diff --git a/SQcircuit/tests/data/phase_coord/zeropi_paper_0.npy b/SQcircuit/tests/data/phase_coord/zeropi_paper_0.npy
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deleted file mode 100644
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deleted file mode 100644
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deleted file mode 100644
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diff --git a/SQcircuit/tests/data/qubits/zeroPi_1 b/SQcircuit/tests/data/qubits/zeroPi_1
deleted file mode 100644
index 436ed9f..0000000
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diff --git a/SQcircuit/tests/data/spectra/zeropi_paper.npy b/SQcircuit/tests/data/spectra/zeropi_paper.npy
new file mode 100644
index 0000000..76db89a
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diff --git a/SQcircuit/tests/test_descriptions.py b/SQcircuit/tests/test_descriptions.py
old mode 100644
new mode 100755
index 785745c..d101000
--- a/SQcircuit/tests/test_descriptions.py
+++ b/SQcircuit/tests/test_descriptions.py
@@ -5,18 +5,22 @@
 
 import SQcircuit as sq
 
+from SQcircuit.settings import set_optim_mode
+
 TESTDIR = os.path.dirname(os.path.abspath(__file__))
-DATADIR = os.path.join(TESTDIR, "data/descriptions")
+DATADIR = os.path.join(TESTDIR, 'data', 'descriptions')
 
 
 def test_zeropi_description():
 
+    set_optim_mode(True)
+
     loop1 = sq.Loop(0.5)
 
-    C = sq.Capacitor(0.15, "GHz")
-    CJ = sq.Capacitor(10, "GHz")
-    JJ = sq.Junction(5, "GHz", loops=[loop1])
-    L = sq.Inductor(0.13, "GHz", loops=[loop1])
+    C = sq.Capacitor(0.15, 'GHz')
+    CJ = sq.Capacitor(10, 'GHz')
+    JJ = sq.Junction(5, 'GHz', loops=[loop1])
+    L = sq.Inductor(0.13, 'GHz', loops=[loop1])
 
     elements = {
         (0, 1): [CJ, JJ],
@@ -35,17 +39,21 @@ def test_zeropi_description():
 
     txt = cr.description(tp='txt', _test=True)
 
-    f = open(DATADIR + '/zero_pi_junctions_txt.txt', 'r')
-    txt_data = f.read().replace("\\n", "\n")
+    with open(os.path.join(DATADIR, 'zero_pi_junctions_txt-a.txt'), 'r') as f:
+        txt_data_a = f.read()
+    with open(os.path.join(DATADIR, 'zero_pi_junctions_txt-b.txt'), 'r') as f:
+        txt_data_b = f.read()
 
-    assert txt == txt_data
+    assert (txt == txt_data_a) or (txt == txt_data_b)
 
     ltx = cr.description(tp='ltx', _test=True)
 
-    f = open(DATADIR + '/zero_pi_junctions_ltx.txt', 'r')
-    ltx_data = (f.read().replace("\\n", "\n")).replace("\\\\", "\\")
+    with open(os.path.join(DATADIR, 'zero_pi_junctions_ltx-a.txt'), 'r') as f:
+        ltx_data_a = f.read()
+    with open(os.path.join(DATADIR, 'zero_pi_junctions_ltx-b.txt'), 'r') as f:
+        ltx_data_b = f.read()
 
-    assert ltx_data == ltx
+    assert (ltx == ltx_data_a) or (ltx == ltx_data_b)
 
     ###########################################################################
     # flux_dist = "inductors"
@@ -55,17 +63,23 @@ def test_zeropi_description():
 
     txt = cr.description(tp='txt', _test=True)
 
-    f = open(DATADIR + '/zero_pi_inductors_txt.txt', 'r')
-    txt_data = f.read().replace("\\n", "\n")
+    with open(os.path.join(DATADIR, 'zero_pi_inductors_txt-a.txt'), 'r') as f:
+        txt_data_a = f.read()
+    with open(os.path.join(DATADIR, 'zero_pi_inductors_txt-b.txt'), 'r') as f:
+        txt_data_b = f.read()
 
-    assert txt == txt_data
+    assert (txt == txt_data_a) or (txt == txt_data_b)
 
     ltx = cr.description(tp='ltx', _test=True)
 
-    f = open(DATADIR + '/zero_pi_inductors_ltx.txt', 'r')
-    ltx_data = (f.read().replace("\\n", "\n")).replace("\\\\", "\\")
+    with open(os.path.join(DATADIR, 'zero_pi_inductors_ltx-a.txt'), 'r') as f:
+        ltx_data_a = f.read()
+    with open(os.path.join(DATADIR, 'zero_pi_inductors_ltx-b.txt'), 'r') as f:
+        ltx_data_b = f.read()
+
+    assert (ltx == ltx_data_a) or (ltx == ltx_data_b)
 
-    assert ltx_data == ltx
+    set_optim_mode(False)
 
 
 def test_loop_description():
@@ -74,9 +88,9 @@ def test_loop_description():
 
     loop1 = sq.Loop(id_str="loop1")
 
-    JJ1 = sq.Junction(1, loops=[loop1], cap=C, id_str="JJ1")
-    JJ2 = sq.Junction(1, loops=[loop1], cap=C, id_str="JJ2")
-    L = sq.Inductor(1, loops=[loop1], cap=C, id_str="ind")
+    JJ1 = sq.Junction(1, loops=[loop1], cap=C, id_str='JJ1')
+    JJ2 = sq.Junction(1, loops=[loop1], cap=C, id_str='JJ2')
+    L = sq.Inductor(1, loops=[loop1], cap=C, id_str='ind')
 
     elements = {
         (0, 1): [JJ1],
@@ -91,8 +105,8 @@ def test_loop_description():
     cr = sq.Circuit(elements, flux_dist='all')
     desc = cr.loop_description(_test=True)
 
-    f = open(DATADIR + '/flux_dist_all.txt', 'r')
-    desc_data = f.read().replace("\\n", "\n")
+    with open(os.path.join(DATADIR, 'flux_dist_all.txt'), 'r') as f:
+        desc_data = f.read()
 
     assert desc == desc_data
 
@@ -103,8 +117,8 @@ def test_loop_description():
     cr = sq.Circuit(elements, flux_dist='inductors')
     desc = cr.loop_description(_test=True)
 
-    f = open(DATADIR + '/flux_dist_inductors.txt', 'r')
-    desc_data = f.read().replace("\\n", "\n")
+    with open(os.path.join(DATADIR, 'flux_dist_inductors.txt'), 'r') as f:
+        desc_data = f.read()
 
     assert desc == desc_data
 
@@ -115,7 +129,38 @@ def test_loop_description():
     cr = sq.Circuit(elements, flux_dist='junctions')
     desc = cr.loop_description(_test=True)
 
-    f = open(DATADIR + '/flux_dist_junctions.txt', 'r')
-    desc_data = f.read().replace("\\n", "\n")
+    with open(os.path.join(DATADIR, 'flux_dist_junctions.txt'), 'r') as f:
+        desc_data = f.read()
 
     assert desc == desc_data
+
+
+def test_elements_description():
+    print()
+
+    loop1 = sq.Loop(0.5)
+
+    C = sq.Capacitor(0.15, 'GHz')
+    CJ1 = sq.Capacitor(10, 'GHz')
+    JJ1 = sq.Junction(5, 'GHz', loops=[loop1])
+    CJ2 = sq.Capacitor(12, 'GHz')
+    JJ2 = sq.Junction(7, 'GHz', loops=[loop1])
+    L = sq.Inductor(0.13, 'GHz', loops=[loop1])
+
+    elements = {
+        (0, 1): [CJ1, JJ1],
+        (0, 2): [L],
+        (3, 0): [C],
+        (1, 2): [C],
+        (1, 3): [L],
+        (2, 3): [CJ2, JJ2],
+    }
+
+    cr = sq.Circuit(elements, flux_dist='junctions')
+
+    txt = cr.element_description(_test=True)
+
+    with open(os.path.join(DATADIR, 'elements.txt'), 'r') as f:
+        test_data = f.read()
+
+    assert txt == test_data
diff --git a/SQcircuit/tests/test_elements.py b/SQcircuit/tests/test_elements.py
old mode 100644
new mode 100755
index d0347ea..a8a5733
--- a/SQcircuit/tests/test_elements.py
+++ b/SQcircuit/tests/test_elements.py
@@ -1,13 +1,33 @@
-"""
-test_elements contains the test cases for the SQcircuit elements
+"""test_elements contains the test cases for the SQcircuit ``elements.py``
 functionalities.
 """
 import pytest
 import numpy as np
+from typing import Union
+
+import torch
+from torch import Tensor
 
 import SQcircuit as sq
 
 from SQcircuit.elements import Capacitor, Inductor, Junction
+from SQcircuit.exceptions import UNIT_ERROR, ModeError
+import SQcircuit.functions as sqf
+from SQcircuit.settings import set_optim_mode, get_optim_mode, SUPPORTED_ENGINES
+
+
+def float_torch_to_python(x: Tensor) -> float:
+    """Helper function to transform the torch float variable to python float
+    variable."""
+
+    return float(x.detach().cpu().numpy())
+
+
+def conditional_to_tensor(x: float) -> Union[float, Tensor]:
+    if get_optim_mode():
+        return torch.tensor(x, dtype=torch.float64)
+    
+    return x
 
 
 ###############################################################################
@@ -16,53 +36,84 @@
 
 
 def test_capacitor_error_massages():
-    error = "The input unit for the capacitor is not correct. " \
-            "Look at the documentation for the correct input format."
-    with pytest.raises(ValueError, match=error):
-        Capacitor(10, "H")
+    with pytest.raises(ValueError, match=UNIT_ERROR):
+        Capacitor(10, 'H')
 
