# Modeling noise in superconducting qubits: experiments on an IBM quantum processor and numerical simulations

Grégoire Misguich, Liran Shirizly, Haggai Landa

► To cite this version:

Grégoire Misguich, Liran Shirizly, Haggai Landa. Modeling noise in superconducting qubits: experiments on an IBM quantum processor and numerical simulations. Open Quantum Many-Body Physics 2024 (OpenQMB 2024), Andrea De Luca; Jacopo De Nardis; Zala Lenarcic; Marco Schirò, Oct 2024, Orsay, France. hal-05006591

HAL Id: hal-05006591

https://hal.science/hal-05006591v1

Submitted on 26 Mar 2025

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# Modeling noise in superconducting qubits
experiments on an IBM quantum processor
and numerical simulations

Liran Shirizly (IBM Quantum, Haifa)

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Haggai Landa (IBM Quantum, Haifa)

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G. Misguich

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Phys. Rev. Lett. 132 (2024)

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STRATEGIE
NATIONALE
QUANTIQUE

<figure><img src="image_10.png" /></figure>

PROGRAMME ET
EQUIPEMENTS
PRIORITAIRES
DE RECHERCHE
QUANTIQUE

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# Outline

* Construct a **noise model** for the dynamics of **idle superconducting qubits**

* Hamiltonian (including cross-talk),

* dissipation/noise (Lindblad)

* coupling to charge parities

* determination of the microscopic **parameters** of the model
(1- and 2- qubits characterization experiments)

* Is the model able to **describe** the dissipative dynamics of a collection of qubits?

* Choose an entangled initial state. ex: **graph state**.

* Measure observables as a function of time. ex: stabilizers

* Simulate the model numerically (requires MPS & MPO)

* Compare numerics & experiment
<figure><img src="image_13.png" /></figure>

# IBM Eagle devices

ibm_sherbrooke

OpenQASM 3

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

* 127-qubit quantum processor based on a transmon superconducting qubit architecture (announced in Nov. 2022).

* qubit connectivity: heavy-hexagonal layout

research.ibm.com/blog/127-qubit-quantum-processor-eagle
Model: Hamiltonian (1)

Idle qubits

$$
H = \frac{1}{2} \sum_{a=1}^{N} \Delta_a (1 - \sigma_a^z)
$$

<figure><img src="image_19.png" /></figure>
Model: Hamiltonian (2)

Idle qubits + charge fluctuations

$$H = \frac{1}{2} \sum_{a=1}^{N} \left( \Delta_a \pm \underbrace{\nu_a}_{=\text{parity } \tilde{\sigma}_a} \right) (1 - \sigma_a^Z)$$

<figure><img src="image_20.png" /></figure>

each qubit's frequency is shifted according to the charge *parity* of the qubit's junction electrodes.

See for instance K. Serniak, M. Hays, G. de Lange, S. Diamond, S. Shankar, L. D. Burkhart, L. Frunzio, M. Houzet, and M. H. Devoret, Phys. Rev. Lett. 121, 157701 (2018).
Model: Hamiltonian (3)

idle qubits + charge fluctuations + cross talk

$$H = \frac{1}{2} \sum_{a=1}^{N} \left( \Delta_a \pm \underbrace{\nu_a}_{=\text{parity } \tilde{\sigma}_a} \right) (1 - \sigma_a^z)$$

$$+ \frac{1}{2} \sum_{1 \le a < b \le N} \zeta_{ab} (1 - \sigma_a^z)(1 - \sigma_b^z)$$

Ising coupling between qubits which are nearest neighbors (in the heavy-hex geometry)

<figure><img src="image_21.png" /></figure>

each qubit's frequency is shifted according to the charge *parity* of the qubit's junction electrodes.

