A group structure arising from Grover walks on complete graphs with self-loops and its application
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Quantum Physics arXiv:2602.13686 (quant-ph) [Submitted on 14 Feb 2026] Title:A group structure arising from Grover walks on complete graphs with self-loops and its application Authors:Tatsuya Tsurii, Naoharu Ito View a PDF of the paper titled A group structure arising from Grover walks on complete graphs with self-loops and its application, by Tatsuya Tsurii and 1 other authors View PDF HTML (experimental) 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. Comments: Subjects: Quantum Physics (quant-ph); Discrete Mathematics (cs.DM); Mathematical Physics (math-ph); Group Theory (math.GR) MSC classes: 05C25, 15A30, 81Q99, 39A23 Cite as: arXiv:2602.13686 [quant-ph] (or arXiv:2602.13686v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2602.13686 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Journal reference: Discrete Mathematics, Volume 349(7), 2026 Related DOI: https://doi.org/10.1016/j.disc.2026.115036 Focus to learn more DOI(s) linking to related resources Submission history From: Tatsuya Tsurii [view email] [v1] Sat, 14 Feb 2026 09:12:46 UTC (13 KB) Full-text links: Access Paper: View a PDF of the paper titled A group structure arising from Grover walks on complete graphs with self-loops and its application, by Tatsuya Tsurii and 1 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-02 Change to browse by: cs cs.DM math math-ph math.GR math.MP References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) Links to Code Toggle Papers with Code (What is Papers with Code?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)