[Paper Review] Quantum Error Mitigation

Zhenyu Cai, Ryan Babbush+6 more|arXiv (Cornell University)|Oct 3, 2022

Quantum Computing Algorithms and Architecture29 citations

TL;DR

This review surveys quantum error mitigation (QEM) methods for NISQ devices, analyzes their principled efficacy, hardware demonstrations, and trade-offs, and discusses open challenges and prospects for achieving quantum advantage with mitigation.

ABSTRACT

For quantum computers to successfully solve real-world problems, it is necessary to tackle the challenge of noise: the errors which occur in elementary physical components due to unwanted or imperfect interactions. The theory of quantum fault tolerance can provide an answer in the long term, but in the coming era of `NISQ' machines we must seek to mitigate errors rather than completely remove them. This review surveys the diverse methods that have been proposed for quantum error mitigation, assesses their in-principle efficacy, and then describes the hardware demonstrations achieved to date. We identify the commonalities and limitations among the methods, noting how mitigation methods can be chosen according to the primary type of noise present, including algorithmic errors. Open problems in the field are identified and we discuss the prospects for realising mitigation-based devices that can deliver quantum advantage with an impact on science and business.

Motivation & Objective

• Motivate the need to tackle noise in near-term quantum devices before full fault tolerance.
• Define and formalize what constitutes quantum error mitigation (QEM) and its targets.
• Assess the in-principle efficacy and practical overheads of QEM methods.
• Survey hardware demonstrations and benchmarking of QEM methods.
• Identify open problems and discuss the potential for mitigation-based quantum advantage.

Proposed method

• Define primary circuit and noisy output states; introduce estimators for observable O on rho0.
• Discuss bias-variance decomposition of error-mitigation estimators and the sampling overhead concept.
• Present and compare multiple QEM techniques (e.g., zero-noise extrapolation, probabilistic error cancellation, measurement error mitigation, symmetry/purity constraints, subspace expansions, N-representability, learning-based approaches).
• Outline how overhead and bias scale with circuit fault rate and experimental conditions.
• Describe metrics for comparing QEM methods, including calibration overhead, mean-square-error, and combinations of methods.

Experimental results

Research questions

• RQ1 How can QEM reduce bias in expectation values while accounting for increased variance and sampling overhead?
• RQ2 How do different QEM methods perform under varying noise types and circuit fault rates?
• RQ3 What are the practical limits (overhead, scalability) for applying QEM on NISQ devices?
• RQ4 How do ZNE, PEC, and other methods compare in calibration requirements and robustness across platforms?
• RQ5 What open problems remain to enable mitigation-based quantum advantage?

Key findings

• QEM reduces bias in estimators but typically increases variance, leading to a bias-variance trade-off.
• The sampling overhead of QEM scales exponentially with circuit fault rate in the worst cases, limiting practicality for large or highly noisy circuits.
• Zero-noise extrapolation is simple and widely implemented, with various extrapolation strategies (polynomial, exponential, Richardson) discussed.
• Probabilistic error cancellation can, in principle, remove bias completely but incurs exponential sampling overhead.
• Measurement error mitigation and other constraints (symmetry, purity, subspace, N-representability) offer alternative routes with different overheads and applicability.
• The framework emphasizes choosing mitigation strategies based on the dominant noise type and circuit structure, and notes open problems and practical integration with other error suppression methods.