Hitting Time of Quantum Walks with Perturbation

TL;DR

This paper analyzes how perturbations affect the hitting time in quantum walks, providing bounds and introducing concepts like delayed perturbed hitting time to understand their impact.

Contribution

It introduces bounds on the perturbed quantum walk hitting time and defines delayed perturbed hitting times, extending the understanding of quantum walk dynamics under perturbations.

Findings

Upper bounds for perturbed quantum hitting time derived

Delayed perturbed hitting time (DPQHT) is greater than a specific difference involving classical bounds

Perturbation effects on quantum walks are quantitatively characterized

Abstract

The hitting time is the required minimum time for a Markov chain-based walk (classical or quantum) to reach a target state in the state space. We investigate the effect of the perturbation on the hitting time of a quantum walk. We obtain an upper bound for the perturbed quantum walk hitting time by applying Szegedy's work and the perturbation bounds with Weyl's perturbation theorem on classical matrix. Based on the definition of quantum hitting time given in MNRS algorithm, we further compute the delayed perturbed hitting time (DPHT) and delayed perturbed quantum hitting time (DPQHT). We show that the upper bound for DPQHT is actually greater than the difference between the square root of the upper bound for a perturbed random walk and the square root of the lower bound for a random walk.