See for instance K. Serniak, M. Hays, G. de Lange, S. Diamond, S. Shankar, L. D. Burkhart, L. Frunzio, M. Houzet, and M. H. Devoret, Phys. Rev. Lett. 121, 157701 (2018).
Model: noise & dissipation

$$ \frac{\partial}{\partial t} \rho = -\frac{i}{\hbar} [H, \rho] + \mathcal{D}[\sigma^{+}] + \mathcal{D}[\sigma^{z}] $$

$$ \mathcal{D}[\sigma^{+}] = \sum_{i} g_{0,i} \left( \sigma_{i}^{+} \rho \sigma_{i}^{-} - \frac{1}{2} \{ \sigma_{i}^{-} \sigma_{i}^{+}, \rho \} \right) $$

$$ \mathcal{D}[\sigma^{z}] = \sum_{i} g_{2,i} (\sigma_{i}^{z} \rho \sigma_{i}^{z} - \rho), $$

Lindblad master equation

* Jump operators corresponding to dissipative/relaxation process $|1\rangle \rightarrow |0\rangle$. $g_0 = 1/T_1$

* Jump operators corresponding to dephasing $g_2 = (1/T_2 - 1/2T_1)/2$

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>
# Determination of model parameters: characterization experiments

<figure><img src="image_24.png" /></figure>

□ 1-qubit experiment [relaxation & Ramsey, for each qubit]
→ determine (fits) $T_1$, $T_2$, ($g_0$ and $g_2$) $\Delta$ and $\nu$

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

□ 2-qubits experiments [for all pairs (a, b) of connected qubits]
→ cross talk coupling $\zeta_{ab}$
Parameters of the model

<table><thead><tr><th>Qubit</th><th>T<sub>1</sub> [µs]</th><th>T<sub>2</sub> [µs]</th><th>ν/2π [Hz]</th><th>Δ/2π [Hz]</th><th>Edge</th><th>ζ/2π [kHz]</th></tr></thead><tbody><tr><td>4</td><td>155 ± 5</td><td>148 ± 4</td><td>2269 ± 19</td><td>-4867 ± 35</td><td>4, 5</td><td>-38.4 ± 0.3</td></tr><tr><td>5</td><td>172 ± 6</td><td>146 ± 3</td><td>6252 ± 22</td><td>-6153 ± 28</td><td>5, 6</td><td>-29.5 ± 0.3</td></tr><tr><td>6</td><td>250 ± 13</td><td>183 ± 4</td><td>11087 ± 19</td><td>-4525 ± 28</td><td>6, 7</td><td>-32.9 ± 0.3</td></tr><tr><td>7</td><td>172 ± 7</td><td>142 ± 6</td><td>1254 ± 37</td><td>-858 ± 35</td><td>7, 8</td><td>-31.1 ± 0.4</td></tr><tr><td>8</td><td>79 ± 2</td><td>76 ± 2</td><td>2200 ± 44</td><td>36103 ± 62</td><td>8, 16</td><td>-45.8 ± 0.4</td></tr><tr><td>16</td><td>163 ± 6</td><td>224 ± 8</td><td>2832 ± 353</td><td>-1400 ± 190</td><td>16, 26</td><td>-32.9 ± 0.3</td></tr><tr><td>26</td><td>157 ± 5</td><td>42 ± 1</td><td>3199 ± 83</td><td>-6127 ± 107</td><td>26, 25</td><td>-31.7 ± 0.3</td></tr><tr><td>25</td><td>132 ± 4</td><td>99 ± 3</td><td>5041 ± 27</td><td>-1071 ± 45</td><td>25, 24</td><td>-29.8 ± 0.4</td></tr><tr><td>24</td><td>173 ± 7</td><td>110 ± 3</td><td>3890 ± 27</td><td>-6428 ± 40</td><td>24, 23</td><td>-30.6 ± 0.4</td></tr><tr><td>23</td><td>221 ± 13</td><td>111 ± 4</td><td>2204 ± 30</td><td>-40798 ± 52</td><td>23, 22</td><td>-38.8 ± 0.4</td></tr><tr><td>22</td><td>100 ± 2</td><td>93 ± 2</td><td>3126 ± 30</td><td>-6368 ± 45</td><td>22, 15</td><td>-142.6 ± 0.4</td></tr><tr><td>15</td><td>229 ± 12</td><td>48 ± 1</td><td>7909 ± 60</td><td>6708 ± 76</td><td>15, 4</td><td>-34.8 ± 0.3</td></tr></tbody></table>

