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arXiv · 2602.13686

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

Tatsuya Tsurii·Naoharu Ito
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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.

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Tatsuya Tsurii, Naoharu Ito. 2026-02-14. A group structure arising from Grover walks on complete graphs with self-loops and its application. https://doi.org/10.1016/j.disc.2026.115036

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