A group structure arising from Grover walks on complete graphs with self-loops and its application
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908834611593216 |
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| author | Tsurii, Tatsuya Ito, Naoharu |
| author_facet | Tsurii, Tatsuya Ito, Naoharu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13686 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A group structure arising from Grover walks on complete graphs with self-loops and its application Tsurii, Tatsuya Ito, Naoharu Quantum Physics Discrete Mathematics Mathematical Physics Group Theory 05C25, 15A30, 81Q99, 39A23 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. |
| title | A group structure arising from Grover walks on complete graphs with self-loops and its application |
| topic | Quantum Physics Discrete Mathematics Mathematical Physics Group Theory 05C25, 15A30, 81Q99, 39A23 |
| url | https://arxiv.org/abs/2602.13686 |