Sensitivity of Quantum Walks with Perturbation

TL;DR

This paper analyzes how perturbations affect the hitting time and stationary distribution of quantum walks, providing bounds on their deviations due to noise, which is crucial for quantum algorithm robustness.

Contribution

It introduces bounds on the impact of perturbations on quantum walk hitting times and stationary distributions using Szegedy's work and classical matrix perturbation theory.

Findings

Bounded the perturbed quantum walk hitting time.

Established an upper bound for the total variation distance.

Applied classical perturbation bounds to quantum sampling.

Abstract

Quantum computers are susceptible to noises from the outside world. We investigate the effect of perturbation on the hitting time of a quantum walk and the stationary distribution prepared by a quantum walk based algorithm. The perturbation comes from quantizing a transition matrix Q with perturbation E (errors). We bound the perturbed quantum walk hitting time from above by applying Szegedy's work and the perturbation bounds with Weyl's perturbation theorem on classical matrix. Based on an efficient quantum sample preparation approach invented in {\em speed-up via quantum sampling} and the perturbation bounds for stationary distribution for classical matrix, we find an upper bound for the total variation distance between the prepared quantum sample and the true quantum sample.