<figure><img src="image_28.png" /></figure>
Average over all charge parities

1 qubit

each parity $\tilde{\sigma}$ is constant during the timescale
of an experiment but it fluctuates randomly
between different experiments/shots

<figure><img src="image_29.png" /></figure>

N qubits: statistical **average over the 2**^N **configurations of the parities**

$$
\rho(t) = \frac{1}{2^N} \sum_{\tilde{\sigma}_a = \pm 1} \rho_{\{\tilde{\sigma}_1, \tilde{\sigma}_2, \dots, \tilde{\sigma}_N\}}(t)
$$

→ **Single simulation** with charge parities in the following mixed state:

$$
\tilde{\rho} = \frac{1}{2^N} \left( \begin{bmatrix} 1/2 & 0 \\ 0 & 1/2 \end{bmatrix} \otimes \begin{bmatrix} 1/2 & 0 \\ 0 & 1/2 \end{bmatrix} \otimes \dots \otimes \begin{bmatrix} 1/2 & 0 \\ 0 & 1/2 \end{bmatrix} \right)
$$
Model: summary

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /></figure>

$$ \rho(t) = \rho_{\text{qubits}}(t) \otimes \underbrace{\text{Id}_2 \otimes \cdots \otimes \text{Id}_2}_{\tilde{\rho}: \text{charge-parities}} $$
Simulations
# Density matrix & MPO

□ Density matrix $\rho$ of an open system (mixed state) with Hilbert space dimension $D = 2^N$

■ $\rho(t): D \times D$ matrix

■ Lindblad evolution $\partial_t \rho = \mathcal{L}[\rho]$
$\mathcal{L}$: Liouvilian (super)operator, $D^2 \times D^2$ matrix

→ Beyond a few spins/qubits
brute force calculations are not possible

<figure><img src="image_32.png" /></figure>

□ Matrix-product operator (MPO) representation

Prosen & Žnidarič J. Stat. Mech. 2009

<figure><img src="image_33.png" /></figure>

□ For state which are “not too correlated” this
offer an important gain (small **bond dimension**)

□ What determines the bond dimension of the MPO is the Operator-Space Entanglement Entropy (OSEE). The OSEE quantifies, in a bipartition A | B of the system, the total amount of correlations between regions A and B.
See Zanardi et al. 2000; Prosen & Pižorn, 2007

□ Alternative representation: MPDO
Verstraete et al. PRL (2004)
lindbladMPO code

<figure><img src="image_34.png" /></figure>

□ Simulation of Lindblad dynamics (open quantum system) for interacting qubits, based on MPS/MPO representation of the many-body density matrix

□ Efficient for states that are “not too correlated” (OSEE) / low spatial dimension

<figure><img src="image_35.png" /></figure>

□ Time evolution: W'' scheme [Zaletel et al. PRB 2015]

☑ long-range interactions

☑ Small Trotter error $\mathcal{L}$: Lindbladian $\exp(\tau\mathcal{L})=\text{MPO}+\mathcal{O}(\tau^5)$ [Bidzhiev & GM, PRB 2017]

<figure><img src="image_36.png" /></figure>

□ Based on the iTensor library (C++ version)

□ Python interface & Qiskit compatibility

<figure><img src="image_37.png" /></figure>

□ Open source :

▪ github.com/qiskit-community/lindbladmpo

▪ H. Landa & GM, SciPostPhysCore.6.2.037
Application:
qubits initialized in a (N = 12) graph state at t = 0
# Graph state

## Definition

<figure><img src="image_38.png" /></figure>

* Graph, $E$: set of edges

* Initialize all qubits to $|+\rangle$

* Apply Control-Z for all pairs of qubits which are 'connected' in the graph

$$ S_a = \underbrace{X_a \prod_{b \in \text{neighbors of } a} Z_b^Z}_{\text{"stabilizer"}} $$

$$ |g\rangle = \prod_{(i,j) \in E} CZ(i,j) |++\cdots+\rangle $$

$$ S_a |g\rangle = |g\rangle $$

## Motivations

* rich entanglement structure (multi-party entanglement)

* measurement-based quantum computations schemes

* error correction codes

* ...

