Categorical Tensor-Graph Semantics for Quantum Algorithms

Quantum Physics arXiv:2607.12128 (quant-ph) [Submitted on 13 Jul 2026] Title:Categorical Tensor-Graph Semantics for Quantum Algorithms Authors:Naihong Hu, Ruining Li, Futao Wang View a PDF of the paper titled Categorical Tensor-Graph Semantics for Quantum Algorithms, by Naihong Hu and 2 other authors View PDF Abstract:This paper systematically investigates quantum computing protocols from the intuitive perspective of categorical tensor-graph semantics within the category FHilb. While conventional Hilbert-space formalisms conceal the structural nature of quantum algorithms behind high-dimensional matrix operations, the topological framework developed herein directly encodes algorithmic functionalities to their graphical skeletons. We provide a comprehensive topological reinterpretation of the Bernstein-Vazirani and the Simon algorithms, demonstrating how topological transformations distill their core mathematical essence and clarify the operational mechanism of oracles. Going beyond standard qubit models, we formalize the qutrit-adapted generalized Deutsch-Jozsa algorithm as well as the generalized single-shot Grover algorithm, explicitly laying out their graphical representations. We further implement CNOT gates via complementary Frobenius structures, trace the diagrammatic genesis of quantum entanglement including Bell and GHZ states, and present a diagrammatic simplification for W-state preparation protocol. By bridging tensor category theory with practical quantum algorithmic design, this work furnishes composable, scalable diagrammatic toolkit essential for automated circuit optimization across the evolving quantum hardware ecosystem. Comments: Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Category Theory (math.CT) MSC classes: Primary 18M05, 18M15, 18M30, Secondary 68Q12, 05E16 Cite as: arXiv:2607.12128 [quant-ph] (or arXiv:2607.12128v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2607.12128 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Naihong Hu [view email] [v1] Mon, 13 Jul 2026 20:21:14 UTC (6,710 KB) Full-text links: Access Paper: View a PDF of the paper titled Categorical Tensor-Graph Semantics for Quantum Algorithms, by Naihong Hu and 2 other authorsView PDFTeX Source view license Current browse context: quant-ph new | recent | 2026-07 Change to browse by: math math-ph math.CT 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?) 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?)