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Enhancing classical simulation with noisy quantum devices

Ruiqi Zhang, Fuchuan Wei, Zhaohui Wei

TL;DR

The paper tackles the challenge of simulating non-Clifford quantum circuits by turning hardware noise into a computational resource. It introduces structure-preserving Monte Carlo (SPMC) to decompose PQCs into Clifford circuits that preserve architecture, and then extends this with Noisy-device-enhanced Classical Simulation (NDE-CS), which learns sparse Clifford representations from noisy hardware data to estimate noiseless observables with far fewer samples. Theoretical results show Pauli-insertion training yields coefficients valid for the noiseless limit under angle-independent Pauli noise, and numerical experiments on second-order Trotterized Ising circuits demonstrate orders-of-magnitude reductions in sampling cost compared to pure classical Monte Carlo, as well as regimes where NDE-CS outperforms Sparse Pauli Dynamics (SPD). Overall, NDE-CS provides a scalable hybrid quantum-classical framework that leverages hardware noise to accelerate the classical simulation of large, deep quantum circuits, with strong potential for extensions to tensor-network-based simulators. The work suggests a practical path toward exploiting near-term quantum devices as computational resources rather than mere error sources, especially as hardware scales up and noise characteristics become more malleable.

Abstract

As quantum devices continue to improve in scale and precision, a central challenge is how to effectively utilize noisy hardware for meaningful computation. Most existing approaches aim to recover noiseless circuit outputs from noisy ones through error mitigation or correction. Here, we show that noisy quantum devices can be directly leveraged as computational resources to enhance the classical simulation of quantum circuits. We introduce the Noisy-device-enhanced Classical Simulation (NDE-CS) protocol, which improves stabilizer-based classical Monte Carlo simulation methods by incorporating data obtained from noisy quantum hardware. Specifically, NDE-CS uses noisy executions of a target circuit together with noisy Clifford circuits to learn how the target circuit can be expressed in terms of Clifford circuits under realistic noise. The same learned relation can then be reused in the noiseless Clifford limit, enabling accurate estimation of ideal expectation values with substantially reduced sampling cost. Numerical simulations on Trotterized Ising circuits demonstrate that NDE-CS achieves orders-of-magnitude reductions in sampling cost compared to the underlying purely classical Monte Carlo approaches from which it is derived, while maintaining the same accuracy. We also compare NDE-CS with Sparse Pauli Dynamics (SPD), a powerful classical framework capable of simulating quantum circuits at previously inaccessible scales, and provide an example where the cost of SPD scales exponentially with system size, while NDE-CS scales much more favorably. These results establish NDE-CS as a scalable hybrid simulation approach for quantum circuits, where noise can be harnessed as a computational asset.

Enhancing classical simulation with noisy quantum devices

TL;DR

The paper tackles the challenge of simulating non-Clifford quantum circuits by turning hardware noise into a computational resource. It introduces structure-preserving Monte Carlo (SPMC) to decompose PQCs into Clifford circuits that preserve architecture, and then extends this with Noisy-device-enhanced Classical Simulation (NDE-CS), which learns sparse Clifford representations from noisy hardware data to estimate noiseless observables with far fewer samples. Theoretical results show Pauli-insertion training yields coefficients valid for the noiseless limit under angle-independent Pauli noise, and numerical experiments on second-order Trotterized Ising circuits demonstrate orders-of-magnitude reductions in sampling cost compared to pure classical Monte Carlo, as well as regimes where NDE-CS outperforms Sparse Pauli Dynamics (SPD). Overall, NDE-CS provides a scalable hybrid quantum-classical framework that leverages hardware noise to accelerate the classical simulation of large, deep quantum circuits, with strong potential for extensions to tensor-network-based simulators. The work suggests a practical path toward exploiting near-term quantum devices as computational resources rather than mere error sources, especially as hardware scales up and noise characteristics become more malleable.

Abstract

As quantum devices continue to improve in scale and precision, a central challenge is how to effectively utilize noisy hardware for meaningful computation. Most existing approaches aim to recover noiseless circuit outputs from noisy ones through error mitigation or correction. Here, we show that noisy quantum devices can be directly leveraged as computational resources to enhance the classical simulation of quantum circuits. We introduce the Noisy-device-enhanced Classical Simulation (NDE-CS) protocol, which improves stabilizer-based classical Monte Carlo simulation methods by incorporating data obtained from noisy quantum hardware. Specifically, NDE-CS uses noisy executions of a target circuit together with noisy Clifford circuits to learn how the target circuit can be expressed in terms of Clifford circuits under realistic noise. The same learned relation can then be reused in the noiseless Clifford limit, enabling accurate estimation of ideal expectation values with substantially reduced sampling cost. Numerical simulations on Trotterized Ising circuits demonstrate that NDE-CS achieves orders-of-magnitude reductions in sampling cost compared to the underlying purely classical Monte Carlo approaches from which it is derived, while maintaining the same accuracy. We also compare NDE-CS with Sparse Pauli Dynamics (SPD), a powerful classical framework capable of simulating quantum circuits at previously inaccessible scales, and provide an example where the cost of SPD scales exponentially with system size, while NDE-CS scales much more favorably. These results establish NDE-CS as a scalable hybrid simulation approach for quantum circuits, where noise can be harnessed as a computational asset.
Paper Structure (30 sections, 4 theorems, 96 equations, 11 figures, 1 algorithm)

