Effect of matrix sparsity and quantum noise on quantum random walk linear solvers

TL;DR

This paper investigates how quantum noise impacts the accuracy of quantum random walk-based solvers for sparse linear systems, revealing noise-induced error increases and proposing a revised algorithm to improve results.

Contribution

It identifies the detrimental effect of quantum noise on sparse matrix solutions and introduces a new algorithm to mitigate invalid quantum random walks, enhancing solver accuracy.

Findings

Quantum noise increases error in sparse matrix solutions.

Invalid quantum random walks cause accuracy degradation.

Revised algorithm improves solution accuracy under noise.

Abstract

We study the effects of quantum noise in hybrid quantum-classical solver for sparse systems of linear equations using quantum random walks, applied to stoquastic Hamiltonian matrices. In an ideal noiseless quantum computer, sparse matrices achieve solution vectors with lower relative error than dense matrices. However, we find quantum noise reverses this effect, with overall error increasing as sparsity increases. We identify invalid quantum random walks as the cause of this increased error and propose a revised linear solver algorithm which improves accuracy by mitigating these invalid walks.