Direct Analysis of Zero-Noise Extrapolation: Polynomial Methods, Error Bounds, and Simultaneous Physical-Algorithmic Error Mitigation

TL;DR

This paper analyzes zero-noise extrapolation in quantum computing, providing bounds on errors, exploring polynomial methods, and proposing strategies for simultaneous error mitigation to improve near-term quantum algorithms.

Contribution

It offers a comprehensive theoretical analysis of polynomial-based zero-noise extrapolation, including error bounds and sample complexity, and introduces a joint error mitigation strategy for quantum algorithms.

Findings

Bias and variance bounds for polynomial extrapolation

Sample complexity estimates for desired precision

A practical strategy for simultaneous circuit and algorithmic error mitigation

Abstract

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 $\varepsilon$, 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.