The average search probabilities of discrete-time quantum walks

• URL: http://arxiv.org/abs/2108.09818v1
• Date: Sun, 22 Aug 2021 18:53:22 GMT
• Title: The average search probabilities of discrete-time quantum walks
• Authors: Hanmeng Zhan
• Abstract summary: 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 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 $n$, the average search probability approaches $1/4$ as the valency goes to infinity.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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