## Experiment: choose a graph compatible with the device qubit topology (heavy-hexagon)

<figure><img src="image_39.png" /></figure>

$N = 3$ linear graph state

<figure><img src="image_40.png" /></figure>

$N = 12$ ring graph state

<figure><img src="image_41.png" /></figure>
Experimental preparation of (N = 12) ring graph state

<figure><img src="image_42.png" /><figcaption>FIG. 11. Quantum circuit initializing a 12-qubits ring graph state. The index shown for each qubit (in the notation $q_i$) is the physical qubit number actually used in the experiment. The first layer of circuits prepares the state $|+\rangle^{\otimes 12}$, then two layers of controlled-Z (CZ) gates follow according to the ring topology. Since CZ isn't a basis gate in ibm_cusco, equivalent transpiled circuits are implemented.</figcaption></figure>
# 12 qubits ring graph state: experiment vs simulation

Hamiltonian and noise parameters fixed by the 1- and 2-qubit characterization experiments → **no adjustable parameter**

<figure><img src="image_43.png" /></figure>

<figure><img src="image_44.png" /></figure>

* $\langle Z_{15}X_4Z_5 \rangle = \text{expectation value of a 3-qubit observable, a graph state stabilizer}$

* 12 qubits + 12 charge parities → $N = 24$ simulated qubits (mixed state) → brute force calculation impossible. Here calculations with MPO

* Good agreement between simulations and experiments → **validation of the model**

time [µs]
12 qubits ring graph state: experiment vs simulation

<figure><img src="image_45.png" /><figcaption>FIG. 4. (a) The dynamics of the 12Q-ring graph state stabilizers’ mean decoherence, as measured in the experiment (points) and extracted from simulations (lines). The presented measure that ideally equals 1 is shown with and without intermediate dynamical decoupling sequences, which cancel the effect of frequency shifts, charge-parity fluctuations, and ZZ coupling, see the text for details. (b) Using the simulations we can see how the fidelity of $\rho(t)$ with the ideal 12Q state is significantly improved when using the DD sequence canceling the coherent Hamiltonian errors.</figcaption></figure>

<figure><img src="image_46.png" /></figure>

Fidelity with respect to the initial graph state
$= \text{Tr}[|g\rangle\langle g|\rho(t)]$

DD: dynamical decoupling
Thank you

Liran Shirizly, GM, Haggai Landa

<figure><img src="image_47.png" /></figure>

arXiv 2308.01860 / Phys. Rev. Lett. 132 (2024)

<figure><img src="image_48.png" /></figure>
# Appendix
Determination of model parameters: $T_1$

initialize the qubit in $|1\rangle$ and fit the probability $p_1(t)$ to be in state $|0\rangle$ by $p_1(t) \sim A \exp(-t/T_1) + B$

<figure><img src="image_49.png" /></figure>

$T1 = 203 \pm 4.55 \ \mu s$

$\text{reduced-}\chi^2 = 1.167$

<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /></figure>
# Determination of model parameters: $T_2$, $\Delta$ and $\nu$
Ramsey experiment

1024 experiments ('shots') per point

* Initialize in $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$

* Measure $P_x = \text{Tr}[\rho(t)|+\rangle\langle+|]$ and $P_y = \text{Tr}[\rho(t)|i\rangle\langle i|]$ as a function of time

