Periodicity of bipartite walks on certain graphs and its connections to periodicity of Grover's walk - Qiuting (Tina) Chen

Title:Periodicity of bipartite walks on certain graphs and its connections to periodicity of Grover's walk

 Abstract: This talk will focus on 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. The talk is based on the following paper https://arxiv.org/abs/2211.02752 .