Error analysis of quantum operators written as a linear combination of permutations

TL;DR

This paper investigates how matrices expressed as linear combinations of permutations respond to quantum bit-flip and phase-flip errors, revealing conditions under which their eigenvalues are resilient, with implications for quantum algorithm robustness.

Contribution

It provides a framework for analyzing eigenvalue perturbations in permutation-based matrices under quantum errors, aiding the design of error-resilient quantum algorithms.

Findings

Eigenvalues are resilient to bit-flip errors with positive coefficients.

Mixed-sign coefficients lead to less resilience but small perturbations at low error rates.

Block encoding with control registers affects error impact analysis.

Abstract

In this paper, we consider matrices given as a linear combination of permutations and analyze the impact of bit and phase-flips on the perturbation of the eigenvalues. When the coefficients in the linear combination are positive, we observe that the eigenvalues of the resulting matrices exhibit resilience to quantum bit-flip errors. In addition, we analyze the bit-flips in combination with positive and negative coefficients and the phase-flips. Although matrices with mixed-sign coefficients show less resilience to the bit-flip and phase-flip errors, the numerical evidence shows that the perturbation of the eigenspectrum is very small when the rate of these errors is small. We also discuss the situation when this matrix is implemented through block encoding and there is a control register. Since any square matrix can be expressed as a linear combination of permutations multiplied by two scaling matrices from the left and right (via Sinkhorn's theorem), this paper gives a framework to study matrix computations in quantum algorithms related to numerical linear algebra. In addition, it can give ideas to design more error-resilient algorithms that may involve quantum registers with different error characteristics.