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arXiv:2310.10037 · Advanced Quantum Technologies

Purity‐Assisted Zero‐Noise Extrapolation for Quantum Error Mitigation

Tian‐Ren Jin, Yun‐Hao Shi, Zheng‐An Wang, Tianming Li, Kai Xu, Heng Fan

Published Sep 2, 2024• 2 citations
arXivHTMLPDF

Abstract

Abstract Quantum error mitigation aims to reduce errors in quantum systems and improve accuracy. Zero‐noise extrapolation (ZNE) is a commonly used method, where noise is amplified, and the target expectation is extrapolated to a noise‐free point. However, ZNE relies on assumptions about error rates based on the error model. In this study, a purity‐assisted zero‐noise extrapolation (pZNE) method is utilized to address limitations in error rate assumptions and enhance the extrapolation process. The pZNE is based on the Pauli diagonal error model implemented using the Pauli twirling technique. Although this method does not significantly reduce the bias of routine ZNE, it extends its effectiveness to a wider range of error rates where routine ZNE may face limitations. In addition, the practicality of the pZNE method is verified through numerical simulations and experiments on the online quantum computation platform, Quafu . Comparisons with routine ZNE and virtual distillation methods show that biases in extrapolation methods increase with error rates and may become divergent at high error rates. The bias of pZNE is slightly lower than routine ZNE, while its error rate threshold surpasses that of routine ZNE. Furthermore, for full density matrix information, the pZNE method is more efficient than the routine ZNE.

