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Quantum Software Tools Programming

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

arXiv
Authors: Tatsuya Tsurii, Naoharu Ito

Year

2026

Paper ID

2647

Status

Preprint

Abstract Read

~2 min

Abstract Words

111

Citations

N/A

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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  • This paper introduces a group-theoretic framework to analyze the algebraic structure of the Grover walk on a complete graph with self-loops.

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References & Citation Signals

[1] DOI https://doi.org/arXiv:2602.13686 [2] arXiv https://arxiv.org/abs/2602.13686 [3] Publisher https://arxiv.org/abs/2602.13686

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