Szegedy's quantum walk with queries

TL;DR

This paper explores a new approach to Szegedy's quantum walk search algorithm using reflections around marked vertices, demonstrating potential for perfect success probability on complete graphs and improvements on other graph types.

Contribution

It introduces a novel method of implementing Szegedy's quantum walk with standard quantum queries, enhancing search success probabilities.

Findings

Achieves probability 1 of finding a marked vertex on complete graphs.

Numerical evidence suggests improved success on 2D grids.

Expresses Szegedy's search operator using standard query model for certain graphs.

Abstract

When searching for a marked vertex in a graph, Szegedy's usual search operator is defined by using the transition probability matrix of the random walk with absorbing barriers at the marked vertices. Instead of using this operator, we analyze searching with Szegedy's quantum walk by using reflections around the marked vertices, that is, the standard form of quantum query. We show we can boost the probability to 1 of finding a marked vertex in the complete graph. Numerical simulations suggests that the success probability can be improved for other graphs, like the two-dimensional grid. We also prove that, for a certain class of graphs, we can express Szegedy's search operator, obtained from the absorbing walk, using the standard query model.