 
 def test_capacitor_energy():
-    cap = Capacitor(10, "GHz")
-    assert cap.energy() == 10
-    # change the default frequency of the SQcirucit
-    sq.set_unit_freq("MHz")
-    # check back the energy
-    assert cap.energy() == 10 * 1000
-    # set back the default frequency back to GHz
-    sq.set_unit_freq("GHz")
-    # check the energy functionality from setting the value
-    val = cap.value()
-    cap2 = Capacitor(val, "F")
-    assert cap2.energy() == 10
-
-
-def test_capacitor_Q():
-    cap = Capacitor(10, "GHz", Q=1)
-    assert cap.Q(-1) == 1
-    assert cap.Q(10) == 1
-    assert cap.Q(0) == 1
-
-    cap = Capacitor(10, "GHz", Q=1e7)
-    assert cap.Q(-1) == 1e7
-    assert cap.Q(10) == 1e7
-    assert cap.Q(0) == 1e7
-
-    Q = lambda omega: omega ** 2
-    cap = Capacitor(10, "GHz", Q=Q)
-    assert cap.Q(2) == 4
+    cap = Capacitor(10, 'GHz')
+    assert cap.get_value('GHz') == 10
+
+    assert cap.get_value('MHz') == 10 * 1000
+
+    val = cap.get_value()
+    cap2 = Capacitor(val, 'F')
+    assert cap2.get_value('GHz') == 10
+
+
+def test_capacitor_q():
+    for engine in SUPPORTED_ENGINES:
+        sq.set_engine(engine)
+
+        # Test constant Q
+        cap = Capacitor(10, 'GHz', Q=1)
+        assert cap.Q(conditional_to_tensor(-1)) == 1
+        assert cap.Q(conditional_to_tensor(10)) == 1
+        assert cap.Q(conditional_to_tensor(0)) == 1
+
+        cap = Capacitor(10, 'GHz', Q=1e7)
+        assert cap.Q(conditional_to_tensor(-1)) == 1e7
+        assert cap.Q(conditional_to_tensor(10)) == 1e7
+        assert cap.Q(conditional_to_tensor(0)) == 1e7
+
+        # Test custom Q function
+        Q = lambda omega: omega ** 2
+        cap = Capacitor(10, 'GHz', Q=Q)
+        assert cap.Q(conditional_to_tensor(2)) == 4
+
+        # Test default Q
+        cap = Capacitor(10, 'GHz', Q='default')
+        assert np.isclose(sqf.to_numpy(cap.Q(conditional_to_tensor(10e9))),
+                          2531813.9189372803)
+        assert np.isclose(sqf.to_numpy(cap.Q(conditional_to_tensor(6e9 * 2 * np.pi))),
+                          1e6)
+
+        cap = sq.Capacitor(3.4, 'GHz', Q='default')
+        assert np.isclose(sqf.to_numpy(cap.Q(conditional_to_tensor(0.56e9))),
+                          19041456.468334354)
 
 
 def test_capacitor_unit():
     cap = Capacitor(10)
-    val = cap.value()
-    assert cap.energy() == 10
-    assert cap.unit == "GHz"
+    val = cap.get_value()
+    assert np.isclose(val, 2e-15)
 
-    sq.set_unit_cap("F")
+    sq.set_unit_cap('F')
     cap = Capacitor(val)
-    assert cap.energy() == 10
-    assert cap.unit == "F"
+    assert cap.get_value('GHz') == 10
+    # this is global, so need to set back
+    sq.set_unit_cap('GHz')
+
+
+def test_capacitor_grad():
+    sq.set_engine('NumPy')
+
+    # First check error massages
+    with pytest.raises(ModeError):
+        Capacitor(10, requires_grad=True)
+
+    with pytest.raises(ModeError):
+        assert not Capacitor(10).requires_grad
+
+    cap_value_no_grad = Capacitor(10).get_value()
+
+    set_optim_mode(True)
+
+    cap_value_with_grad = Capacitor(10, requires_grad=True).get_value()
+
+    assert cap_value_no_grad == float_torch_to_python(cap_value_with_grad)
+
+    set_optim_mode(False)
 
 
 ###############################################################################
@@ -71,80 +122,138 @@ def test_capacitor_unit():
 
 
 def test_inductor_error_massages():
-    error = "The input unit for the inductor is not correct. " \
-            "Look at the documentation for the correct input format."
-    with pytest.raises(ValueError, match=error):
-        Inductor(10, "F")
+    with pytest.raises(ValueError, match=UNIT_ERROR):
+        Inductor(10, 'F')
 
 
 def test_inductor_energy():
-    ind = Inductor(10, "GHz")
-    assert ind.energy() == 10
-    # change the default frequency of the SQcirucit
-    sq.set_unit_freq("MHz")
-    # check back the energy
-    assert ind.energy() == 10 * 1000
-    # set back the default frequency back to GHz
-    sq.set_unit_freq("GHz")
-    # check the energy functionality from setting the value
-    val = ind.value()
-    ind2 = Inductor(val, "H")
-    assert ind2.energy() == 10
-
-
-def test_inductor_Q():
-    ind = Inductor(10, "GHz", Q=1)
-    assert ind.Q(-1, 0) == 1
-    assert ind.Q(10, 12) == 1
-    assert ind.Q(0, 11) == 1
-
-    ind = Inductor(10, "GHz", Q=1e7)
-    assert ind.Q(-1, 5) == 1e7
-    assert ind.Q(10, 6) == 1e7
-    assert ind.Q(0, 1) == 1e7
-
-    ind = Inductor(10, "GHz")
-    assert np.isclose(ind.Q(2 * np.pi * 0.5e9, 10), 500e6)
-    assert np.isclose(ind.Q(2 * np.pi * 0.5e9, 1), 500e6)
-
-    Q = lambda omega, T: omega ** 2
-    ind = Inductor(10, "GHz", Q=Q)
-    assert ind.Q(2, 10) == 4
+    ind = Inductor(10, 'GHz')
+    assert ind.get_value('GHz') == 10
+
+    val = ind.get_value()
+    ind2 = Inductor(val, 'H')
+    assert ind2.get_value('GHz') == 10
+
+
+def test_inductor_q():
+    for engine in SUPPORTED_ENGINES:
+        sq.set_engine(engine)
+
+        # Test constant Q
+        ind = Inductor(10, 'GHz', Q=1)
+        assert ind.Q(conditional_to_tensor(-1), 0) == 1
+        assert ind.Q(conditional_to_tensor(10), 12) == 1
+        assert ind.Q(conditional_to_tensor(0), 11) == 1
+
+        ind = Inductor(10, 'GHz', Q=1e7)
+        assert ind.Q(conditional_to_tensor(-1), 5) == 1e7
+        assert ind.Q(conditional_to_tensor(10), 6) == 1e7
+        assert ind.Q(conditional_to_tensor(0), 1) == 1e7
+
+        # Test custom function Q
+        Q = lambda omega, T: omega ** 2
+        ind = Inductor(10, 'GHz', Q=Q)
+        assert ind.Q(conditional_to_tensor(2), 10) == 4
+
+        # Test default Q
+        ind = Inductor(10, 'GHz')
+        assert np.isclose(sqf.to_numpy(ind.Q(conditional_to_tensor(2 * np.pi * 0.5e9), 10)),
+                          500e6)
+        assert np.isclose(sqf.to_numpy(ind.Q(conditional_to_tensor(2 * np.pi * 0.5e9), 1)),
+                          500e6)
+
+        ind = Inductor(10, 'GHz', Q='default')
+        assert np.isclose(sqf.to_numpy(ind.Q(conditional_to_tensor(100), 0.015)),
+                          559912673679772.0)
 
 
 def test_inductor_unit():
     ind = Inductor(10)
-    val = ind.value()
-    assert ind.energy() == 10
-    assert ind.unit == "GHz"
+    val = ind.get_value()
+    assert np.isclose(val, 1.6346e-8)
 
-    sq.set_unit_ind("H")
+    sq.set_unit_ind('H')
     ind = Inductor(val)
-    assert ind.energy() == 10
-    assert ind.unit == "H"
+    assert ind.get_value('GHz') == 10
+    # this is global, so need to set back
+    sq.set_unit_ind('GHz')
+
+
+def test_inductor_grad():
+    sq.set_engine('NumPy')
+
+    # First check error massages
+    with pytest.raises(ModeError):
+        Inductor(10, requires_grad=True)
+
+    with pytest.raises(ModeError):
+        assert not Inductor(10).requires_grad
+
+    ind_value_no_grad = Inductor(10).get_value()
+
+    set_optim_mode(True)
+
+    ind_value_with_grad = Inductor(10, requires_grad=True).get_value()
 
+    assert ind_value_no_grad == float_torch_to_python(ind_value_with_grad)
+
+    set_optim_mode(False)
 
 ###############################################################################
 # Josephson Junction Tests
 ###############################################################################
 
+
 def test_junction_error_massages():
-    error = "The input unit for the Josephson Junction is not correct. " \
-            "Look at the documentation for the correct input format."
-    with pytest.raises(ValueError, match=error):
-        Junction(10, "F")
+    with pytest.raises(ValueError, match=UNIT_ERROR):
+        Junction(10, 'F')
+
+
+def test_junction_y():
+    for engine in SUPPORTED_ENGINES:
+        sq.set_engine(engine)
 
+        # Test custom Y
+        y_func = lambda omega, T: omega * T
+        JJ = Junction(10, 'GHz', Y=y_func)
+        assert JJ.Y(conditional_to_tensor(10), 2) == 20
 
-def test_junction_Y():
-    y_func = lambda omega, T: omega*T
-    JJ = Junction(10, "GHz", Y=y_func)
-    assert JJ.Y(10, 2) == 20
+        # Test default Y
+        JJ = Junction(10, 'GHz', Y='default')
+        assert np.isclose(sqf.to_numpy(JJ.Y(conditional_to_tensor(1e9 * 2 * np.pi), 0.015)),
+                          1.6206254741014418e+16)
+        JJ = Junction(0.142, 'GHz')
+        assert np.isclose(sqf.to_numpy(JJ.Y(conditional_to_tensor(50e9), 0.015)),
+                          793439739057240.9)
 
 
 def test_junction_unit():
     JJ = Junction(10)
-    assert JJ.unit == "GHz"
+    val = JJ.get_value()
+    assert np.isclose(val, 10 * 2 * np.pi * 1e9)
 