<figure><img src="image_52.png" /></figure>

* Fitting function

$$P_x = A e^{-t/T_2} [b \cos((\Delta + \omega_s + \nu)t + \phi) + (1-b) \cos((\Delta + \omega_s - \nu)t + \phi)] + B$$

$$P_y = A e^{-t/T_2} [b \sin((\Delta + \omega_s + \nu)t + \phi) + (1-b) \sin((\Delta + \omega_s - \nu)t + \phi)] + B.$$

* Remarks:

* in practice $b \sim 0.5$ (equal probabilities to have even or odd charge)

* $\phi$, $A$ and $B$: account for SPAM errors (ideally $\phi = 0$, $A = 0$ and $B = 0$)
# Determination of model parameters: $T_2 \Delta$ and $\nu$ Ramsey experiment

<figure><img src="image_53.png" /></figure>

## Fitting function

$$P_x = A e^{-t/T_2} [b \cos((\Delta + \omega_s + \nu)t + \phi) + (1-b) \cos((\Delta + \omega_s - \nu)t + \phi)] + B$$

$$P_y = A e^{-t/T_2} [b \sin((\Delta + \omega_s + \nu)t + \phi) + (1-b) \sin((\Delta + \omega_s - \nu)t + \phi)] + B.$$

## Remarks:

* in practice $b \sim 0.5$ (equal probabilities to have even or odd charge)

* $\phi$, $A$ and $B$: account for SPAM errors (ideally $\phi = 0 A = 0$ and $B = 0$)
Model: noise & dissipation

ibm_sherbrooke

OpenQASM 3

<figure><img src="image_54.png" /></figure>

<figure><img src="image_55.png" /></figure>
# Determination of model parameters: couplings $\zeta_{ab}$

* Hamiltonian (reminder)

$$H = \frac{1}{2} \sum_{1 \le a < b \le N} \zeta_{ab} (1 - \sigma_a^z)(1 - \sigma_b^z) + \frac{1}{2} \sum_a (\Delta_a + \tilde{\sigma}_a v_a)(1 - \sigma_a^z)$$

* Remarks

* $\Delta$ and $v$ are known from the single-qubit characterization experiments

* $H$ is diagonal in the $\sigma^z$ basis → trivial eigenstates

* For every pair $(a, b)$ of connected qubit:

* Solve the dynamics ($H$ above) for 2 qubits

* Trace out one qubit
    → statistical mixture of oscillations with frequencies

$$\Delta_1 + \nu_1, \Delta_1 - \nu_1, \Delta_1 + \nu_1 + 2\zeta_{1,2} \text{ and } \Delta_1 - \nu_1 + 2\zeta_{1,2}$$

* Isolate the pair $(a, b)$ in the device and run characterization experiments
    → fits to get $\zeta_{ab}$

<figure><img src="image_56.png" /></figure>
Coupling to charge parities: time dependence of $v$

<figure><img src="image_57.png" /><figcaption>FIG. 7. Charge-parity frequency splitting $v$, as a function of time for several qubits on ibm_cusco. Each point is obtained from a different Ramsey experiment, the lines are added for clarity. Error bars indicate the uncertainty in the fitted values of $v$. The duration of each experiment is about 1 minute, and the time axis indicates the time that passed relative to the first experiment's completion time. The presented data shows typical intervals between jumps and typical ranges of $v$ values.</figcaption></figure>
Coupling to charge parities: time dependence of $\nu$

<figure><img src="image_58.png" /><figcaption>FIG. 8. Charge-parity frequency splitting $\nu$, as a function of time for several qubits on a test Falcon device, similar to the ones available through [27]. Each point is obtained from different Ramsey experiment, the lines are added for clarity. Error bars indicated the uncertainty in the fitted value of $\nu$.</figcaption></figure>
3 qubits: experiment vs simulation

Hamiltonian and noise parameters fixed by the previous 1- and 2-qubit characterization experiments
→ no adjustable parameter

<figure><img src="image_59.png" /></figure>
3 qubits: experiment vs simulation

Hamiltonian and noise parameters fixed by the previous 1- and 2-qubit characterization experiments
→ no adjustable parameter

<figure><img src="image_60.png" /></figure>