Repository: PennyLaneAI/pennylane

Issue:
[BUG] Matrix method of `qml.exp` doesn't utilize `wire_order`
### Expected behavior

Matrix method of `qml.exp` should utilize `wire_order` in calculating both `diagonalizing_mat` and `eigenval_mat`. 

### Actual behavior

Since `qml.eigvals` doesn't support `wire_order` currently, the matrix method of  `qml.exp` skips using `wire_order` and hence throws errors for observables with missing wires.

### Additional information

_No response_

### Source code

```shell
import pennylane as qml
from pennylane import numpy as np

op = qml.exp(qml.PauliZ(wires=[0]) @ qml.PauliY(wires=[1]) @ qml.PauliZ(wires=[4]), coeff=np.array(0.25))
qml.matrix(op)
```


### Tracebacks

```shell
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
Input In [21], in <cell line: 1>()
----> 1 qml.matrix(op)

File ~/workspace/.conda/envs/pennylane/lib/python3.9/site-packages/pennylane/transforms/op_transforms.py:213, in op_transform.__call__(self, *targs, **tkwargs)
    210     obj, *targs = targs
    212 if isinstance(obj, (qml.operation.Operator, qml.tape.QuantumTape)) or callable(obj):
--> 213     return self._create_wrapper(obj, *targs, **tkwargs)
    215 # Input is not an operator nor a QNode nor a quantum tape nor a qfunc.
    216 # Assume Python decorator syntax:
    217 #
   (...)
    229 # Prepend the input to the transform args,
    230 # and create a wrapper function.
    231 if obj is not None:

File ~/workspace/.conda/envs/pennylane/lib/python3.9/site-packages/pennylane/transforms/op_transforms.py:412, in op_transform._create_wrapper(self, obj, wire_order, *targs, **tkwargs)
    409     if wire_order is not None:
    410         tkwargs["wire_order"] = wire_order
--> 412     wrapper = self.fn(obj, *targs, **tkwargs)
    414 elif isinstance(obj, qml.tape.QuantumTape):
    415     # Input is a quantum tape. Get the quantum tape.
    416     tape, verified_wire_order = self._make_tape(obj, wire_order)

File ~/workspace/.conda/envs/pennylane/lib/python3.9/site-packages/pennylane/transforms/op_transforms.py:274, in op_transform.fn(self, obj, *args, **kwargs)
    265     return self.tape_fn(obj.expand(), *args, **kwargs)
    267 except (
    268     AttributeError,
    269     qml.operation.OperatorPropertyUndefined,
   (...)
    272     # if obj.expand() does not exist, a required operation property was not found,
    273     # or the tape transform function does not exist, simply raise the original exception
--> 274     raise e1 from None

File ~/workspace/.conda/envs/pennylane/lib/python3.9/site-packages/pennylane/transforms/op_transforms.py:258, in op_transform.fn(self, obj, *args, **kwargs)
    240 """Evaluate the underlying operator transform function.
    241 
    242 If a corresponding tape transform for the operator has been registered
   (...)
    255     any: the result of evaluating the transform
    256 """
    257 try:
--> 258     return self._fn(obj, *args, **kwargs)
    260 except Exception as e1:  # pylint: disable=broad-except
    262     try:
    263         # attempt to decompose the operation and call
    264         # the tape transform function if defined

File ~/workspace/.conda/envs/pennylane/lib/python3.9/site-packages/pennylane/ops/functions/matrix.py:127, in matrix(op, wire_order)
    124 if isinstance(op, qml.Hamiltonian):
    125     return qml.utils.sparse_hamiltonian(op, wires=wire_order).toarray()
--> 127 return op.matrix(wire_order=wire_order)

File ~/workspace/.conda/envs/pennylane/lib/python3.9/site-packages/pennylane/ops/op_math/exp.py:180, in Exp.matrix(self, wire_order)
    178     diagonalizing_mat = qml.matrix(self.diagonalizing_gates)()
    179     eigvals_mat = math.diag(self.eigvals())
--> 180     mat = diagonalizing_mat.conj().T @ eigvals_mat @ diagonalizing_mat
    181     return expand_matrix(mat, wires=self.wires, wire_order=wire_order)
    182 except OperatorPropertyUndefined:

File ~/workspace/.conda/envs/pennylane/lib/python3.9/site-packages/pennylane/numpy/tensor.py:155, in tensor.__array_ufunc__(self, ufunc, method, *inputs, **kwargs)
    151 args = [i.unwrap() if hasattr(i, "unwrap") else i for i in inputs]
    153 # call the ndarray.__array_ufunc__ method to compute the result
    154 # of the vectorized ufunc
--> 155 res = super().__array_ufunc__(ufunc, method, *args, **kwargs)
    157 if ufunc.nout == 1:
    158     res = (res,)

ValueError: matmul: Input operand 1 has a mismatch in its core dimension 0, with gufunc signature (n?,k),(k,m?)->(n?,m?) (size 8 is different from 2)
```


