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

Pegah Mohammadipour, Xiantao Li

TL;DR

This paper extends the analysis to polynomial least squares-based extrapolation, which mitigates measurement noise and avoids overfitting and proposes 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.

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.

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

TL;DR

This paper extends the analysis to polynomial least squares-based extrapolation, which mitigates measurement noise and avoids overfitting and proposes 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.

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 , 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.

Paper Structure

This paper contains 21 sections, 16 theorems, 127 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main Results and Assumptions
  3. Notations and Terminologies.
  4. Organization.
  5. Richardson's Extrapolation
  6. Zero-Noise Extrapolation via Least Squares
  7. Incorporating the Trotter Error
  8. Numerical Experiments
  9. The distribution of the nodes.
  10. Extrapolation using least-squares.
  11. The choice of the degree of the approximating polynomial.
  12. Extrapolation of Trotter error.
  13. Simultaneous mitigation of circuit and algorithmic errors.
  14. Summary and Discussions
  15. Acknowledgments
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\{x_j\}_{j = 0}^{n}$ be a set of equidistant nodes given by xj-unif, and let $p_n(x)$ be the degree-$n$ polynomial interpolating the function $f$ at these points. Under assumption eq:gevrey, for $0 < \varepsilon < e^{-e}$, if $M \le B^{-\frac{B}{B-1}}$ and $n = \Omega\bigl(\frac{\log(1/\varepsi

Figures (7)

  • Figure 1: A simple illustration of Richardson’s extrapolation for error mitigation. A polynomial $p_n(x)$ is constructed by interpolating $f(x)$ at the points $x_0, x_1, \dots, x_n$ and then evaluated at $x = 0$, corresponding to the zero-noise limit.
  • Figure 2: A simple illustration of Richardson's extrapolation method for quantum error mitigation using equidistant noise scales. The ideal zero-noise expectation value is $0.191826$. In comparison, the extrapolated zero-noise expectation value is $0.188129$.
  • Figure 3: Richardson extrapolation using equidistant and Chebyshev nodes. The ideal zero-noise expectation value is $0.191826$, while the extrapolated expectation values using equidistant nodes and Cheyshev nodes are $0.17516$, and $0.0244759$, respectively.
  • Figure 4: ZNE from the least-square approach \ref{['eq:least_sq_statement']} using equidistant and Chebyshev nodes. The extrapolated expectation values using equidistant nodes and Cheyshev nodes are $0.193625$ and $0.195975$, respectively, while the ideal expectation value is $0.191826$.
  • Figure 5: The change of extrapolation error with respect to the degree of the approximating polynomial in the least-square approach \ref{['eq:least_sq_statement']}.
  • ...and 2 more figures

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1: Importance Sampling
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • ...and 25 more