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arXiv:2311.08183 · Communications Physics

Circuit-noise-resilient virtual distillation

Xiao-Yue Xu, Chen Ding, Shuo Zhang, Wan‐Su Bao, He-Liang Huang

Published Oct 9, 2024• 1 citation
arXivHTMLPDF

Abstract

Abstract Quantum error mitigation (QEM) is crucial for near-term quantum devices, as noise inherently exists in physical quantum systems and undermines the accuracy of quantum algorithms. A typical QEM method, called Virtual Distillation (VD), aims to mitigate errors for expectation values of observables and achieve effective exponential suppression using multiple copies of the noisy state. However, imperfect VD circuit implementation may yield negative mitigation outcomes, potentially more severe than those achieved without QEM. To address this, we introduce Circuit-Noise-Resilient Virtual Distillation (CNR-VD). This method, featuring a calibration procedure that utilizes easily-prepared input states, refines the outcomes of VD when its circuit is contaminated by noise, seeking to recover the results of an ideally conducted VD circuit. Simulation results demonstrate that the CNR-VD estimator effectively reduces deviations induced by noise in the VD circuit, showcasing improvements in accuracy by an order of magnitude at most compared to the original VD. Meanwhile, CNR-VD enables positive mitigation effects, even in the presence of higher noise levels where the original VD performs worse than the unmitigated estimation. The proposed CNR-VD significantly enhances the noise-resilience of VD, and thus is anticipated to elevate the performance of quantum algorithm implementations on near-term quantum devices.