### System information

```shell
Name: PennyLane
Version: 0.26.0.dev0
Summary: PennyLane is a Python quantum machine learning library by Xanadu Inc.
Home-page: https://github.com/XanaduAI/pennylane
Author: 
Author-email: 
License: Apache License 2.0
Location: /home/jovyan/workspace/.conda/envs/pennylane/lib/python3.9/site-packages
Requires: appdirs, autograd, autoray, cachetools, networkx, numpy, pennylane-lightning, retworkx, scipy, semantic-version, toml
Required-by: PennyLane-Cirq, PennyLane-Lightning, PennyLane-qiskit, pennylane-qulacs, PennyLane-SF

Platform info:           Linux-5.4.196-108.356.amzn2.x86_64-x86_64-with-glibc2.31
Python version:          3.9.13
Numpy version:           1.23.2
Scipy version:           1.9.1
Installed devices:
- default.gaussian (PennyLane-0.26.0.dev0)
- default.mixed (PennyLane-0.26.0.dev0)
- default.qubit (PennyLane-0.26.0.dev0)
- default.qubit.autograd (PennyLane-0.26.0.dev0)
- default.qubit.jax (PennyLane-0.26.0.dev0)
- default.qubit.tf (PennyLane-0.26.0.dev0)
- default.qubit.torch (PennyLane-0.26.0.dev0)
- default.qutrit (PennyLane-0.26.0.dev0)
- lightning.qubit (PennyLane-Lightning-0.25.1)
- cirq.mixedsimulator (PennyLane-Cirq-0.24.0)
- cirq.pasqal (PennyLane-Cirq-0.24.0)
- cirq.qsim (PennyLane-Cirq-0.24.0)
- cirq.qsimh (PennyLane-Cirq-0.24.0)
- cirq.simulator (PennyLane-Cirq-0.24.0)
- qiskit.aer (PennyLane-qiskit-0.24.0)
- qiskit.basicaer (PennyLane-qiskit-0.24.0)
- qiskit.ibmq (PennyLane-qiskit-0.24.0)
- qiskit.ibmq.circuit_runner (PennyLane-qiskit-0.24.0)
- qiskit.ibmq.sampler (PennyLane-qiskit-0.24.0)
- qulacs.simulator (pennylane-qulacs-0.24.0)
- strawberryfields.fock (PennyLane-SF-0.20.1)
- strawberryfields.gaussian (PennyLane-SF-0.20.1)
- strawberryfields.gbs (PennyLane-SF-0.20.1)
- strawberryfields.remote (PennyLane-SF-0.20.1)
- strawberryfields.tf (PennyLane-SF-0.20.1)
```


### Existing GitHub issues

- [X] I have searched existing GitHub issues to make sure the issue does not already exist.

Discussion:
codecov[bot]: # [Codecov](https://codecov.io/gh/PennyLaneAI/pennylane/pull/3057?src=pr&el=h1&utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=PennyLaneAI) Report
> :exclamation: No coverage uploaded for pull request base (`v0.26.0-rc0@f09e09a`). [Click here to learn what that means](https://docs.codecov.io/docs/error-reference?utm_medium=referral&utm_source=github&utm_content=comment&utm_campaign=pr+comments&utm_term=PennyLaneAI#section-missing-base-commit).
> The diff coverage is `n/a`.

> :exclamation: Current head b09bc7b differs from pull request most recent head a46d84c. Consider uploading reports for the commit a46d84c to get more accurate results

```diff
@@              Coverage Diff               @@
##             v0.26.0-rc0    #3057   +/-   ##
==============================================
  Coverage               ?   99.67%           
==============================================
  Files                  ?      272           
  Lines                  ?    23136           
  Branches               ?        0           
==============================================
  Hits                   ?    23060           
  Misses                 ?       76           
  Partials               ?        0           
```



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Patch:
diff --git a/.pre-commit-config.yaml b/.pre-commit-config.yaml
index bcb981d48..2cfedfce7 100644
--- a/.pre-commit-config.yaml
+++ b/.pre-commit-config.yaml
@@ -18,4 +18,3 @@ repos:
     hooks:
     - id: black
       exclude: ^doc/
-      language_version: python3.8
diff --git a/doc/releases/changelog-0.26.0.md b/doc/releases/changelog-0.26.0.md
index 3946bf8cf..72813e555 100644
--- a/doc/releases/changelog-0.26.0.md
+++ b/doc/releases/changelog-0.26.0.md
@@ -491,6 +491,10 @@
 
