Quantum Hitting Time according to a given distribution

TL;DR

This paper investigates quantum hitting time in Szegedy quantum walks, proving quadratic speedup over classical hitting time under certain conditions, and explores using general distributions instead of stationary ones.

Contribution

It provides a detailed proof of quadratic speedup for time-reversible walks and examines the impact of using general distributions on quantum hitting time.

Findings

Quadratic speedup established for time-reversible Szegedy walks.

Using general distributions affects quantum hitting time, supported by numerical experiments.

The proof is presented in a linear algebra framework accessible to the community.

Abstract

In this work we focus on the notion of quantum hitting time for discrete-time Szegedy quantum walks, compared to its classical counterpart. Under suitable hypotheses, quantum hitting time is known to be of the order of the square root of classical hitting time: this quadratic speedup is a remarkable example of the computational advantages associated with quantum approaches. Our purpose here is twofold. On one hand, we provide a detailed proof of quadratic speedup for time-reversible walks within the Szegedy framework, in a language that should be familiar to the linear algebra community. Moreover, we explore the use of a general distribution in place of the stationary distribution in the definition of quantum hitting time, through theoretical considerations and numerical experiments.