-    sq.set_unit_JJ("MHz")
-    JJ = Junction(10)
-    assert JJ.unit == "MHz"
+    sq.set_unit_JJ("Hz")
+    JJ = Junction(val / 2 / np.pi)
+    assert np.isclose(JJ.get_value('GHz'),  10 * 2 * np.pi)
+    # this is global, so need to set back
+    sq.set_unit_JJ('GHz')
+
+
+def test_junction_grad():
+    sq.set_engine('NumPy')
+    # First check error massages
+    with pytest.raises(ModeError):
+        Junction(10, requires_grad=True)
+
+    with pytest.raises(ModeError):
+        assert not Junction(10).requires_grad
+
+    junc_value_no_grad = Junction(10).get_value()
+
+    set_optim_mode(True)
+
+    junc_value_with_grad = Junction(10, requires_grad=True).get_value()
+
+    assert junc_value_no_grad == float_torch_to_python(junc_value_with_grad)
+
+    set_optim_mode(False)
diff --git a/SQcircuit/tests/test_elements_grad.py b/SQcircuit/tests/test_elements_grad.py
new file mode 100644
index 0000000..7885ae2
--- /dev/null
+++ b/SQcircuit/tests/test_elements_grad.py
@@ -0,0 +1,77 @@
+import SQcircuit as sq
+from SQcircuit import ENV
+import SQcircuit.functions as sqf
+from SQcircuit.settings import set_optim_mode
+from SQcircuit.tests.test_grad import function_grad_test
+
+
+def fluxonium_numpy(trunc_num, phi_ext=0):
+    set_optim_mode(False)
+    loop = sq.Loop(phi_ext)
+    C_numpy = sq.Capacitor(3.6, 'GHz', Q='default')
+    L_numpy = sq.Inductor(0.46, 'GHz', Q='default', loops=[loop])
+    JJ_numpy = sq.Junction(10.2, 'GHz', Y='default', loops=[loop])
+    circuit_numpy = sq.Circuit({(0, 1): [C_numpy, L_numpy, JJ_numpy], }, flux_dist='junctions')
+    circuit_numpy.set_trunc_nums([trunc_num, ])
+    return circuit_numpy
+
+
+def fluxonium_torch(trunc_num, phi_ext=0):
+    set_optim_mode(True)
+    loop = sq.Loop(phi_ext, requires_grad=True)
+    C_torch = sq.Capacitor(3.6, 'GHz', Q='default', requires_grad=True)
+    L_torch = sq.Inductor(0.46, 'GHz', Q='default', loops=[loop], requires_grad=True)
+    JJ_torch = sq.Junction(10.2, 'GHz', Y='default', loops=[loop], requires_grad=True)
+    circuit_torch = sq.Circuit({(0, 1): [C_torch, L_torch, JJ_torch], }, flux_dist='junctions')
+    circuit_torch.set_trunc_nums([trunc_num, ])
+    return circuit_torch
+
+
+def test_capacitor_q_grad():
+    cr_numpy = fluxonium_numpy(120, 0.1)
+    cr_torch = fluxonium_torch(120, 0.1)
+
+    test_q_cap = lambda cr: cr.elements[(0, 1)][0].Q(
+        sqf.abs(cr.efreqs[1] - cr.efreqs[0]),
+    )
+
+    function_grad_test(
+        cr_numpy,
+        test_q_cap,
+        cr_torch,
+        test_q_cap
+    )
+
+
+def test_ind_q_grad():
+    cr_numpy = fluxonium_numpy(120, 0.1)
+    cr_torch = fluxonium_torch(120, 0.1)
+
+    test_q_ind = lambda cr: cr.elements[(0, 1)][1].Q(
+        sqf.abs(cr.efreqs[1] - cr.efreqs[0]),
+        ENV['T']
+    )
+
+    function_grad_test(
+        cr_numpy,
+        test_q_ind,
+        cr_torch,
+        test_q_ind
+    )
+
+
+def test_junction_y_grad():
+    cr_numpy = fluxonium_numpy(120, 0.1)
+    cr_torch = fluxonium_torch(120, 0.1)
+
+    test_y = lambda cr: cr.elements[(0, 1)][2].Y(
+        sqf.abs(cr.efreqs[1] - cr.efreqs[0]),
+        ENV['T']
+    )
+
+    function_grad_test(
+        cr_numpy,
+        test_y,
+        cr_torch,
+        test_y
+    )
diff --git a/SQcircuit/tests/test_functions.py b/SQcircuit/tests/test_functions.py
new file mode 100644
index 0000000..c01f7b9
--- /dev/null
+++ b/SQcircuit/tests/test_functions.py
@@ -0,0 +1,13 @@
+import torch
+from torch.autograd import gradcheck
+
+import SQcircuit.functions as sqf
+
+
+def test_k0e():
+    x = torch.rand(20, requires_grad=True, dtype=torch.double)
+    assert gradcheck(sqf.k0e, x, eps=1e-6, atol=1e-4)
+
+def test_log_k0():
+    x = torch.rand(20, requires_grad=True, dtype=torch.double)
+    assert gradcheck(sqf.log_k0, x, eps=1e-6, atol=1e-4)
diff --git a/SQcircuit/tests/test_grad.py b/SQcircuit/tests/test_grad.py
new file mode 100755
index 0000000..2af8a60
--- /dev/null
+++ b/SQcircuit/tests/test_grad.py
@@ -0,0 +1,672 @@
+"""
+test_elements contains the test cases for the SQcircuit elements
+functionalities.
+"""
+
+import numpy as np
+import torch
+
+from SQcircuit.elements import Capacitor, Junction, Inductor, Loop
+from SQcircuit.settings import get_optim_mode, set_optim_mode
+from SQcircuit.circuit import Circuit, unt
+import SQcircuit.functions as sqf
+from SQcircuit.tests.conftest import (
+    create_fluxonium_numpy,
+    create_fluxonium_torch,
+    create_transmon_numpy,
+    create_transmon_torch,
+    create_flux_transmon_numpy,
+    create_flux_transmon_torch,
+    create_fluxonium_torch_flux,
+)
+
+
+trunc_num = 120
+eigen_count = 20
+tolerance = 2e-2
+
+all_units = unt.farad_list | unt.freq_list | unt.henry_list
+
+
+def max_ratio(a, b):
+    return np.max([np.abs(b / a), np.abs(a / b)])
+
+
+def function_grad_test(
+    circuit_numpy: Circuit,
+    function_numpy,
+    circuit_torch: Circuit,
+    function_torch,
+    num_eigenvalues=20,
+    delta=1e-4
+):
+    """General test function for comparing linear approximation with 
+    gradient computed with PyTorch backpropagation.
+
+    Parameters
+    ----------
+        circuit_numpy:
+            Numpy circuit for which linear approximation will be calculated.
+        function_numpy:
+            Function to call on the numpy circuit. This should match the 
+            expected output of `function_torch`.
+        circuit_torch:
+            Equivalent circuit to `circuit_numpy`, but constructed in PyTorch.
+        function_torch:
+            Equivalent function to `function_numpy`, but written in PyTorch.
+        num_eigenvalues:
+            Integer specifying the number of eigenvalues to use.
+        delta:
+            Perturbation dx to each parameter value in `circuit_numpy` to
+            compute linear gradient df/dx~(f(x+dx)-f(x))/dx.
+    """
+    set_optim_mode(False)
+    circuit_numpy.diag(num_eigenvalues)
+    numpy_val = function_numpy(circuit_numpy)
+
+    set_optim_mode(True)
+    circuit_torch.diag(num_eigenvalues)
+    tensor_val = function_torch(circuit_torch)
+    tensor_val.backward()
+    
+    assert np.isclose(tensor_val.detach().numpy(), numpy_val)
+
+    for edge, elements_by_edge in circuit_numpy.elements.items():
+        for element_idx, element_numpy in enumerate(elements_by_edge):
+            print(f'Checking gradient of {element_numpy} on edge {edge}.')
+            set_optim_mode(False)
+            scale_factor = (
+                1 / (2 * np.pi) if isinstance(element_numpy, Junction) else 1
+            )
+
+            # Calculate f(x+delta)
+            elem_value = element_numpy.get_value(u=element_numpy.value_unit)
+            element_numpy.set_value(
+                scale_factor * elem_value * (1 + delta),
+                element_numpy.value_unit
+            )
+            circuit_numpy.update()
+            circuit_numpy.diag(num_eigenvalues)
+            val_plus = function_numpy(circuit_numpy)
+
+            # Calculate f(x-delta)
+            element_numpy.set_value(
+                scale_factor * elem_value * (1 - delta),
+                element_numpy.value_unit
+            )
+            circuit_numpy.update()
+            circuit_numpy.diag(num_eigenvalues)
+            val_minus = function_numpy(circuit_numpy)
+
+            # Calculate gradient
+            grad_numpy = (val_plus - val_minus) / (
+                2 
+                * delta 
+                * elem_value 
+                * scale_factor 
+                * all_units[element_numpy.value_unit]
+            )
+
+            # Reset circuit
+            element_numpy.set_value(
+                scale_factor * elem_value,
+                element_numpy.value_unit
+            )
+
+            set_optim_mode(True)
+            torch_el = circuit_torch.elements[edge][element_idx]
+            if torch_el in circuit_torch.parameters_elems:
+                grad_torch = torch_el.grad
+                if grad_torch is None:
+                    grad_torch = 0
+                else:
+                    grad_torch = grad_torch.detach().numpy()
+
+                # TODO: Modify Element class so that following
+                #  gradient scaling is not necessary
+                if (
+                    isinstance(element_numpy, Capacitor)
+                    and element_numpy.value_unit in unt.freq_list
+                ):
+                    grad_factor = (
+                        -unt.e**2
+                        / 2
+                        / element_numpy._value**2
+                        / (2*np.pi*unt.hbar)
+                    )
+                    grad_torch /= grad_factor
+                elif (
+                    isinstance(element_numpy, Inductor)
+                    and element_numpy.value_unit in unt.freq_list
+                ):
+                    grad_factor = (
+                        -(unt.Phi0/2/np.pi)**2
+                        / element_numpy._value**2
+                        / (2*np.pi*unt.hbar)
+                    )
+                    grad_torch /= grad_factor
+                if isinstance(element_numpy, Junction):
+                    grad_torch *= (2 * np.pi)
+                print(f"grad torch: {grad_torch}, grad numpy: {grad_numpy}")
+
+                assert np.sign(grad_torch) == np.sign(grad_numpy)
+                assert max_ratio(grad_torch, grad_numpy) <= 1 + tolerance
+
+    for loop_idx, loop in enumerate(circuit_numpy.loops):
+        set_optim_mode(False)
+
+        loop_flux = loop.internal_value
+        # assert 0 + delta < loop_flux < 1 - delta
+
+        # Calculate f(x+delta)
+        loop.internal_value = loop_flux + delta * 2 * np.pi
+        circuit_numpy.diag(num_eigenvalues)
+        val_plus = function_numpy(circuit_numpy)
+
+        # Calculate f(x-delta)
+        loop.internal_value = loop_flux - delta * 2 * np.pi
+        circuit_numpy.diag(num_eigenvalues)
+        val_minus = function_numpy(circuit_numpy)
+
+        # Calculate gradient, being careful for flux insensitivity
+        if val_plus == val_minus:
+            grad_numpy = 0
+        else:
+            grad_numpy = (val_plus - val_minus) / (2 * delta * 2 * np.pi)
+
+        # Reset circuit
+        loop.internal_value = loop_flux
+
+        torch_loop = circuit_torch.loops[loop_idx]
+        set_optim_mode(True)
+        if torch_loop in circuit_torch.parameters_elems:
+            grad_torch = torch_loop.internal_value.grad
+            if grad_torch is None:
+                grad_torch = 0
+            else:
+                grad_torch = grad_torch.detach().numpy()
+
+            print(f"loop #: {loop_idx}")
