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arXiv · 2211.02752

Periodicity of bipartite walk on biregular graphs with conditional spectra

Qiuting Chen
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Abstract

In this paper we study a class of discrete quantum walks, known as bipartite walks. These include the well-known Grover's walks. Any discrete quantum walk is given by the powers of a unitary matrix $U$ indexed by arcs or edges of the underlying graph. The walk is periodic if $U^k=I$ for some positive integer $k$. Kubota has given a characterization of periodicity of Grover's walk when the walk is defined on a regular bipartite graph with at most five eigenvalues. We extend Kubota's results--if a biregular graph $G$ has eigenvalues whose squares are algebraic integers with degree at most two, we characterize periodicity of the bipartite walk over $G$ in terms of its spectrum. We apply periodicity results of bipartite walks to get a characterization of periodicity of Grover's walk on regular graphs.

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Qiuting Chen. 2022-11-04. Periodicity of bipartite walk on biregular graphs with conditional spectra. https://arxiv.org/abs/2211.02752

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