The average search probabilities of discrete-time quantum walks

TL;DR

This paper analyzes the average search probabilities of discrete-time quantum walks on various regular graphs, deriving formulas and asymptotic behaviors, notably showing that the probability approaches 1/4 for large valency in certain graph families.

Contribution

It provides a spectral analysis linking quantum walk search probabilities to graph adjacency matrices and derives asymptotic results for specific classes of regular graphs.

Findings

Average search probability approaches 1/4 for large valency in certain graph families.

Derived formulas relate search probabilities to adjacency matrices of augmented and subgraphs.

Established conditions under which the probability limit is approached based on graph parameters.

Abstract

We study the average probability that a discrete-time quantum walk finds a marked vertex on a graph. We first show that, for a regular graph, the spectrum of the transition matrix is determined by the weighted adjacency matrix of an augmented graph. We then consider the average search probability on a distance regular graph, and find a formula in terms of the adjacency matrix of its vertex-deleted subgraph. In particular, for any family of (1) complete graphs, or (2) strongly regular graphs, or (3) distance regular graphs of a fixed parameter $d$, varying valency $k$ and varying size $n$, such that $k^{d-1}/n$ vanishes as $k$ increases, the average search probability approaches $1/4$ as the valency goes to infinity. We also present a more relaxed criterion, in terms of the intersection array, for this limit to be approached by distance regular graphs.