This paper contains 30 sections, 4 theorems, 96 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Circuit Model and Noise Model
  4. Monte Carlo and Stabilizer-Based Methods for Quantum Circuit Simulation
  5. Structure-Preserving Monte Carlo Simulation
  6. Decomposition of parameterized quantum circuits
  7. Structure-preserving Monte Carlo sampling
  8. Classical simulation under noise
  9. Leveraging noisy quantum devices for classical simulation
  10. Motivating observations
  11. Noisy-device-enhanced classical simulation protocol (NDE-CS)
  12. Part I: Selecting the relevant Clifford circuits
  13. Part II: Estimating the coefficients using a noisy quantum device
  14. Algorithmic summary
  15. Theoretical analysis
  16. ...and 15 more sections

Key Result

lemma 1

For the channel $\mathcal{R}_{Z}(\theta)$ corresponding to the rotation gate $R_Z(\theta)$, one has the decomposition where $\mathcal{I}, \mathcal{Z}, \mathcal{S}$ denote the channels associated with the gates $I$, $Z$, and $S$, respectively. For $\theta \in [0,\pi/4]$, this decomposition achieves the minimal possible $\ell_1$-norm among all exact decompositions.

Figures (11)

  • Figure 1: A single-qubit PQC with observable $X + Z$ used in Observation 3.
  • Figure 2: Framework for the NDE-CS protocol. The objective is to estimate the noiseless expectation value of a given observable for a target non-Clifford circuit. The blue arrows indicate the training stage: we construct pairs consisting of the target non-Clifford circuit and associated Clifford circuits. Both types of circuits are run on noisy quantum hardware to obtain measurement data, forming a training dataset for a classical machine-learning model. The model learns a mapping from Clifford-circuit expectations to the corresponding expectations of the target non-Clifford circuit from noisy measurement data. The green arrows denote the inference stage: the trained model takes as input the classically simulated expectations of the Clifford circuits (obtained efficiently via a stabilizer simulator) and outputs a prediction for the target non-Clifford expectation value. Through this framework, noisy quantum devices enhance classical simulation by providing the data needed to learn an effective mapping from Clifford-circuit expectations to target-circuit expectations.
  • Figure 3: Circuit structure for numerical simulations of the second-order Ising Hamiltonian. (a) The number of qubits is $n$, and the Trotter step number is $N$, with $\theta = -\frac{2J_{ij}T}{N}$, $\phi = \frac{2Th_i}{N}$, and the observable $M_Z=\sum_i Z_i$. (b) To mimic realistic superconducting hardware, the rotation gates in (a) are compiled into the native basis consisting of $\{R_Z, \mathrm{C}Z, H\}$ gates.
  • Figure 4: Errors of the NDE-CS protocol for the $n=16$, $N=5$ Trotter-step Ising circuit $\mathcal{C}(\bm{\theta})_{16,5}$ under different numbers of sampled Clifford circuits $M_\mathrm{C}$ and Pauli insertion patterns $M_\mathrm{P}$. (a) Absolute error $\varepsilon_{\mathrm{abs}} =|\langle O\rangle^{(M_\mathrm{C},M_\mathrm{P})} -\operatorname{Tr} \left( O\,\mathcal{C}(\bm{\theta})(\rho) \right)|$. (b) Relative error $\varepsilon_{\mathrm{rel}} =|\langle O\rangle^{(M_\mathrm{C},M_\mathrm{P})} -\operatorname{Tr} \left( O\,\mathcal{C}(\bm{\theta})(\rho) \right)| /|\operatorname{Tr} \left( O\,\mathcal{C}(\bm{\theta})(\rho) \right)|$. The horizontal and vertical axes in (a) and (b) represent $M_\mathrm{P}$ and $M_\mathrm{C}$, respectively, and each data point corresponds to the mean of 20 independent NDE-CS runs. Dark-blue regions in (b) correspond to $\varepsilon_{\mathrm{rel}}\leq 1\%$, showing that such $(M_\mathrm{C},M_\mathrm{P})$ configurations are sufficient to achieve sub-percent accuracy in estimating the noiseless expectation. (c) Line plots of the relative error $\varepsilon_{\mathrm{rel}}$ as a function of $M_\mathrm{C}$ for several fixed $M_\mathrm{P}$ values. Each curve exhibits monotonic convergence: as both $M_\mathrm{C}$ and $M_\mathrm{P}$ increase, the relative error systematically decreases and eventually stabilizes below the $1\%$ threshold.
  • Figure 5: Simulation of the 16-qubit, 5-step Ising circuit $\mathcal{C}(\bm{\theta})_{16,5}$ using the SMC method. The figure shows the relation between the number of samples and the simulation relative error $\varepsilon_{\mathrm{rel}}$. As illustrated, the relative error decreases proportionally to the inverse square root of the sample number, i.e., $\varepsilon_{\mathrm{rel}} \propto 1/\sqrt{N_{\mathrm{sample}}}$, demonstrating the expected Monte Carlo convergence behavior.
  • ...and 6 more figures

Theorems & Definitions (5)

  • lemma 1: benninkUnbiasedSimulationNearClifford2017
  • lemma 2: Optimal Clifford decomposition of rotation gate
  • Theorem 1: Effect of Pauli insertions
  • Theorem 2: Rewrite of Theorem \ref{['thm:Pauli_insertion_theory']}
  • proof