 <h3>Bug fixes</h3>
 
+* Fixes a bug with `qml.ops.Exp` operators when the coefficient is autograd but the diagonalizing gates
+  don't act on all wires.
+  [(#3057)](https://github.com/PennyLaneAI/pennylane/pull/3057)
+
 * Fixed a bug where the tape transform `single_qubit_fusion` computed wrong rotation angles
   for specific combinations of rotations.
   [(#3024)](https://github.com/PennyLaneAI/pennylane/pull/3024)
diff --git a/pennylane/ops/op_math/exp.py b/pennylane/ops/op_math/exp.py
index 35bb2163f..fde45907c 100644
--- a/pennylane/ops/op_math/exp.py
+++ b/pennylane/ops/op_math/exp.py
@@ -168,14 +168,17 @@ class Exp(SymbolicOp):
     def matrix(self, wire_order=None):
 
         coeff_interface = math.get_interface(self.coeff)
-        if coeff_interface == "autograd":
+        if coeff_interface == "autograd" and math.requires_grad(self.coeff):
             # math.expm is not differentiable with autograd
             # So we try to do a differentiable construction if possible
             #
             # This won't catch situations when the base matrix is autograd,
             # but at least this provides as much trainablility as possible
             try:
-                diagonalizing_mat = qml.matrix(self.diagonalizing_gates)()
+                if len(self.diagonalizing_gates()) == 0:
+                    eigvals_mat = math.diag(self.eigvals())
+                    return expand_matrix(eigvals_mat, wires=self.wires, wire_order=wire_order)
+                diagonalizing_mat = qml.matrix(self.diagonalizing_gates, wire_order=self.wires)()
                 eigvals_mat = math.diag(self.eigvals())
                 mat = diagonalizing_mat.conj().T @ eigvals_mat @ diagonalizing_mat
                 return expand_matrix(mat, wires=self.wires, wire_order=wire_order)


Test patch:
diff --git a/tests/ops/op_math/test_exp.py b/tests/ops/op_math/test_exp.py
index bd54d4c3d..e6be0ec20 100644
--- a/tests/ops/op_math/test_exp.py
+++ b/tests/ops/op_math/test_exp.py
@@ -139,14 +139,38 @@ class TestMatrix:
         assert qml.math.allclose(op.matrix(), isingxx.matrix())
 
     @pytest.mark.autograd
-    def test_matrix_autograd_rx(self):
+    @pytest.mark.parametrize("requires_grad", (True, False))
+    def test_matrix_autograd_rx(self, requires_grad):
         """Test the matrix comparing to the rx gate."""
-        phi = np.array(1.234)
+        phi = np.array(1.234, requires_grad=requires_grad)
         exp_rx = Exp(qml.PauliX(0), -0.5j * phi)
         rx = qml.RX(phi, 0)
 
         assert qml.math.allclose(exp_rx.matrix(), rx.matrix())
 
+    @pytest.mark.autograd
+    @pytest.mark.parametrize("requires_grad", (True, False))
+    def test_matrix_autograd_rz(self, requires_grad):
+        """Test the matrix comparing to the rz gate. This is a gate with an
+        autograd coefficient but empty diagonalizing gates."""
+        phi = np.array(1.234, requires_grad=requires_grad)
+        exp_rz = Exp(qml.PauliZ(0), -0.5j * phi)
+        rz = qml.RZ(phi, 0)
+
+        assert qml.math.allclose(exp_rz.matrix(), rz.matrix())
+
+    @pytest.mark.autograd
+    @pytest.mark.parametrize("requires_grad", (True, False))
+    def test_tensor_with_pauliz_autograd(self, requires_grad):
+        """Test the matrix for the case when the coefficient is autograd and
+        the diagonalizing gates don't act on every wire for the matrix."""
+        phi = qml.numpy.array(-0.345, requires_grad=requires_grad)
+        base = qml.PauliZ(0) @ qml.PauliY(1)
+        autograd_op = Exp(base, phi)
+        mat = qml.math.expm(phi * qml.matrix(base))
+
+        assert qml.math.allclose(autograd_op.matrix(), mat)
+
     @pytest.mark.autograd
     def test_base_no_diagonalizing_gates_autograd_coeff(self):
         """Test the matrix when the base matrix doesn't define the diagonalizing gates."""