Beyond Penrose tensor diagrams with the ZX calculus: Applications to quantum computing, quantum machine learning, condensed matter physics, and quantum gravity

Quantum Physics arXiv:2511.06012 (quant-ph) [Submitted on 8 Nov 2025] Title:Beyond Penrose tensor diagrams with the ZX calculus: Applications to quantum computing, quantum machine learning, condensed matter physics, and quantum gravity Authors:Quanlong Wang, Richard D. P. East, Razin A. Shaikh, Lia Yeh, Boldizsár Poór, Bob Coecke View a PDF of the paper titled Beyond Penrose tensor diagrams with the ZX calculus: Applications to quantum computing, quantum machine learning, condensed matter physics, and quantum gravity, by Quanlong Wang and 5 other authors View PDF Abstract:We introduce the Spin-ZX calculus as an elevation of Penrose's diagrams and associated binor calculus to the level of a formal diagrammatic language. The power of doing so is illustrated by the variety of scientific areas we apply it to: permutational quantum computing, quantum machine learning, condensed matter physics, and quantum gravity. Respectively, we analyse permutational computing transition amplitudes, evaluate barren plateaus for SU(2) symmetric ansätze, study properties of AKLT states, and derive the minimum quantised volume in loop quantum gravity. Our starting point is the mixed-dimensional ZX calculus, a purely diagrammatic language that has been proven to be complete for finite-dimensional Hilbert spaces. That is, any equation that can be derived in the Hilbert space formalism, can also be derived in the mixed-dimensional ZX calculus. We embed the Spin-ZX calculus inside the mixed-dimensional ZX calculus, rendering it a quantum information flavoured diagrammatic language for the quantum theory of angular momentum, i.e. SU(2) representation theory. We diagrammatically derive the fundamental spin coupling objects - such as Clebsch-Gordan coefficients, symmetrising mappings between qubits and spin spaces, and spin Hamiltonians - under this embedding. Our results establish the Spin-ZX calculus as a powerful tool for representing and computing with SU(2) systems graphically, offering new insights into foundational relationships and paving the way for new diagrammatic algorithms for theoretical physics. Comments: Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2511.06012 [quant-ph] (or arXiv:2511.06012v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2511.06012 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Boldizsár Poór [view email] [v1] Sat, 8 Nov 2025 13:42:53 UTC (1,480 KB) Full-text links: Access Paper: View a PDF of the paper titled Beyond Penrose tensor diagrams with the ZX calculus: Applications to quantum computing, quantum machine learning, condensed matter physics, and quantum gravity, by Quanlong Wang and 5 other authorsView PDFTeX Source view license Current browse context: quant-ph new | recent | 2025-11 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?)