+            print(f"grad torch: {grad_torch}, grad numpy: {grad_numpy}")
+            assert np.sign(grad_torch) == np.sign(grad_numpy)
+            assert max_ratio(grad_torch, grad_numpy) <= 1 + tolerance
+
+    set_optim_mode(True)
+    circuit_torch.zero_grad()
+
+
+def first_eigendifference_numpy(circuit):
+    return circuit._efreqs[1] - circuit._efreqs[0]
+
+
+def first_eigendifference_torch(circuit):
+    eigenvals, _ = circuit.diag(eigen_count)
+    return (eigenvals[1] - eigenvals[0]) * 2 * np.pi * 1e9
+
+
+def test_omega_transmon():
+    """Verify gradient of first eigendifference omega_1-omega_0 
+    in transmon circuit with linearized value."""
+
+    # Create circuits
+    circuit_numpy = create_transmon_numpy(trunc_num)
+    circuit_torch = create_transmon_torch(trunc_num)
+
+    function_grad_test(
+        circuit_numpy,
+        first_eigendifference_numpy,
+        circuit_torch,
+        first_eigendifference_torch
+    )
+    set_optim_mode(False)
+
+
+def test_omega_flux_transmon():
+    """Verify gradient of first eigendifference omega_1-omega_0 
+    in transmon circuit with linearized value."""
+
+    flux_points = [1e-2, 0.25, 0.5 - 1e-2]
+    for flux_point in flux_points:
+        print('flux point:', flux_point)
+        circuit_numpy = create_flux_transmon_numpy(trunc_num, flux_point)
+        circuit_torch = create_flux_transmon_torch(trunc_num, flux_point)
+
+        function_grad_test(
+            circuit_numpy,
+            first_eigendifference_numpy,
+            circuit_torch,
+            first_eigendifference_torch
+        )
+    set_optim_mode(False)
+
+
+def test_omega_flux_fluxonium():
+    """Verify gradient of first eigendifference omega_1-omega_0 
+    in fluxonium circuit with gradient for loop."""
+
+    flux_points = [1e-2, 0.25, 0.5 - 1e-2]
+    for flux_point in flux_points:
+        print('flux point:', flux_point)
+        circuit_numpy = create_fluxonium_numpy(trunc_num, flux_point)
+        circuit_torch = create_fluxonium_torch_flux(trunc_num, flux_point)
+
+        function_grad_test(
+            circuit_numpy,
+            first_eigendifference_numpy,
+            circuit_torch,
+            first_eigendifference_torch
+        )
+    set_optim_mode(False)
+
+
+def t1_inv(circuit):
+    return (
+        circuit.dec_rate('capacitive', (0, 1)) 
+        + circuit.dec_rate('inductive', (0, 1))
+        + circuit.dec_rate('quasiparticle', (0, 1))
+    )
+
+
+def test_t1_transmon():
+    """Compare gradient of T1 decoherence due to capacitive, inductive, and 
+    quasiparticle noise in transmon circuit with linearized value."""
+
+    # Create circuits
+    circuit_numpy = create_transmon_numpy(trunc_num)
+    circuit_torch = create_transmon_torch(trunc_num)
+
+    function_grad_test(
+        circuit_numpy,
+        t1_inv,
+        circuit_torch,
+        t1_inv,
+        delta=1e-6
+    )
+    set_optim_mode(False)
+
+
+def test_t1_transmon_flux():
+    """Compare gradient of T1 decoherence due to capacitive, inductive, and 
+    quasiparticle noise in transmon circuit with linearized value."""
+
+    flux_points = [1e-2, 0.25, 0.5 - 1e-2]
+    for flux_point in flux_points:
+        print('flux point:', flux_point)
+        circuit_numpy = create_flux_transmon_numpy(trunc_num, flux_point)
+        circuit_torch = create_flux_transmon_torch(trunc_num, flux_point)
+
+        function_grad_test(
+            circuit_numpy,
+            t1_inv,
+            circuit_torch,
+            t1_inv,
+            delta=1e-6
+        )
+    set_optim_mode(False)
+
+
+def test_grad_multiple_steps():
+    """Sample ability of PyTorch to successfully update circuit parameters and 
+    iteratively decrease loss for simple LC resonator frequency optimization 
+    task."""
+    set_optim_mode(True)
+    cap_unit, ind_unit = 'pF', 'uH'
+
+    # Test C differentiation
+    cap = Capacitor(7.746, cap_unit, Q=1e6, requires_grad=True)
+    ind = Inductor(81.67, ind_unit)
+    elements = {
+        (0, 1): [cap, ind],
+    }
+    cr = Circuit(elements)
+    cr.set_trunc_nums([10, ])
+    eigenvalues, _ = cr.diag(2)
+    optimizer = torch.optim.SGD(cr.parameters, lr=1)
+    omega_target = torch.tensor(20e6 / 1e9)  # convert to GHz
+    loss = 0
+    for idx in range(10):
+        print(
+            f"Parameter values (C [pF] and L [uH]): "
+            f"{cap.get_value().detach().numpy()}, " 
+            f"{ind.get_value().detach().numpy()}"
+            f"\n"
+        )
+        optimizer.zero_grad()
+        eigenvalues, _ = cr.diag(2)
+        omega = eigenvalues[1] - eigenvalues[0]
+        loss = (omega - omega_target) ** 2 / omega_target ** 2
+        loss.backward()
+        cap._value.grad *= (cap._value**2)
+        optimizer.step()
+        cr.update()
+
+    assert loss.item() <= 6e-3
+
+
+def test_grad_fluxonium():
+    """Verify gradient values on more complex circuit, first resonant 
+    eigendifference in fluxonium. As opposed to previous test with transmon 
+    circuit, note that this also involves linear inductor and loop."""
+    set_optim_mode(True)
+    loop1 = Loop()
+    cap = Capacitor(3.6, 'GHz', Q=1e6, requires_grad=True)
+    ind = Inductor(
+        0.46, 'GHz', Q=500e6, loops=[loop1], requires_grad=True
+    )
+    junc = Junction(
+        10.2, 'GHz', cap=cap,
+        A=1e-7, x=3e-06, loops=[loop1], requires_grad=True
+    )
+
+    # define the circuit
+    elements = {
+        (0, 1): [cap, ind, junc]
+    }
+    cr = Circuit(elements, flux_dist='all')
+    cr.set_trunc_nums([trunc_num])
+    loop1.set_flux(0)
+    eigenvalues, _ = cr.diag(2)
+    optimizer = torch.optim.SGD(cr.parameters, lr=1e-1)
+    # convert to GHz (initial value: ~8.2 GHz)
+    omega_target = torch.tensor(2e9 / 1e9)
+    loss = 0
+    for idx in range(10):
+        cap_value = cap.get_value().detach().numpy()
+        ind_value = ind.get_value().detach().numpy()
+        junc_value = junc.get_value().detach().numpy()
+        print(
+            f"Parameter values (C [F], L [H], JJ [Hz]): "
+            f"{cap_value, ind_value, junc_value}\n"
+        )
+        optimizer.zero_grad()
+        eigenvalues, _ = cr.diag(2)
+        omega = (eigenvalues[1] - eigenvalues[0])
+        loss = (omega - omega_target) ** 2 / omega_target ** 2
+        loss.backward()
+        cap._value.grad *= (cap._value) ** 2
+        ind._value.grad *= (ind._value) ** 2
+        junc._value.grad *= (junc._value) ** 2
+        optimizer.step()
+        cr.update()
+    assert loss.item() <= 5e-3
+    set_optim_mode(False)
+
+
+def test_spectrum_fluxonium():
+    """Verify gradient of first eigendifference omega_1-omega_0 in fluxonium 
+    circuit with linearized value."""
+
+    # Create circuits
+    circuit_numpy = create_fluxonium_numpy(trunc_num)
+    circuit_torch = create_fluxonium_torch(trunc_num)
+
+    function_grad_test(
+        circuit_numpy,
+        first_eigendifference_numpy,
+        circuit_torch,
+        first_eigendifference_torch
+    )
+    set_optim_mode(False)
+
+
+def test_t1_fluxonium():
+    """Verify gradient of fluxonium for T1 noise sources, including capacitive,
+    inductive, and quasiparticle decoherence."""
+
+    # Create circuits
+    circuit_numpy = create_fluxonium_numpy(trunc_num)
+    circuit_torch = create_fluxonium_torch(trunc_num)
+
+    def t1_inv_capacitive(circuit):
+        return circuit.dec_rate('capacitive', (0, 1))
+
+    def t1_inv_inductive(circuit):
+        return circuit.dec_rate('inductive', (0, 1))
+
+    def t1_inv_quasiparticle(circuit):
+        return circuit.dec_rate('quasiparticle', (0, 1))
+
+    # Test capacitive T1 decoherence
+    print('capacitive')
+    function_grad_test(
+        circuit_numpy,
+        t1_inv_capacitive,
+        circuit_torch,
+        t1_inv_capacitive,
+        delta=1e-6
+    )
+
+    # Test inductive T1 decoherence
+    print('inductive')
+    function_grad_test(
+        circuit_numpy,
+        t1_inv_inductive,
+        circuit_torch,
+        t1_inv_inductive,
+        delta=1e-6
+    )
+
+    # Test quasiparticle T1 decoherence
+    print('quasiparticle')
+    function_grad_test(
+        circuit_numpy,
+        t1_inv_quasiparticle,
+        circuit_torch,
+        t1_inv_quasiparticle,
+        delta=1e-9
+    )
+    set_optim_mode(False)
+
+
+def flux_sensitivity_function(
+    sensitivity_function,
+    flux_point=0.4,
+    delta=0.1
+):
+    def flux_sensitivity(circuit):
+        sensitivities = []
+
+        for loop_idx, _ in enumerate(circuit.loops):
+            first_harmonic_values = []
+            for offset in (-delta, 0, delta):
+                circuit.loops[loop_idx].set_flux(flux_point + offset)
+                circuit.diag(2)
+                first_harmonic = sensitivity_function(circuit)
+                first_harmonic_values.append(first_harmonic)
+            f_minus, f_0, f_plus = first_harmonic_values
+
+            sens = ((f_0 - f_minus) / delta)**2 + ((f_plus - f_0) / delta)**2
+            # Normalize loss function
+            sens /= sqf.abs(f_0 / delta) ** 2
+            sensitivities.append(sens)
+
+        if get_optim_mode():
+            return torch.mean(torch.stack(sensitivities))
+        else:
+            return np.mean(sensitivities)
+
+    return flux_sensitivity
+
+
+def test_flux_sensitivity():
+
+    # Create circuits
+    circuit_numpy = create_fluxonium_numpy(trunc_num)
+    circuit_torch = create_fluxonium_torch(trunc_num)
+
+    function_grad_test(
+        circuit_numpy,
+        flux_sensitivity_function(first_eigendifference_numpy),
+        circuit_torch,
+        flux_sensitivity_function(first_eigendifference_torch),
+        delta=1e-4
+    )
+
+
+def test_anharmonicity():
+    def anharmonicity_numpy(circuit):
+        b = circuit._efreqs[2] - circuit._efreqs[1]
+        a = circuit._efreqs[1] - circuit._efreqs[0]
+        return b / a
+
+    def anharmonicity_torch(circuit):
+        eigenvals, _ = circuit.diag(eigen_count)
+        b = (eigenvals[2] - eigenvals[1]) * 2 * np.pi * 1e9
+        a = (eigenvals[1] - eigenvals[0]) * 2 * np.pi * 1e9
+        return b / a
+
+    # Create circuits
+    circuit_numpy = create_fluxonium_numpy(trunc_num)
+    circuit_torch = create_fluxonium_torch(trunc_num)
+
+    function_grad_test(
+        circuit_numpy,
+        anharmonicity_numpy,
+        circuit_torch,
+        anharmonicity_torch,
+        delta=1e-4
+    )
+
+
+def test_t2_cc():
+    # flux_points = [1e-2, 0.25, 0.5 - 1e-2, 0.5 + 1e-2, 0.75]
+    flux_points = [1e-2, 0.25, 0.5 - 1e-2]
+
+    for phi_ext in flux_points:
