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

• URL: http://arxiv.org/abs/2602.13686v1
• Date: Sat, 14 Feb 2026 09:12:46 GMT
• Title: A group structure arising from Grover walks on complete graphs with self-loops and its application
• Abstract summary: Group-theoretic characterization reveals underlying symmetries in the time evolution of the Grover walk.<n>This paper introduces a group-theoretic framework to analyze the algebraic structure of the Grover walk on a complete graph with self-loops.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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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