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arXiv:2102.06056

Qubit-efficient exponential suppression of errors

Piotr Czarnik, Andrew Arrasmith, Łukasz Cincio, Patrick J. Coles

Published Feb 11, 2021• 14 citations
arXivHTMLPDF

Abstract

Achieving a practical advantage with near-term quantum computers hinges on having effective methods to suppress errors. Recent breakthroughs have introduced methods capable of exponentially suppressing errors by preparing multiple noisy copies of a state and virtually distilling a more purified version. Here we present an alternative method, the Resource-Efficient Quantum Error Suppression Technique (REQUEST), that adapts this breakthrough to much fewer qubits by making use of active qubit resets, a feature now available on commercial platforms. Our approach exploits a space/time trade-off to achieve a similar error reduction using only $2N+1$ qubits as opposed to $MN+1$ qubits, for $M$ copies of an $N$ qubit state. Additionally, we propose a method using near-Clifford circuits to find the optimal number of these copies in the presence of realistic noise, which limits this error suppression. We perform a numerical comparison between the original method and our qubit-efficient version with a realistic trapped-ion noise model. We find that REQUEST can reproduce the exponential suppression of errors of the virtual distillation approach, while out-performing virtual distillation when fewer than $3N+1$ qubits are available. Finally, we examine the scaling of the number of shots $N_S$ required for REQUEST to achieve useful corrections. We find that $N_S$ remains reasonable well into the quantum advantage regime where $N$ is hundreds of qubits.

DOI
10.48550/arxiv.2102.06056

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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.
Suppressing noise in physical systems is of fundamental importance.As quantum computers mature, quantum error correcting codes (QECs) will be adopted in order to suppress errors to any desired level.However in the noisy, intermediate-scale quantum (NISQ) era, the complexity and scale required to adopt even the smallest QEC is prohibitive: a single logical qubit needs to be encoded into many thousands of physical qubits.Here we show that, for the crucial case of estimating expectation values of observables (key to almost all NISQ algorithms) one can indeed achieve an effective exponential suppression.We take n independently prepared circuit outputs to create a state whose symmetries prevent errors from contributing bias to the expected value.The approach is very well suited for current and near-term quantum devices as it is modular in the main computation and requires only a shallow circuit that bridges the n copies immediately prior to measurement.Using no more than four circuit copies, we confirm error suppression below 10 -6 for circuits consisting of several hundred noisy gates (2-qubit gate error 0.5%) in numerical simulations validating our approach.
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.
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.
Error mitigation has elevated quantum computing to the scale of hundreds of qubits and tens of layers; however, yet larger scales (deeper circuits) are needed to fully exploit the potential of quantum computing to solve practical problems otherwise intractable. Here we demonstrate three key results that pave the way for the leap from quantum utility to quantum advantage: (1) we present a thorough derivation of random and systematic errors associated to the most advanced error mitigation strategies, including probabilistic error cancellation (PEC), zero noise extrapolation (ZNE) with probabilistic error amplification, and tensor-network error mitigation (TEM); (2) we prove that TEM (i) has the lowest sampling overhead among all three techniques under realistic noise, (ii) is optimal, in the sense that it saturates the universal lower cost bound for error mitigation, and (iii) is therefore the most promising approach to quantum advantage; (3) we propose a concrete notion of practical quantum advantage in terms of the universality of algorithms, stemming from the commercial need for a problem-independent quantum simulation device. We also establish a connection between error mitigation, relying on additional measurements, and error correction, relying on additional qubits, by demonstrating that TEM with a sufficient bond dimension works similarly to an error correcting code of distance 3. We foresee that the interplay and trade-off between the two resources will be the key to a smooth transition between error mitigation and error correction, and hence between near-term and fault-tolerant quantum computers. Meanwhile, we argue that quantum computing with optimal error mitigation, relying on modest classical computer power for tensor network contraction, has the potential to reach larger scales in accurate simulation than classical methods alone.