+        set_optim_mode(False)
+        circuit_numpy = create_fluxonium_numpy(trunc_num, phi_ext)
+
+        # Create torch circuit
+        set_optim_mode(True)
+        circuit_torch = create_fluxonium_torch(trunc_num, phi_ext)
+
+        function_grad_test(
+            circuit_numpy,
+            lambda cr: cr.dec_rate('cc', states=(0, 1)),
+            circuit_torch,
+            lambda cr: cr.dec_rate('cc', states=(0, 1)),
+            num_eigenvalues=50,
+            delta=1e-6
+        )
+
+
+def test_t2_charge():
+    # charge_offsets = [
+    # 1e-2, 0.2, 0.4, 0.5 - 1e-2, 0.5 + 1e-2, 0.6, 0.8, 1-1e-2
+    # ]
+    charge_offsets = [1e-2, 0.3, 0.5 - 1e-2, 0.8]
+
+    for ng in charge_offsets:
+        circuit_numpy = create_transmon_numpy(trunc_num)
+        circuit_numpy.set_charge_offset(1, ng)
+
+        circuit_torch = create_transmon_torch(trunc_num)
+        circuit_torch.set_charge_offset(1, ng)
+
+        function_grad_test(
+            circuit_numpy,
+            lambda cr: cr.dec_rate('charge', states=(0, 1)),
+            circuit_torch,
+            lambda cr: cr.dec_rate('charge', states=(0, 1)),
+            num_eigenvalues=50,
+            delta=1e-6
+        )
+
+
+def test_t2_flux():
+    # flux_points = [1e-2, 0.25, 0.5 - 1e-2, 0.5 + 1e-2, 0.75]
+    flux_points = [1e-2, 0.25, 0.5 - 1e-2]
+
+    for phi_ext in flux_points:
+        circuit_numpy = create_fluxonium_numpy(trunc_num, phi_ext)
+        circuit_torch = create_fluxonium_torch(trunc_num, phi_ext)
+
+        function_grad_test(
+            circuit_numpy,
+            lambda cr: cr.dec_rate('flux', states=(0, 1)),
+            circuit_torch,
+            lambda cr: cr.dec_rate('flux', states=(0, 1)),
+            num_eigenvalues=50,
+            delta=1e-6
+        )
+
+
+def test_t2_cc_phi_ext():
+    # flux_points = [1e-2, 0.25, 0.5 - 1e-2, 0.5 + 1e-2, 0.75]
+    flux_points = [1e-2, 0.25, 0.5 - 1e-2]
+
+    for phi_ext in flux_points:
+        set_optim_mode(False)
+        circuit_numpy = create_fluxonium_numpy(trunc_num, phi_ext)
+
+        # Create torch circuit
+        set_optim_mode(True)
+        circuit_torch = create_fluxonium_torch_flux(trunc_num, phi_ext)
+
+        function_grad_test(
+            circuit_numpy,
+            lambda cr: cr.dec_rate('cc', states=(0, 1)),
+            circuit_torch,
+            lambda cr: cr.dec_rate('cc', states=(0, 1)),
+            num_eigenvalues=50,
+            delta=1e-6
+        )
+
+
+def test_t2_charge_phi_ext():
+    # charge_offsets = [
+    # 1e-2, 0.2, 0.4, 0.5 - 1e-2, 0.5 + 1e-2, 0.6, 0.8, 1-1e-2
+    # ]
+    charge_offsets = [1e-2, 0.3, 0.5 - 1e-2, 0.8]
+
+    phi_ext = 0.5-1e-3
+
+    for ng in charge_offsets:
+        circuit_numpy = create_flux_transmon_numpy(trunc_num, phi_ext)
+        circuit_numpy.set_charge_offset(1, ng)
+
+        circuit_torch = create_flux_transmon_torch(trunc_num, phi_ext)
+        circuit_torch.set_charge_offset(1, ng)
+
+        function_grad_test(
+            circuit_numpy,
+            lambda cr: cr.dec_rate('charge', states=(0, 1)),
+            circuit_torch,
+            lambda cr: cr.dec_rate('charge', states=(0, 1)),
+            num_eigenvalues=50,
+        )
+
+
+def test_t2_flux_phi_ext():
+    # flux_points = [1e-2, 0.25, 0.5 - 1e-2, 0.5 + 1e-2, 0.75]
+    flux_points = [1e-2, 0.25, 0.5 - 1e-2]
+
+    for phi_ext in flux_points:
+        circuit_numpy = create_fluxonium_numpy(trunc_num, phi_ext)
+        circuit_torch = create_fluxonium_torch_flux(trunc_num, phi_ext)
+
+        function_grad_test(
+            circuit_numpy,
+            lambda cr: cr.dec_rate('flux', states=(0, 1)),
+            circuit_torch,
+            lambda cr: cr.dec_rate('flux', states=(0, 1)),
+            num_eigenvalues=50,
+            delta=1e-6
+        )
+
+# def test_T2_flux_JJL():
+#     flux_points = [0.5] #, 0.25, 0.5 - 1e-2, 0.5 + 1e-2, 0.75]
+#
+#     for phi_ext in flux_points:
+#         print('phi_ext', phi_ext)
+#         circuit_numpy = create_JJL_numpy(45, phi_ext)
+#         circuit_torch = create_JJL_torch(45, phi_ext)
+#
+#         function_grad_test(
+#             circuit_numpy,
+#             first_eigendifference_numpy,
+#             circuit_torch,
+#             first_eigendifference_torch,
+#             num_eigenvalues=50,
+#             delta=1e-4
+#         )
diff --git a/SQcircuit/tests/test_noise.py b/SQcircuit/tests/test_noise.py
old mode 100644
new mode 100755
diff --git a/SQcircuit/tests/test_papers.py b/SQcircuit/tests/test_papers.py
new file mode 100644
index 0000000..3b61c7a
--- /dev/null
+++ b/SQcircuit/tests/test_papers.py
@@ -0,0 +1,217 @@
+"""Tests example code in published papers."""
+
+import os
+
+import numpy as np
+import torch
+
+import SQcircuit as sq
+import SQcircuit.functions as sqf
+from SQcircuit.tests.test_grad import function_grad_test
+from SQcircuit.tests.conftest import (
+    create_JJL_numpy,
+    create_JJL_torch,
+)
+
+TESTDIR = os.path.dirname(os.path.abspath(__file__))
+PHASEDIR = os.path.join(TESTDIR, 'data', 'phase_coord')
+SPECDIR = os.path.join(TESTDIR, 'data', 'spectra')
+
+def _test_paper_1_a() -> None:
+    target_freqs = np.array([0.69721321, 1.43920434, 1.90074875, 2.05449921])
+
+    sq.set_engine('NumPy')
+
+    loop1 = sq.Loop(value=0)
+
+    C = sq.Capacitor(value=0.15, unit="GHz")
+    CJ = sq.Capacitor(value=10, unit="GHz")
+    L = sq.Inductor(value=0.13, unit="GHz", loops = [loop1])
+    JJ = sq.Junction(value=5, unit="GHz", loops=[loop1])
+
+    elements = {
+        (0, 1): [CJ, JJ],
+        (0, 2): [L],
+        (0, 3): [C],
+        (1, 2): [C],
+        (1, 3): [L],
+        (2, 3): [CJ, JJ]
+    }
+
+    cr = sq.Circuit(elements)
+
+    cr.set_charge_offset(mode=3, ng=1)
+
+    cr.set_trunc_nums([25, 1, 25])
+
+    efreqs, _ = cr.diag(n_eig=5)
+
+    ## Perform test
+    efreqs_minus_ground = efreqs[1:] - efreqs[0]
+    assert np.allclose(efreqs_minus_ground, target_freqs)
+    ##
+
+    cr.set_charge_offset(mode=3, ng=0)
+
+    # phi = np.linspace(0, 1,100)
+    # # spectrum of the circuit
+    # n_eig=5
+    # spec = np.zeros((n_eig, len(phi)))
+    # for i in range(len(phi)):
+    #     # set the external flux for the loop
+    #     loop1.set_flux(phi[i])
+    #     # diagonalize the circuit
+    #     spec[:, i], _ = cr.diag(n_eig)
+
+    # ## Perform test
+    # with open(os.path.join(SPECDIR, 'zeropi_paper.npy'), 'rb') as f:
+    #     test_spec = np.load(f)
+    # assert np.allclose(test_spec, spec)
+    # ##
+
+    # duplicates test_phase_coords, except for truncation numbers
+    loop1.set_flux(0)
+    _, _ = cr.diag(n_eig=2)
+    # create a range for each mode
+    phi1 = np.pi*np.linspace(-1,1,100)
+    phi2 = 0
+    phi3 = np.pi*np.linspace(-0.5,1.5,100)
+    # the ground state
+    state0 = cr.eig_phase_coord(k=0, grid=[phi1, phi2, phi3])
+    # the first excited state
+    state1 = cr.eig_phase_coord(k=1, grid=[phi1, phi2, phi3])
+
+    ##
+    with open(os.path.join(PHASEDIR, 'zeropi_paper_0.npy'), 'rb') as f:
+        test_state0 = np.load(f)
+
+    assert np.allclose(np.abs(test_state0), np.abs(state0), rtol=1e-4, atol=1e-3)
+
+    with open(os.path.join(PHASEDIR, 'zeropi_paper_1.npy'), 'rb') as f:
+        test_state1 = np.load(f)
+    assert np.allclose(np.abs(test_state1), np.abs(state1), rtol=1e-4, atol=1e-3)
+    ##
+
+
+def test_paper_1_b() -> None:
+    # define the loop of the circuit
+    loop1 = sq.Loop()
+
+    # define the circuit’s elements
+    C = sq.Capacitor(3.6, "GHz", Q=1e6)
+    L = sq.Inductor(0.46, "GHz", loops=[loop1])
+    JJ = sq.Junction(10.2, "GHz", cap=C, A=5e-7, loops=[loop1])
+
+    # define the fluxonium circuit
+    elements = {(0, 1): [L, JJ]}
+    cr = sq.Circuit(elements, flux_dist="all")
+
+    # set the truncation numbers
+    cr.set_trunc_nums([100])
+
+    # external flux for sweeping over
+    phi = np.linspace(0, 1, 300)
+
+    # T_1 and T_phi
+    T_1 = np.zeros_like(phi)
+    T_phi = np.zeros_like(phi)
+
+    for i in range(len(phi)):
+        # set the external flux for the loop
+        loop1.set_flux(phi[i])
+            
+        # diagonalize the circuit
+        _, _ = cr.diag(n_eig=2)
+
+        # get the T_1 for capacitive loss
+        T_1[i] = 1/cr.dec_rate(dec_type="capacitive", states=(1,0))
+    
+        # get the T_phi for cc noise
+        T_phi[i] = 1/cr.dec_rate(dec_type="cc", states=(1,0))
+
+
+def test_paper_2_a() -> None:
+    ## First, do the code verbatim
+
+    # Set the engine
+    sq.set_engine('PyTorch')
+
+    # Define the transmon elements
+    C = sq.Capacitor(12, 'fF', requires_grad=True)
+    JJ = sq.Junction(10, 'GHz')
+    elements = {(0, 1): [C, JJ]}
+
+    # Define the transmon circuit
+    transmon = sq.Circuit(elements)
+
+    # Set truncation number
+    transmon.set_trunc_nums([100])
+
+    # Diagonalize the circuit
+    efreqs, evecs = transmon.diag(n_eig=10)
+
+    # Calculate the matrix element
+    g_10 = 1 / C.get_value() * torch.abs(
+        evecs[1].conj()
+        @ transmon.charge_op(1, 'original')
+        @ evecs[0]
+    )
+
+    # Backpropagation step
+    g_10.backward()
+
+    # Access the gradient
+    print(C.grad)
+
+
+    ## Now, check that it's correct
+    # First clear old grad
+    C.grad = None
+
+    # Build equivalent numpy circuit
+    sq.set_engine('NumPy')
+    C = sq.Capacitor(12, 'fF')
+    JJ = sq.Junction(10, 'GHz')
+    elements = {(0, 1): [C, JJ]}
+    transmon_numpy = sq.Circuit(elements)
+    transmon_numpy.set_trunc_nums([100])
+    transmon_numpy.diag(n_eig=10)
+
+    #  Function to get matrix element
+    def g_10(cr):
+        evecs = cr.evecs
+        C = cr.elements[(0, 1)][0]
+        return 1 / C.get_value() * sqf.abs(
+            sqf.dag(evecs[1])
+            @ transmon.charge_op(1, 'original')
+            @ evecs[0]
+        )
+
+    function_grad_test(
+        transmon_numpy,
+        g_10,
+        transmon,
+        g_10,
+        delta=1e-6
+    )
+
+def test_paper_2_b() -> None:
+    epsilon_star = 1e-5
+
+    circuit = create_JJL_numpy(30, 0.1)
+    circuit.diag(10)
+
+    circuit.check_convergence(
+        t = 2,
+        threshold = epsilon_star
+    )
+
+    circuit = create_JJL_torch(30, 0.1)
+    circuit.diag(10)
+
+    circuit.check_convergence(
+        t = 2,
+        threshold = epsilon_star
+    )
+
+    # TOOD: test epsilon matches predicted value
diff --git a/SQcircuit/tests/test_phase_coord.py b/SQcircuit/tests/test_phase_coord.py
old mode 100644
new mode 100755
index d28060b..b9ed23f
--- a/SQcircuit/tests/test_phase_coord.py
+++ b/SQcircuit/tests/test_phase_coord.py
@@ -4,9 +4,10 @@
 import dill as pickle
 
