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Maximum Likelihood Quantum Error Mitigation for Algorithms with a Single Correct Output

Dror Baron, Hrushikesh Pramod Patil, Huiyang Zhou

TL;DR

This work tackles scalable quantum error mitigation in the low-shot, high-noise regime by introducing qubit-wise majority vote (QMV). It proves that QMV is the maximum-likelihood estimator for quantum algorithms with a single correct output under symmetric i.i.d. noise, and it derives a shot bound of $S=O( rac{\ln(n)}{\epsilon^2})$ (with $\epsilon=0.5-p$), enabling logarithmic scaling in the number of qubits. The approach extends to asymmetric noise with weighted votes, adaptive measurement subsetting (AMS), and even multiple outputs via sliding-window strategies, and it is validated on real IBM devices where QMV often requires fewer shots than competing methods and can recover correct results even when not observed. The results demonstrate that exploiting problem structure (a single correct output) yields substantial practical gains in error mitigation, with clear implications for BV-type algorithms and larger random circuits on NISQ hardware.

Abstract

Quantum error mitigation is an important technique to reduce the impact of noise in quantum computers. With more and more qubits being supported on quantum computers, there are two emerging fundamental challenges. First, the number of shots required for quantum algorithms with large numbers of qubits needs to increase in order to obtain a meaningful distribution or expected value of an observable. Second, although steady progress has been made in improving the fidelity of each qubit, circuits with a large number of qubits are likely to produce erroneous results. This low-shot, high-noise regime calls for highly scalable error mitigation techniques. In this paper, we propose a simple and effective mitigation scheme, qubit-wise majority vote, for quantum algorithms with a single correct output. We show that our scheme produces the maximum likelihood (ML) estimate under certain assumptions, and bound the number of shots required. Our experimental results on real quantum devices confirm that our proposed approach requires fewer shots than existing ones, and can sometimes recover the correct answers even when they are not observed from the measurement results.

Maximum Likelihood Quantum Error Mitigation for Algorithms with a Single Correct Output

TL;DR

This work tackles scalable quantum error mitigation in the low-shot, high-noise regime by introducing qubit-wise majority vote (QMV). It proves that QMV is the maximum-likelihood estimator for quantum algorithms with a single correct output under symmetric i.i.d. noise, and it derives a shot bound of (with ), enabling logarithmic scaling in the number of qubits. The approach extends to asymmetric noise with weighted votes, adaptive measurement subsetting (AMS), and even multiple outputs via sliding-window strategies, and it is validated on real IBM devices where QMV often requires fewer shots than competing methods and can recover correct results even when not observed. The results demonstrate that exploiting problem structure (a single correct output) yields substantial practical gains in error mitigation, with clear implications for BV-type algorithms and larger random circuits on NISQ hardware.

Abstract

Quantum error mitigation is an important technique to reduce the impact of noise in quantum computers. With more and more qubits being supported on quantum computers, there are two emerging fundamental challenges. First, the number of shots required for quantum algorithms with large numbers of qubits needs to increase in order to obtain a meaningful distribution or expected value of an observable. Second, although steady progress has been made in improving the fidelity of each qubit, circuits with a large number of qubits are likely to produce erroneous results. This low-shot, high-noise regime calls for highly scalable error mitigation techniques. In this paper, we propose a simple and effective mitigation scheme, qubit-wise majority vote, for quantum algorithms with a single correct output. We show that our scheme produces the maximum likelihood (ML) estimate under certain assumptions, and bound the number of shots required. Our experimental results on real quantum devices confirm that our proposed approach requires fewer shots than existing ones, and can sometimes recover the correct answers even when they are not observed from the measurement results.
Paper Structure (21 sections, 1 theorem, 45 equations, 1 figure, 7 tables)

This paper contains 21 sections, 1 theorem, 45 equations, 1 figure, 7 tables.

Table of Contents

  1. Introduction
  2. Problem formulation and prior art
  3. Problem formulation
  4. Quantum error mitigation
  5. Estimation
  6. Maximum likelihood mitigation
  7. Qubit-wise majority vote (QMV)
  8. Reduction in number of shots
  9. Asymmetric noise
  10. Adaptive measurement subsetting
  11. Multiple correct outputs
  12. Derivations
  13. Proof of Theorem \ref{['th:main']}
  14. Number of shots for QMV
  15. Experimental results
  16. ...and 6 more sections

Key Result

Theorem 1

Consider a quantum algorithm that has a single correct output, $x_0\in{{\cal{X}}_n}$, where the measurements obtained by all shots are statistically independent, and the probabilities of flipping any entry, $i\in\{1,\ldots,n\}$, are symmetric and less than $0.5$, $\Pr(y_i=1|z_i=0)=\Pr(y_i=0|z_i=1)=p

Figures (1)

  • Figure 1: Example for the procedure of the proposed qubit-wise majority vote (QMV) and adaptive measurement subsetting (AMS) approaches on an example 4 qubit random circuit. For AMS the same example is repeated and the threshold is selected to be 0.01.

Theorems & Definitions (6)

  • Example 1
  • Theorem 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5