DOI
10.1002/qute.202400150

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Abstract Quantum error mitigation aims to reduce errors in quantum systems and improve accuracy. Zero‐noise extrapolation (ZNE) is a commonly used method, where noise is amplified, and the target expectation is extrapolated to a noise‐free point. However, ZNE relies on assumptions about error rates based on the error model. In this study, a purity‐assisted zero‐noise extrapolation (pZNE) method is utilized to address limitations in error rate assumptions and enhance the extrapolation process. The pZNE is based on the Pauli diagonal error model implemented using the Pauli twirling technique. Although this method does not significantly reduce the bias of routine ZNE, it extends its effectiveness to a wider range of error rates where routine ZNE may face limitations. In addition, the practicality of the pZNE method is verified through numerical simulations and experiments on the online quantum computation platform, Quafu . Comparisons with routine ZNE and virtual distillation methods show that biases in extrapolation methods increase with error rates and may become divergent at high error rates. The bias of pZNE is slightly lower than routine ZNE, while its error rate threshold surpasses that of routine ZNE. Furthermore, for full density matrix information, the pZNE method is more efficient than the routine ZNE.
Zero-noise extrapolation (ZNE) is a widely used quantum error mitigation technique that artificially amplifies circuit noise and then extrapolates the results to the noise-free circuit. A common ZNE approach is Richardson extrapolation, which relies on polynomial interpolation. Despite its simplicity, efficient implementations of Richardson extrapolation face several challenges, including approximation errors from the non-polynomial behavior of noise channels, overfitting due to polynomial interpolation, and exponentially amplified measurement noise. This paper provides a comprehensive analysis of these challenges, presenting bias and variance bounds that quantify approximation errors. Additionally, for any precision ε , our results offer an estimate of the necessary sample complexity. We further extend the analysis to polynomial least squares-based extrapolation, which mitigates measurement noise and avoids overfitting. Finally, we propose a strategy for simultaneously mitigating circuit and algorithmic errors in the Trotter-Suzuki algorithm by jointly scaling the time step size and the noise level. This strategy provides a practical tool to enhance the reliability of near-term quantum computations. We support our theoretical findings with numerical experiments.
Quantum computers in the NISQ era are prone to noise. A range of quantum error mitigation techniques has been proposed to address this issue. Zero-noise extrapolation (ZNE) stands out as a promising one. ZNE involves increasing the noise levels in a circuit and then using extrapolation to infer the zero noise case from the noisy results obtained. This paper presents a novel ZNE approach that does not require circuit folding or noise scaling to mitigate depolarizing and/or decoherence noise. To mitigate depolarizing noise, we propose leveraging the extreme/infinite noisy case, which allows us to avoid circuit folding. Specifically, the circuit output with extreme noise becomes the maximally mixed state. We show that using circuit-reliability metrics, simple linear extrapolation can effectively mitigate depolarizing noise. With decoherence noise, different states decay into the all-zero state at a rate that depends on the number of excited states and time. Therefore, we propose a state- and latency-aware exponential extrapolation that does not involve folding or scaling. When dealing with a quantum system affected by both decoherence and depolarizing noise, we propose to use our two mitigation techniques in sequence: first applying decoherence error mitigation, followed by depolarizing error mitigation. A common limitation of ZNE schemes is that if the circuit of interest suffers from high noise, scaling-up noise levels could not provide useful data for extrapolation. We propose using circuit-cut techniques to break a large quantum circuit into smaller sub-circuits to overcome this limitation. This way, the noise levels of the sub-circuits are lower than the original circuit, and ZNE can become more effective in mitigating their noises.
We present a simple, malleable and low-overhead approach for improving generic biased quantum error mitigation (QEM) methods, achieving up to 15% fidelity improvements over standard QEM on 100-qubit circuits with up to 2000 entangling gates. We do so by constructing verifiable benchmark circuits which mirror the application circuit's native-gate structure and thus noise profile. These circuits can be used to benchmark and mitigate the bias of the underlying error mitigation method, requiring only the application circuit and hardware native gate set. We present two methods for generating benchmark circuits; one is agnostic to the target hardware at the expense of a small overhead of single-qubit gates, while the other is specific to the IBM superconducting hardware and has no gate overhead. As a corollary, we introduce benchmarked-noise zero-noise extrapolation (bnZNE) as a simple adaptation of zero-noise extrapolation (ZNE), one of the most popular error mitigation methods. We consider as an example the bias-mitigated ZNE and bnZNE of Trotterized Hamiltonian simulations, observing that our approaches outperform standard ZNE using both small-scale classical simulations and 100-qubit utility-scale experiments on the IBM superconducting hardware. We consider the measurement of both single-site observables as well as two-site correlations along a one-dimensional qubit chain. We also provide a software package for implementing the error mitigation techniques used in this research.
Quantum error mitigation (QEM) and quantum error correction (QEC) are two research areas that are often considered as distinct entities, and the problem of combining the two approaches in a non-trivial way has only recently started to be explored. In this paper, we explore a paradigm at the intersection of the two, based on the error mitigation technique of Zero-Noise Extrapolation (ZNE), that uses the distance of an error correcting code as a noise parameter. This is distinct from some alternative approaches, as QEC is here used as a subroutine inside the QEM framework, while other proposals use QEM as a subroutine inside QEC experiments. Intuitively, we exploit the fact that a reduction in the physical noise level is analogous to an increase in the code distance, as both of them result in a decrease in the logical error rate. As such, the extrapolation to zero noise in the case of ZNE becomes comparable to the extrapolation to infinite distance in the case of this method. We describe how to calculate expectation values from a fault-tolerant computation, and we gain some analytical intuition for our ansatz choice. We explore the performance of the considered method to reduce the errors in a range of expectation values for a realistic circuit-level noise model and realistic device imperfections on the rotated surface code, and we particularly show that the performance of the method holds even in the case of non-stabiliser input states.
Zero-noise extrapolation (ZNE) is an increasingly popular technique for mitigating errors in noisy quantum computations without using additional quantum resources. We review the fundamentals of ZNE and propose several improvements to noise scaling and extrapolation, the two key components in the technique. We introduce unitary folding and parameterized noise scaling. These are digital noise scaling frameworks, i.e. one can apply them using only gate-level access common to most quantum instruction sets. We also study different extrapolation methods, including a new adaptive protocol that uses a statistical inference framework. Benchmarks of our techniques show error reductions of 18X to 24X over non-mitigated circuits and demonstrate ZNE's effectiveness at larger qubit numbers than have been tested previously. In addition to presenting new results, this work is a self-contained introduction to the practical use of ZNE by quantum programmers.
Recent hundred-qubit processors represent a significant hardware advancement, but current limitations prevent effective quantum error correction (QEC), necessitating reliance on quantum error mitigation (QEM) to enhance result fidelity from quantum computers. Our paper introduces a noise-aware folding technique that enhances Zero-Noise Extrapolation (ZNE) by leveraging the noise characteristics of target quantum hardware to fold circuits more efficiently. Unlike traditional ZNE approaches assuming uniform error distribution, our method redistributes noise using calibration data based on hardware noise models. By employing a noise-adaptive compilation method combined with our proposed folding mechanism, we enhance the ZNE accuracy of quantum gate-based computing using superconducting quantum computers. This paper highlights the uniqueness of our method, summarizes noise accumulation, presents the scaling algorithm, and compares the reliability of our method with those of existing models using linear extrapolation model. Experimental results show that compared to existing folding methods, our approach achieved up to a 23.6% improvement on quantum computer simulators and up to a 44.8% improvement on real quantum computers, with statistical confidence intervals indicating stable performance gains across circuit scales.