 import SQcircuit as sq
+from SQcircuit.settings import set_optim_mode
 
 TESTDIR = os.path.dirname(os.path.abspath(__file__))
-DATADIR = os.path.join(TESTDIR, "data/phase_coord")
+DATADIR = os.path.join(TESTDIR, 'data', 'phase_coord')
 
 ###############################################################################
 # zeropi phase coord representation test
@@ -15,12 +16,14 @@
 
 def test_phase_coord_zeropi():
 
+    set_optim_mode(False)
+
     loop1 = sq.Loop()
 
-    C = sq.Capacitor(0.15, "GHz")
-    CJ = sq.Capacitor(10, "GHz")
-    JJ = sq.Junction(5, "GHz", loops=[loop1])
-    L = sq.Inductor(0.13, "GHz", loops=[loop1])
+    C = sq.Capacitor(0.15, 'GHz')
+    CJ = sq.Capacitor(10, 'GHz')
+    JJ = sq.Junction(5, 'GHz', loops=[loop1])
+    L = sq.Inductor(0.13, 'GHz', loops=[loop1])
 
     elements = {
         (0, 1): [CJ, JJ],
@@ -45,15 +48,15 @@ def test_phase_coord_zeropi():
     # the ground state
     state0 = zrpi.eig_phase_coord(0, grid=[phi1, phi2])
 
-    with open(DATADIR + '/zeropi_0', 'rb') as inp:
+    with open(os.path.join(DATADIR, 'zeropi_0'), 'rb') as inp:
         state0_data = pickle.load(inp)
 
-    assert np.allclose(state0, state0_data, rtol=1e-4, atol=1e-3)
+    assert np.allclose(np.abs(state0), np.abs(state0_data), rtol=1e-4, atol=1e-3)
 
     # the first excited state
     state1 = zrpi.eig_phase_coord(1, grid=[phi1, phi2])
 
-    with open(DATADIR + '/zeropi_1', 'rb') as inp:
+    with open(os.path.join(DATADIR, 'zeropi_1'), 'rb') as inp:
         state1_data = pickle.load(inp)
 
-    assert np.allclose(state1, state1_data, rtol=1e-4, atol=1e-3)
+    assert np.allclose(np.abs(state1), np.abs(state1_data), rtol=1e-4, atol=1e-3)
diff --git a/SQcircuit/tests/test_qubit_grad.py b/SQcircuit/tests/test_qubit_grad.py
new file mode 100755
index 0000000..dceb79b
--- /dev/null
+++ b/SQcircuit/tests/test_qubit_grad.py
@@ -0,0 +1,242 @@
+"""
+test_qubits_grad.py check the gradient calculation of SQcircuit for several
+qubits.
+"""
+
+import numpy as np
+
+import SQcircuit as sq
+from SQcircuit.settings import set_optim_mode
+
+
+def _get_grad_val_test(val1, val2, delta):
+    return (val2 - val1) / delta
+
+
+def _fluxonium(
+    c_val: float,
+    l_val: float,
+    j_val: float,
+    flux: float,
+) -> sq.Circuit:
+    """Return the eigenfrequency of the Fluxonium qubit.
+
+    Parameters
+    ----------
+        c_val:
+            Capacitor value in unit of "fF".
+        l_val:
+            Inductor value in unit of "uH".
+        j_val:
+            Junction value in unit of "GHz".
+        flux:
+            External flux of qubit.
+    """
+
+    loop1 = sq.Loop(flux)
+
+    C = sq.Capacitor(c_val, 'fF')
+    L = sq.Inductor(l_val, 'uH', loops=[loop1])
+    JJ = sq.Junction(j_val, 'GHz', loops=[loop1])
+
+    # define the circuit
+    elements = {
+        (0, 1): [C, L, JJ]
+    }
+
+    cr = sq.Circuit(elements)
+
+    cr.set_trunc_nums([100])
+
+    return cr
+
+
+def _get_fluxonium_efreqs(
+    c_val: float,
+    l_val: float,
+    j_val: float,
+    flux: float,
+    m: int = 1,
+) -> float:
+    """Return the eigenfrequency of the Fluxonium qubit.
+
+    Parameters
+    ----------
+        c_val:
+            Capacitor value in unit of "fF".
+        l_val:
+            Inductor value in unit of "uH".
+        j_val:
+            Junction value in unit of "GHz".
+        flux:
+            External flux of qubit.
+        m:
+            Index of desired eigenfrequency.
+    """
+
+    cr = _fluxonium(c_val, l_val, j_val, flux)
+
+    efreq, evec = cr.diag(n_eig=10)
+
+    return efreq[m] - efreq[0]
+
+
+def test_fluxonium_grad():
+    set_optim_mode(False)
+    c_val = 3
+    l_val = 0.353
+    j_val = 10.2
+    flux = 0.25
+
+    delta = 1e-7
+
+    efreq1 = _get_fluxonium_efreqs(c_val, l_val, j_val, flux)
+    efreq2 = _get_fluxonium_efreqs(c_val + delta, l_val, j_val, flux)
+    efreq3 = _get_fluxonium_efreqs(c_val, l_val + delta, j_val, flux)
+    efreq4 = _get_fluxonium_efreqs(c_val, l_val, j_val + delta, flux)
+    efreq5 = _get_fluxonium_efreqs(c_val, l_val, j_val, flux + delta)
+
+    gr_C = _get_grad_val_test(efreq1, efreq2, delta)
+    gr_L = _get_grad_val_test(efreq1, efreq3, delta)
+    gr_J = _get_grad_val_test(efreq1, efreq4, delta)
+    gr_P = _get_grad_val_test(efreq1, efreq5, delta)
+
+    cr = _fluxonium(c_val, l_val, j_val, flux)
+    cr.diag(10)
+
+    C = cr.elements[(0, 1)][0]
+    L = cr.elements[(0, 1)][1]
+    JJ = cr.elements[(0, 1)][2]
+    loop = cr.loops[0]
+
+    freq_scale = 2 * np.pi * 1e9
+    scale = 2 * np.pi
+    assert np.isclose(cr.get_partial_omega(C, m=1)/freq_scale/1e15, gr_C)
+    assert np.isclose(cr.get_partial_omega(L, m=1)/freq_scale/1e6, gr_L)
+    assert np.isclose(cr.get_partial_omega(JJ, m=1), gr_J, atol=1e-6)
+    assert np.isclose(cr.get_partial_omega(loop, m=1)/freq_scale*scale, gr_P)
+
+
+def _zeropi(
+    c_val: float,
+    cj_val: float,
+    l_val: float,
+    j_val: float,
+    flux: float,
+) -> sq.Circuit:
+    """Return the eigenfrequency of the Fluxonium qubit.
+
+    Parameters
+    ----------
+        c_val:
+            Capacitor value in unit of "fF".
+        cj_val:
+            Junction capacitor value in unit of "fF".
+        l_val:
+            Inductor value in unit of "uH".
+        j_val:
+            Junction value in unit of "GHz".
+        flux:
+            External flux of qubit.
+    """
+
+    loop1 = sq.Loop(flux)
+
+    C = sq.Capacitor(c_val, 'fF')
+    CJ = sq.Capacitor(cj_val, "fF")
+    L = sq.Inductor(l_val, 'uH', loops=[loop1])
+    JJ = sq.Junction(j_val, 'GHz', loops=[loop1])
+
+    # define the circuit
+    elements = {(0, 1): [CJ, JJ],
+                (0, 2): [L],
+                (0, 3): [C],
+                (1, 2): [C],
+                (1, 3): [L],
+                (2, 3): [CJ, JJ]}
+
+    cr = sq.Circuit(elements)
+
+    cr.set_trunc_nums([45, 45])
+
+    return cr
+
+
+def _get_zeropi_efreqs(
+    c_val: float,
+    cj_val: float,
+    l_val: float,
+    j_val: float,
+    flux: float,
+    m: int = 1,
+) -> float:
+    """Return the eigenfrequency of the Fluxonium qubit.
+
+    Parameters
+    ----------
+        c_val:
+            Capacitor value in unit of "fF".
+        l_val:
+            Inductor value in unit of "uH".
+        j_val:
+            Junction value in unit of "GHz".
+        flux:
+            External flux of qubit.
+        m:
+            Index of desired eigenfrequency.
+    """
+
+    cr = _zeropi(c_val, cj_val, l_val, j_val, flux)
+
+    efreq, evec = cr.diag(n_eig=10)
+
+    return efreq[m] - efreq[0]
+
+
+def test_zeropi_grad():
+    set_optim_mode(False)
+    c_val = 129
+    cj_val = 1.93
+    l_val = 1.257
+    j_val = 5
+    flux = 0.25
+    delta = 1e-7
+
+    efreq1 = _get_zeropi_efreqs(c_val, cj_val, l_val, j_val, flux)
+    efreq2 = _get_zeropi_efreqs(c_val+delta, cj_val, l_val, j_val, flux)
+    efreq3 = _get_zeropi_efreqs(c_val, cj_val+delta, l_val, j_val, flux)
+    efreq4 = _get_zeropi_efreqs(c_val, cj_val, l_val+delta, j_val, flux)
+    efreq5 = _get_zeropi_efreqs(c_val, cj_val, l_val, j_val+delta, flux)
+    efreq6 = _get_zeropi_efreqs(c_val, cj_val, l_val, j_val, flux+delta)
+
+    gr_C = _get_grad_val_test(efreq1, efreq2, delta)
+    gr_CJ = _get_grad_val_test(efreq1, efreq3, delta)
+    gr_L = _get_grad_val_test(efreq1, efreq4, delta)
+    gr_J = _get_grad_val_test(efreq1, efreq5, delta)
+    gr_P = _get_grad_val_test(efreq1, efreq6, delta)
+
+    cr = _zeropi(c_val, cj_val, l_val, j_val, flux)
+    cr.diag(10)
+
+    CJ = cr.elements[(0, 1)][0]
+    JJ = cr.elements[(0, 1)][1]
+    C = cr.elements[(0, 3)][0]
+    L = cr.elements[(0, 2)][0]
+    loop = cr.loops[0]
+
+    freq_scale = 2 * np.pi * 1e9
+    scale = 2 * np.pi
+    atol = 1e-2
+
+    print(gr_CJ)
+    print(cr.get_partial_omega(CJ, m=1)/freq_scale/1e15)
+
+    assert np.isclose(cr.get_partial_omega(C, m=1)/freq_scale/1e15, gr_C,
+                      atol=atol)
+    assert np.isclose(cr.get_partial_omega(CJ, m=1)/freq_scale/1e15, gr_CJ,
+                      atol=atol)
+    assert np.isclose(cr.get_partial_omega(L, m=1)/freq_scale/1e6, gr_L,
+                      atol=atol)
+    assert np.isclose(cr.get_partial_omega(JJ, m=1), gr_J, atol=atol)
+    assert np.isclose(cr.get_partial_omega(loop, m=1)/freq_scale*scale, gr_P,
+                      atol=atol)
diff --git a/SQcircuit/tests/test_qubits.py b/SQcircuit/tests/test_qubits.py
old mode 100644
new mode 100755
index 139b660..92e9f89
--- a/SQcircuit/tests/test_qubits.py
+++ b/SQcircuit/tests/test_qubits.py
@@ -1,75 +1,316 @@
 """
 test_qubits contains the test cases for well-known qubits.
 """
+from typing import Optional
 
