A group structure arising from Grover walks on complete graphs with self-loops and its application

TL;DR

This paper develops a group-theoretic framework to analyze the algebraic structure and symmetries of Grover walks on complete graphs with self-loops, revealing their periodic behavior.

Contribution

It introduces a novel algebraic approach using quotient groups to understand the structure and symmetries of Grover walks on complete graphs.

Findings

The quotient group is isomorphic to a finite cyclic group.

The group structure depends on the parity of the number of vertices.

The framework uncovers symmetries in the walk's time evolution.

Abstract

This paper introduces a group-theoretic framework to analyze the algebraic structure of the Grover walk on a complete graph with self-loops. We construct a group generated by the Grover matrix and a diagonal matrix whose entries are powers of a complex root of unity. We then characterize the resulting quotient group, which is defined using a subgroup formed by commutators involving these matrices. We show that this quotient group is isomorphic to a finite cyclic group whose structure depends on the parity of the number of vertices. This group-theoretic characterization reveals underlying symmetries in the time evolution of the Grover walk and provides an algebraic framework for understanding its periodic behavior.