From Betti Numbers to Persistence Diagrams: A Hybrid Quantum Algorithm for Topological Data Analysis

Quantum Physics arXiv:2512.02081 (quant-ph) [Submitted on 1 Dec 2025] Title:From Betti Numbers to Persistence Diagrams: A Hybrid Quantum Algorithm for Topological Data Analysis Authors:Dong Liu View a PDF of the paper titled From Betti Numbers to Persistence Diagrams: A Hybrid Quantum Algorithm for Topological Data Analysis, by Dong Liu View PDF HTML (experimental) Abstract:Persistence diagrams serve as a core tool in topological data analysis, playing a crucial role in pathological monitoring, drug discovery, and materials design. However, existing quantum topological algorithms, such as the LGZ algorithm, can only efficiently compute summary statistics like Betti numbers, failing to provide persistence diagram information that tracks the lifecycle of individual topological features, severely limiting their practical value. This paper proposes a novel quantum-classical hybrid algorithm that achieves, for the first time, the leap from "quantum computation of Betti numbers" to "quantum acquisition of practical persistence diagrams." The algorithm leverages the LGZ quantum algorithm as an efficient feature extractor, mining the harmonic form eigenvectors of the combinatorial Laplacian as well as Betti numbers, constructing specialized topological kernel functions to train a quantum support vector machine (QSVM), and learning the mapping from quantum topological features to persistence diagrams. The core contributions of this algorithm are: (1) elevating quantum topological computation from statistical summaries to pattern recognition, greatly expanding its application value; (2) obtaining more practical topological information in the form of persistence diagrams for real-world applications while maintaining the exponential speedup advantage of quantum computation; (3) proposing a novel hybrid paradigm of "classical precision guiding quantum efficiency." This method provides a feasible pathway for the practical implementation of quantum topological data analysis. Comments: Subjects: Quantum Physics (quant-ph); Machine Learning (cs.LG) Cite as: arXiv:2512.02081 [quant-ph] (or arXiv:2512.02081v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2512.02081 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Dong Liu [view email] [v1] Mon, 1 Dec 2025 00:40:29 UTC (11 KB) Full-text links: Access Paper: View a PDF of the paper titled From Betti Numbers to Persistence Diagrams: A Hybrid Quantum Algorithm for Topological Data Analysis, by Dong LiuView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2025-12 Change to browse by: cs cs.LG 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?)