 import numpy as np
 
 import SQcircuit as sq
 
-from SQcircuit.tests.conftest import QubitTest
+from SQcircuit import Circuit
 
 
-class TestZeroPi(QubitTest):
-    """
-    class for testing the zero-pi qubit
-    """
-    @classmethod
-    def setup_class(cls):
-        cls.fileName = "zeroPi_1"
+def assert_qubit_freq_sweep(
+    circuit: Circuit,
+    target_freqs: np.ndarray,
+    target_decays: Optional[dict] = None,
+    sweep_type: str = 'flux',
+    n_points: int = 3,
+) -> None:
+    """This function assumes that circuit has only either one loop or one
+    charge island."""
 
+    params = np.linspace(0, 0.5, n_points)
+    spec = np.zeros(n_points)
 
-class TestInductivelyShunted(QubitTest):
-    """
-    class for testing the zero-pi qubit
-    """
-    @classmethod
-    def setup_class(cls):
-        cls.fileName = "inductivelyShunted_1"
+    decays = {}
+    if target_decays is not None:
+        for dec_channel in target_decays.keys():
+            decays[dec_channel] = np.zeros(n_points)
 
+    for i, param in enumerate(params):
 
-class TestFluxonium(QubitTest):
-    """
-    class for testing the zero-pi qubit
-    """
-    @classmethod
-    def setup_class(cls):
-        cls.fileName = "Fluxonium_1"
+        if sweep_type == 'flux':
+            circuit.loops[0].set_flux(param)
+        elif sweep_type == 'charge':
+            circuit.set_charge_offset(1, param)
 
+        # diagonalize the circuit
+        efreqs, _ = circuit.diag(2)
 
-class TestFluxonium2(QubitTest):
-    """
-    class for testing the zero-pi qubit
-    """
-    @classmethod
-    def setup_class(cls):
-        cls.fileName = "Fluxonium_2"
+        spec[i] = efreqs[1] - efreqs[0]
 
+        for dec_channel in decays.keys():
+            decays[dec_channel][i] = circuit.dec_rate(
+                dec_type=dec_channel, states=(1, 0)
+            )
+
+    assert np.allclose(spec, target_freqs, rtol=0.01)
+
+    for dec_channel in decays.keys():
+        assert np.allclose(
+            decays[dec_channel], target_decays[dec_channel], atol=1e-5
+        )
+
+
+def test_zero_pi() -> None:
+
+    target_freqs = np.array([0.69721321, 0.45628775, 0.0239571])
+
+    loop1 = sq.Loop()
+
+    # define the circuit ’s elements
+    C = sq.Capacitor(0.15, "GHz")
+    CJ = sq.Capacitor(10, "GHz")
+    JJ = sq.Junction(5, "GHz", loops=[loop1])
+    L = sq.Inductor(0.13, "GHz", loops=[loop1])
+
+    # define the circuit
+    elements = {
+        (0, 1): [CJ, JJ],
+        (0, 2): [L],
+        (0, 3): [C],
+        (1, 2): [C],
+        (1, 3): [L],
+        (2, 3): [CJ, JJ]
+    }
+
+    zrpi = sq.Circuit(elements)
+
+    zrpi.set_trunc_nums([25, 13])
+
+    assert_qubit_freq_sweep(zrpi, target_freqs)
 
-class TestTransmon(QubitTest):
-    """
-    class for testing the zero-pi qubit
-    """
-    @classmethod
-    def setup_class(cls):
-        cls.fileName = "Transmon_1"
+
+def test_inductively_shunted() -> None:
+
+    target_freqs = np.array([6.65474896, 3.73775924, 1.24085844])
+
+    loop1 = sq.Loop()
+
+    # define the circuit ’s elements
+    C_r = sq.Capacitor(20.3, "fF")
+    C_q = sq.Capacitor(5.3, "fF")
+    L_r = sq.Inductor(15.6, "nH")
+    L_q = sq.Inductor(386, "nH", loops=[loop1])
+    L_s = sq.Inductor(4.5, "nH", loops=[loop1])
+    JJ = sq.Junction(6.2, "GHz", loops=[loop1])
+
+    # define the circuit
+    elements = {
+        (0, 1): [C_r],
+        (1, 2): [L_r],
+        (0, 2): [L_s],
+        (2, 3): [L_q],
+        (0, 3): [JJ, C_q]
+    }
+
+    cr = sq.Circuit(elements)
+
+    cr.set_trunc_nums([1, 9, 23])
+
+    assert_qubit_freq_sweep(cr, target_freqs)
+
+
+def test_fluxonium_with_added_node() -> None:
+
+    target_freqs = np.array([5.94930425, 4.83738897, 1.75964137])
+
+    # fluxonium with charge island(added by Yudan)
+    loop1 = sq.Loop()
+    C = sq.Capacitor(11, 'fF', Q=1e6)
+    Cg = sq.Capacitor(0.5, 'fF', Q=1e6)
+    L = sq.Inductor(1, 'GHz', Q=500e6, loops=[loop1])
+    JJ = sq.Junction(3, 'GHz', cap=C, A=5e-7, loops=[loop1])
+
+    elements = {
+        (0, 1): [Cg],
+        (1, 2): [L, JJ],
+        (0, 2): [Cg]
+    }
+
+    cr = sq.Circuit(elements)
+
+    cr.set_trunc_nums([30])
+
+    assert_qubit_freq_sweep(cr, target_freqs)
+
+
+def test_fluxonium() -> None:
+
+    target_freqs = np.array([8.21271281, 4.21408414, 0.63935901])
+
+    target_decays = {
+        'capacitive': np.array([7230.42954319, 1831.72786237, 1935.58917645]),
+        'inductive': np.array([2.84034766,   2.73297225, 125.4597119]),
+        'cc': np.array([4209.46216758,  961.34657151, 2953.53540182]),
+        'quasiparticle': np.array([9.09495148, 3.06603484, 0.]),
+        'flux': np.array([8.80960173e-09, 4.53860939e+05, 3.63215029e-05]),
+    }
+
+    loop1 = sq.Loop()
+
+    # define the circuit elements
+    cap = sq.Capacitor(3.6, 'GHz', Q=1e6)
+    ind = sq.Inductor(0.46, 'GHz', Q=500e6, loops=[loop1])
+    junc = sq.Junction(
+        10.2, 'GHz', cap=cap, A=1e-7, x=3e-06, loops=[loop1]
+    )
+
+    # define the circuit
+    elements = {
+        (0, 1): [ind, junc]
+    }
+
+    cr = sq.Circuit(elements, flux_dist='all')
+
+    cr.set_trunc_nums([100])
+
+    assert_qubit_freq_sweep(cr, target_freqs, target_decays)
+
+
+def test_transmon() -> None:
+    
+    target_freqs = np.array([81.15024681, 42.87817012, 14.96712909])
+
+    target_decays = {
+        'capacitive': np.array([209194.40115, 245791.03928, 961486.30626]),
+        'inductive': np.array([0, 0, 0]),
+        'cc': np.array([6251.40470818, 15286.3367821, 41186.35210817]),
+    }
+    
+    cap = sq.Capacitor(20, 'GHz')
+    junc = sq.Junction(15.0, 'GHz', A=1e-7)
+
+    elements = {
+        (0, 1): [cap, junc]
+    }
+
+    cr = sq.Circuit(elements)
+
+    cr.set_trunc_nums([100])
+
+    assert_qubit_freq_sweep(
+        cr, target_freqs, target_decays, sweep_type='charge'
+    )
 
 
 def test_resonator():
     """
     function for testing simple resonator.
     """
-    C = sq.Capacitor(1 / 2 / np.pi, 'pF', Q=1e6)
-    L = sq.Inductor(1 / 2 / np.pi, 'uH', Q=500e6)
+    ind = sq.Capacitor(1 / 2 / np.pi, 'pF', Q=1e6)
+    cap = sq.Inductor(1 / 2 / np.pi, 'uH', Q=500e6)
 
-    circuitElements = {
-        (0, 1): [L, C],
-    }
+    elements = {(0, 1): [ind, cap]}
 
-    cr = sq.Circuit(circuitElements)
+    cr = sq.Circuit(elements)
 
     assert np.isclose(cr.omega/2/np.pi/1e9, 1)[0]
 
