Regular graphs to induce even periodic Grover walks

TL;DR

This paper characterizes regular graphs that induce even-periodic Grover walks, revealing that most such graphs are cycle graphs when the period is twice an odd integer, using Galois theory for the proof.

Contribution

It extends the characterization of graphs inducing periodic Grover walks to even periods, especially for regular graphs with periods of the form 2l where l is odd.

Findings

Most regular graphs inducing 2l-periodic Grover walks are cycle graphs.

The proof employs Galois theory to establish the characterization.

Provides insight into the structure of graphs related to quantum walk periodicity.

Abstract

The interest of this paper is a characterization of graphs that induce periodic Grover walks with given periods. In previous studies, Yoshie has shown that the only graphs that induce odd periodic Grover walks are cycle graphs. However, this problem is largely unsolved for even periods. In this study, we show that regular graphs that induce $2l$-periodic Grover walks are also cycle graphs in most cases, where $l$ is an odd integer. The proof uses Galois theory.