Graphical Algebraic Geometry: From Ideals and Varieties to Quantum Calculi
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Quantum Physics arXiv:2605.13993 (quant-ph) [Submitted on 13 May 2026] Title:Graphical Algebraic Geometry: From Ideals and Varieties to Quantum Calculi Authors:Dichuan Gao, Razin A. Shaikh, Aleks Kissinger View a PDF of the paper titled Graphical Algebraic Geometry: From Ideals and Varieties to Quantum Calculi, by Dichuan Gao and 2 other authors View PDF Abstract:We introduce Graphical Algebraic Geometry (GAG), a family of diagrammatic languages extending the Graphical Linear Algebra programme. We construct several languages within this family and prove that they are universal and complete for the corresponding (co)span semantics of commutative algebras and affine varieties. This framework provides clear graphical representations of algebraic structures -- such as polynomials, ideals, and varieties -- enabling intuitive yet rigorous diagrammatic reasoning. We showcase two practical viewpoints on GAG. First, we show that instances of counting constraint satisfaction problem (#CSP) are recast as rewrite problems of closed diagrams in GAG. This means that deciding rewritability in GAG is #P-hard, and GAG can be viewed as a complete and compositional rewrite system for networks of polynomial constraints. Second, we characterize the qudit ZH calculus, a diagrammatic language for quantum computation, as an extension of Graphical Algebraic Geometry. This establishes the correspondence that Graphical Algebraic Geometry is to the ZH calculus what Graphical Linear Algebra is to the ZX calculus. Using this construction, we show that computing amplitudes in qudit ZH requires only a constant number of queries to a GAG oracle. Comments: Subjects: Quantum Physics (quant-ph); Logic in Computer Science (cs.LO); Category Theory (math.CT) Cite as: arXiv:2605.13993 [quant-ph] (or arXiv:2605.13993v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.13993 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Dichuan Gao [view email] [v1] Wed, 13 May 2026 18:05:02 UTC (345 KB) Full-text links: Access Paper: View a PDF of the paper titled Graphical Algebraic Geometry: From Ideals and Varieties to Quantum Calculi, by Dichuan Gao and 2 other authorsView PDFTeX Source view license Current browse context: quant-ph new | recent | 2026-05 Change to browse by: cs cs.LO math math.CT 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?)