     # check if cr.description() run without error
-
     cr.description(_test=True)
 
+
+def test_coupled_fluxonium_transmon():
+    """Function for testing coupled fluxonium to the transmon."""
+
+    target_efreqs = np.array([
+        0., 2.07145177, 2.34187898, 3.09228726, 4.92917697, 5.19354651,
+        5.87412504, 7.49054673, 7.74403853, 8.2146904
+    ])
+
+    loop1 = sq.Loop()
+
+    # define the circuit elements
+    C = sq.Capacitor(1, 'GHz')
+    L = sq.Inductor(1, 'GHz', loops=[loop1])
+    JJ = sq.Junction(1, 'GHz', loops=[loop1])
+    JJ2 = sq.Junction(1, 'GHz')
+    C_c = sq.Capacitor(1, 'GHz')
+
+    elements = {
+        (0, 1): [L, JJ, C],
+        (0, 2): [JJ2, C],
+        (1, 2): [C_c],
+    }
+
+    cr = sq.Circuit(elements)
+
+    cr.set_trunc_nums([30, 10])
+
+    efreqs, _ = cr.diag(10)
+
+    efreqs = efreqs - efreqs[0]
+
+    assert np.allclose(efreqs, target_efreqs)
+
+
+def test_loop_issue():
+    ec = 1
+    ecj = 3
+    ej = 5
+    el = 0.5
+
+    circuit_dict = {}
+
+    loop1 = sq.Loop(id_str="loop1")
+    loop2 = sq.Loop(0.5, id_str="loop2")
+
+    circuit_dict[(3, 0)] = [sq.Capacitor(ec, "GHz", id_str="C01")]
+    circuit_dict[(3, 1)] = [sq.Capacitor(ec, "GHz", id_str="C02")]
+    circuit_dict[(3, 2)] = [sq.Capacitor(ec, "GHz", id_str="C03")]
+    circuit_dict[(0, 1)] = [sq.Junction(
+        ej, "GHz", id_str="J12", loops=[loop1],
+        cap=sq.Capacitor(ecj, "GHz", id_str="JC12")
+    )]
+    circuit_dict[(0, 2)] = [sq.Junction(
+        ej, "GHz", id_str="J13", loops=[loop1],
+        cap=sq.Capacitor(ecj, "GHz", id_str="JC13")
+    )]
+    circuit_dict[(1, 2)] = [
+        sq.Junction(
+            ej, "GHz", id_str="J23", loops=[loop1, loop2],
+            cap=sq.Capacitor(ecj, "GHz", id_str="JC23")
+        ),
+        sq.Inductor(el, "GHz", id_str="J23", loops=[loop2])
+    ]
+
+    circ_1 = sq.Circuit(circuit_dict)
+    circ_1.set_trunc_nums([20, 15])
+    efreqs, _ = circ_1.diag(2)
+    q_freq_1 = efreqs[1] - efreqs[0]
+
+    circuit_dict = {}
+
+    loop1 = sq.Loop(id_str="loop1")
+    loop2 = sq.Loop(0.5, id_str="loop2")
+
+    circuit_dict[(0, 1)] = [sq.Capacitor(ec, "GHz", id_str="C01")]
+    circuit_dict[(0, 2)] = [sq.Capacitor(ec, "GHz", id_str="C02")]
+    circuit_dict[(0, 3)] = [sq.Capacitor(ec, "GHz", id_str="C03")]
+    circuit_dict[(1, 2)] = [sq.Junction(
+        ej, "GHz", id_str="J12", loops=[loop1],
+        cap=sq.Capacitor(ecj, "GHz", id_str="JC12")
+    )]
+    circuit_dict[(1, 3)] = [sq.Junction(
+        ej, "GHz", id_str="J13", loops=[loop1],
+        cap=sq.Capacitor(ecj, "GHz", id_str="JC13")
+    )]
+    circuit_dict[(2, 3)] = [
+        sq.Junction(
+            ej, "GHz", id_str="J23", loops=[loop1, loop2],
+            cap=sq.Capacitor(ecj, "GHz", id_str="JC23")
+        ),
+        sq.Inductor(el, "GHz", id_str="J23", loops=[loop2])
+    ]
+
+    circ_2 = sq.Circuit(circuit_dict)
+
+    circ_2.set_trunc_nums([20, 15])
+    efreqs, _ = circ_2.diag(2)
+    q_freq_2 = efreqs[1] - efreqs[0]
+
+    assert np.isclose(q_freq_1, 0.010243745)
+    assert np.isclose(q_freq_1, q_freq_2)
+
diff --git a/SQcircuit/tests/test_systems.py b/SQcircuit/tests/test_systems.py
new file mode 100755
index 0000000..91577ea
--- /dev/null
+++ b/SQcircuit/tests/test_systems.py
@@ -0,0 +1,189 @@
+"""test_system contains the test cases for the SQcircuit ``systems.py``
+functionalities.
+"""
+import torch
+
+import SQcircuit as sq
+from SQcircuit.settings import set_optim_mode
+
+
+def is_close(x, y, err) -> bool:
+
+    # print(abs((x - y) / y))
+
+    return abs((x - y) / y) < err
+
+
+def test_coupled_fluxonium():
+    """Test forward and backward pass of system modules for coupled
+    Fluxonium."""
+
+    set_optim_mode(True)
+
+    loop1 = sq.Loop()
+    loop2 = sq.Loop()
+
+    # define the circuit elements
+    C = sq.Capacitor(5.3806, 'fF', requires_grad=True)
+    L1 = sq.Inductor(0.46, 'GHz', loops=[loop1], requires_grad=True)
+    JJ1 = sq.Junction(10.2, 'GHz', loops=[loop1], requires_grad=True)
+
+    L2 = sq.Inductor(0.46, 'GHz', loops=[loop2])
+    JJ2 = sq.Junction(10.2, 'GHz', loops=[loop2])
+
+    C_c = sq.Capacitor(3.6, 'GHz')
+
+    # define the circuit
+    elements = {
+        (0, 1): [L1, JJ1, C],
+        (0, 2): [L2, JJ2, C],
+        (1, 2): [C_c]
+    }
+
+    cr = sq.Circuit(elements)
+    cr.set_trunc_nums([30, 30])
+
+    efreqs, _ = cr.diag(n_eig=10)
+
+    omega_sq = efreqs[1] - efreqs[0]
+
+    omega_sq.backward()
+
+    c_sq_grad = C._value.grad
+    C._value.grad = None
+    l_sq_grad = L1._value.grad
+    L1._value.grad = None
+    jj_sq_grad = JJ1._value.grad
+    JJ1._value.grad = None
+
+    # First fluxonium
+    flxnm1 = sq.Circuit({
+        (0, 1): [L1, JJ1, C]
+    })
+    flxnm1.set_trunc_nums([200])
+    flxnm1.diag(n_eig=40)
+
+    # Second fluxonium
+    flxnm2 = sq.Circuit({
+        (0, 1): [L2, JJ2, C]
+    })
+    flxnm2.set_trunc_nums([200])
+    flxnm2.diag(n_eig=40)
+
+    couplings = [
+        sq.Couple(flxnm1, 1, flxnm2, 1, C_c),
+    ]
+
+    sys = sq.System(couplings)
+
+    H = sys.hamiltonian()
+
+    efreqs = torch.linalg.eigvalsh(H)
+
+    omega_sys = efreqs[1] - efreqs[0]
+
+    omega_sys.backward()
+
+    c_sys_grad = C._value.grad
+    l_sys_grad = L1._value.grad
+    jj_sys_grad = JJ1._value.grad
+
+    assert is_close(omega_sq, omega_sys, 1e-2)
+    # ToDo: Fixing these tests that are failing
+    # assert is_close(c_sq_grad, c_sys_grad, 2e-2)
+    # assert is_close(l_sq_grad, l_sys_grad, 2e-2)
+    assert is_close(jj_sq_grad, jj_sys_grad, 20e-2)
+
+    set_optim_mode(False)
+
+
+def test_three_coupled_transmon():
+    set_optim_mode(True)
+
+    # define the circuit elements
+    C_1 = sq.Capacitor(70, "fF", requires_grad=True)
+    JJ_1 = sq.Junction(8.3472, "GHz")
+    C_2 = sq.Capacitor(72, "fF")
+    JJ_2 = sq.Junction(8.5493, "GHz", requires_grad=True)
+    C_c = sq.Capacitor(200, "fF")
+    JJ_c = sq.Junction(33.5661, "GHz", requires_grad=True)
+    C_1c = sq.Capacitor(4, "fF", requires_grad=True)
+    C_2c = sq.Capacitor(4.2, "fF")
+    C_12 = sq.Capacitor(0.1, "fF")
+
+    elements = {
+        (0, 1): [C_1, JJ_1],
+        (0, 2): [C_2, JJ_2],
+        (0, 3): [C_c, JJ_c],
+        (1, 2): [C_12],
+        (1, 3): [C_1c],
+        (2, 3): [C_2c],
+    }
+
+    cr = sq.Circuit(elements)
+    cr.set_trunc_nums([10, 10, 10])
+
+    efreqs, _ = cr.diag(n_eig=2)
+
+    omega_sq = efreqs[1] - efreqs[0]
+
+    omega_sq.backward()
+
+    c1_sq_grad = C_1._value.grad
+    C_1._value.grad = None
+    c1c_sq_grad = C_1c._value.grad
+    C_1c._value.grad = None
+    jj2_sq_grad = JJ_2._value.grad
+    JJ_2._value.grad = None
+    jjc_sq_grad = JJ_c._value.grad
+    JJ_c._value.grad = None
+
+    # First qubit
+    qubit_1 = sq.Circuit({
+        (0, 1): [C_1, JJ_1]
+    })
+    qubit_1.set_trunc_nums([30])
+    qubit_1.diag(n_eig=10)
+
+    # Second qubit
+    qubit_2 = sq.Circuit({
+        (0, 1): [C_2, JJ_2]
+    })
+    qubit_2.set_trunc_nums([30])
+    qubit_2.diag(n_eig=10)
+
+    # coupler qubit
+    qubit_c = sq.Circuit({
+        (0, 1): [C_c, JJ_c]
+    })
+    qubit_c.set_trunc_nums([30])
+    qubit_c.diag(n_eig=10)
+
+    couplings = [
+        sq.Couple(qubit_1, 1, qubit_c, 1, C_1c),
+        sq.Couple(qubit_2, 1, qubit_c, 1, C_2c),
+        sq.Couple(qubit_1, 1, qubit_2, 1, C_12),
+    ]
+
+    sys = sq.System(couplings)
+
+    H = sys.hamiltonian()
+
+    efreqs = torch.linalg.eigvalsh(H)
+
+    omega_sys = efreqs[1] - efreqs[0]
+
+    omega_sys.backward()
+
+    c1_sys_grad = C_1._value.grad
+    c1c_sys_grad = C_1c._value.grad
+    jj2_sys_grad = JJ_2._value.grad
+    jjc_sys_grad = JJ_c._value.grad
+
+    assert is_close(omega_sq, omega_sys, 2e-2)
+    assert is_close(c1_sq_grad, c1_sys_grad, 2e-2)
+    assert is_close(c1c_sq_grad, c1c_sys_grad, 2e-2)
+    assert is_close(jj2_sq_grad, jj2_sys_grad, 2e-2)
+    assert is_close(jjc_sq_grad, jjc_sys_grad, 25e-2)
+
+    set_optim_mode(False)