The average search probabilities of discrete-time quantum walks

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 vertexdeleted subgraph. In particular, for any family of • complete graphs, or• strongly regular graphs, or• 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.