DOI
10.1038/s42005-024-01815-2

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Abstract Quantum error mitigation (QEM) is crucial for near-term quantum devices, as noise inherently exists in physical quantum systems and undermines the accuracy of quantum algorithms. A typical QEM method, called Virtual Distillation (VD), aims to mitigate errors for expectation values of observables and achieve effective exponential suppression using multiple copies of the noisy state. However, imperfect VD circuit implementation may yield negative mitigation outcomes, potentially more severe than those achieved without QEM. To address this, we introduce Circuit-Noise-Resilient Virtual Distillation (CNR-VD). This method, featuring a calibration procedure that utilizes easily-prepared input states, refines the outcomes of VD when its circuit is contaminated by noise, seeking to recover the results of an ideally conducted VD circuit. Simulation results demonstrate that the CNR-VD estimator effectively reduces deviations induced by noise in the VD circuit, showcasing improvements in accuracy by an order of magnitude at most compared to the original VD. Meanwhile, CNR-VD enables positive mitigation effects, even in the presence of higher noise levels where the original VD performs worse than the unmitigated estimation. The proposed CNR-VD significantly enhances the noise-resilience of VD, and thus is anticipated to elevate the performance of quantum algorithm implementations on near-term quantum devices.
Virtual distillation is a technique that aims to mitigate errors in noisy quantum computers. It works by preparing multiple copies of a noisy quantum state, bridging them through a circuit, and conducting measurements. As the number of copies increases, this process allows for the estimation of the expectation value with respect to a state that approaches the ideal pure state rapidly. However, virtual distillation faces a challenge in realistic scenarios: preparing multiple copies of a quantum state and bridging them through a circuit in a noisy quantum computer will significantly increase the circuit size and introduce excessive noise, which will degrade the performance of virtual distillation. To overcome this challenge, we propose an error mitigation strategy that uses circuit-cutting technology to cut the entire circuit into fragments. With this approach, the fragments responsible for generating the noisy quantum state can be executed on a noisy quantum device, while the remaining fragments are efficiently simulated on a noiseless classical simulator. By running each fragment circuit separately on quantum and classical devices and recombining their results, we can reduce the noise accumulation and enhance the effectiveness of the virtual distillation technique. Our strategy has good scalability in terms of both runtime and computational resources. We demonstrate our strategy’s effectiveness through noisy simulation and experiments on a real quantum device.
Virtual distillation has been proposed as an error mitigation protocol for estimating the expectation values of observables in quantum algorithms. It proceeds by creating a cyclic permutation of M noisy copies of a quantum state using a sequence of controlled-swap gates. If the noise does not shift the dominant eigenvector of the density operator away from the ideal state, then the error in expectation-value estimation can be exponentially reduced with M . In practice, subsequent error mitigation techniques are required to suppress the effect of noise in the cyclic permutation circuit itself, leading to increased experimental complexity. Here, we perform a careful analysis of the effect of uncorrelated, identical noise in the cyclic permutation circuit and find that the estimation of expectation value of observables are robust against dephasing noise. We support the analytical result with numerical simulations and find that 67% of errors are reduced for M=2 , with physical dephasing error probabilities as high as 10% . Our results imply that a broad class of quantum algorithms can be implemented with higher accuracy in the near-term with qubit platforms where non-dephasing errors are suppressed, such as superconducting bosonic qubits and Rydberg atoms.
Virtual distillation is a promising error-mitigation technique that exploits multiple copies of a noisy quantum state to estimate observables as if measured on a purified state. Although originally introduced in the context of bosonic many-body systems under the name of virtual cooling, its development and applications have largely focused on qubit-based quantum computation. Here, we establish a framework for virtual distillation in bosonic quantum information processing and continuous-variable quantum computing. Building on a diagonalization of cyclic shift operators implemented with passive linear-optical interferometers, we derive experimentally accessible protocols for estimating virtually distilled expectation values of observables relevant to bosonic architectures. In particular, we show how to recover noise-mitigated expectation values of number operators, phase-shift operators, and arbitrary quadratures from multi-copy measurements. For number operators, we further demonstrate the estimation of virtually distilled correlators of arbitrary order through the characteristic function of the photon-number distribution. We apply the framework to states affected by photon loss and dephasing, two of the dominant noise mechanisms in bosonic quantum computation, and quantify the resulting suppression of noise contributions. Our results extend virtual distillation beyond its original setting and provide a practical route toward error-mitigated measurements in bosonic quantum processors using experimentally available linear-optical resources.
Contemporary quantum computers have relatively high levels of noise, making it difficult to use them to perform useful calculations, even with a large number of qubits.Quantum error correction is expected to eventually enable fault-tolerant quantum computation at large scales, but until then, it will be necessary to use alternative strategies to mitigate the impact of errors.We propose a near-term friendly strategy to mitigate errors by entangling and measuring M copies of a noisy state ρ.This enables us to estimate expectation values with respect to a state with dramatically reduced error ρ M =Trðρ M Þ without explicitly preparing it, hence the name "virtual distillation."As M increases, this state approaches the closest pure state to ρ exponentially quickly.We analyze the effectiveness of virtual distillation and find that it is governed in many regimes by the behavior of this pure state (corresponding to the dominant eigenvector of ρ).We numerically demonstrate that virtual distillation is capable of suppressing errors by multiple orders of magnitude and explain how this effect is enhanced as the system size grows.Finally, we show that this technique can improve the convergence of randomized quantum algorithms, even in the absence of device noise.
Shadow estimation is a method for deducing numerous properties of an unknown quantum state through a limited set of measurements, which suffers from noises in quantum devices. In this paper, we introduce an error-mitigated shadow estimation approach based on virtual distillation, tailored for applications in near-term quantum devices. Our methodology leverages the qubit reset technique, thereby reducing the associated qubit overhead. Crucially, our approach ensures that the required qubit resources remain independent of the desired accuracy and avoid an exponential measurement overhead, marking a substantial advancement in practical applications. Furthermore, our technique accommodates a mixed Clifford and Pauli-type shadow, which can result in a reduction in the number of required measurements across various scenarios. We also study the trade-off between circuit depth and measurement overhead quantitatively. Through numerical simulations, we substantiate the efficacy of our error mitigation method, establishing its utility in enhancing the robustness of shadow estimations on near-term quantum devices.
In contrast to monolithic devices, modular, networked quantum architectures are based on interconnecting smaller quantum hardware nodes using quantum communication links and offer a promising approach to scalability. Virtual distillation (VD) is a technique that can, under ideal conditions, suppress errors exponentially as the number of quantum state copies increases. However, additional gate operations required for VD introduce further errors, which may limit its practical effectiveness. In this work, we analyze three practical implementations of VD that correspond to edge cases that maximize space-time trade-offs. Specifically, we consider an implementation that minimizes the number of qubits but introduces significantly deeper quantum circuits and contrast it with implementations that parallelize the preparation of copies using additional qubits, including a constant-depth implementation. We rigorously characterize their circuit depth and gate count requirements and develop explicit architectures for implementing them in networked quantum systems—while also detailing implementations in early fault-tolerant quantum architectures. We numerically compare the performance of the three implementations under realistic noise characteristics of networked ion trap systems and conclude the following. First, VD effectively suppresses errors even for very noisy states. Second, the constant-depth implementation consistently outperforms the implementation that minimizes the number of qubits. Finally, the approach is highly robust to errors in remote entangling operations, with noise in local gates being the main limiting factor to its performance.