Periodicity of bipartite walk on biregular graphs with conditional spectra

TL;DR

This paper investigates the periodicity of bipartite quantum walks on biregular graphs, extending previous results to graphs with specific spectral properties and applying these findings to Grover's walks.

Contribution

It extends Kubota's characterization of periodicity from regular bipartite graphs to biregular graphs with certain algebraic integer eigenvalues.

Findings

Characterization of bipartite walk periodicity based on spectral properties.

Extension of periodicity criteria to biregular graphs with algebraic integer eigenvalues.

Application of bipartite walk results to Grover's walk on regular graphs.

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.