Limitation of Quantum Walk Approach to the Maximum Matching Problem

Quantum Physics arXiv:2510.26246 (quant-ph) [Submitted on 30 Oct 2025] Title:Limitation of Quantum Walk Approach to the Maximum Matching Problem Authors:Alcides Gomes Andrade Júnior, Akira Matsubayashi View a PDF of the paper titled Limitation of Quantum Walk Approach to the Maximum Matching Problem, by Alcides Gomes Andrade J\'unior and 1 other authors View PDF HTML (experimental) Abstract:The Maximum Matching problem has a quantum query complexity lower bound of $\Omega(n^{3/2})$ for graphs on $n$ vertices represented by an adjacency matrix. The current best quantum algorithm has the query complexity $O(n^{7/4})$, which is an improvement over the trivial bound $O(n^2)$. Constructing a quantum algorithm for this problem with a query complexity improving the upper bound $O(n^{7/4})$ is an open problem. The quantum walk technique is a general framework for constructing quantum algorithms by transforming a classical random walk search into a quantum search, and has been successfully applied to constructing an algorithm with a tight query complexity for another problem. In this work we show that the quantum walk technique fails to produce a fast algorithm improving the known (or even the trivial) upper bound on the query complexity. Specifically, if a quantum walk algorithm designed with the known technique solves the Maximum Matching problem using $O(n^{2-\epsilon})$ queries with any constant $\epsilon>0$, and if the underlying classical random walk is independent of an input graph, then the guaranteed time complexity is larger than any polynomial of $n$. Comments: Subjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC) ACM classes: F.2.3 Cite as: arXiv:2510.26246 [quant-ph] (or arXiv:2510.26246v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2510.26246 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Akira Matsubayashi [view email] [v1] Thu, 30 Oct 2025 08:29:44 UTC (17 KB) Full-text links: Access Paper: View a PDF of the paper titled Limitation of Quantum Walk Approach to the Maximum Matching Problem, by Alcides Gomes Andrade J\'unior and 1 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-10 Change to browse by: cs cs.CC 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?)