A diagrammatic field theory of quantum error correction

Quantum Physics arXiv:2607.08911 (quant-ph) [Submitted on 9 Jul 2026] Title:A diagrammatic field theory of quantum error correction Authors:Steven Rayan View a PDF of the paper titled A diagrammatic field theory of quantum error correction, by Steven Rayan View PDF HTML (experimental) Abstract:We develop a field-theoretic framework for quantum error correction centred on fusion-space codes in unitary fusion categories. Admissible clusters determine total-charge sectors and orthogonal footprint projectors recording locally visible data left by error histories. The central distinction is between diagnostic footprint algebras and syndrome-admissible commuting algebras: the latter can be measured without revealing logical information and resolve chosen error representatives into measured sectors. For such algebras, exact correctability is equivalent to fibrewise Knill--Laflamme conditions, yielding a measure-then-recover factorization. Under a contractible-vacuum locality hypothesis, closed neutral composites give a categorical sufficient criterion for scalar action on the code. In the Ising theory, four $\sigma$ punctures show that pair-charge footprints can be complementary logical diagnostics and realize an exact one-qubit Clifford shadow. A proper six-$\sigma$ code instead admits a syndrome-admissible pair-charge measurement and exact recovery from an explicit Majorana bilinear error. A second bilinear has the same measured footprint but differs by a logical bit flip, producing a concrete nontrivial footprint fibre and genuine decoding ambiguity. We also formulate conformal-block likelihood data and compute geometry-dependent Ising four-point weights. For growing code families, we prove a conditional Peierls-type threshold theorem: bounded connected-region growth, local stochastic noise, local neutralizability of small residual components, and componentwise decoder balance imply $\Pr_L(\mathrm{fail})\le C|\Omega_L|e^{-cL}$ below a nonzero constant error rate. We conclude with representation-theoretic and algebro-geometric directions involving tube and Hopf algebras, Yangian-type structures, Higgs bundles, spectral curves, Jacobians, and abelian varieties. Comments: Subjects: Quantum Physics (quant-ph); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Algebraic Geometry (math.AG); Representation Theory (math.RT) MSC classes: 81P73, 18M20, 57K16, 81T45, 81T40, 18M30, 81P70 Cite as: arXiv:2607.08911 [quant-ph] (or arXiv:2607.08911v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2607.08911 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Steven Rayan [view email] [v1] Thu, 9 Jul 2026 20:10:13 UTC (82 KB) Full-text links: Access Paper: View a PDF of the paper titled A diagrammatic field theory of quantum error correction, by Steven RayanView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-07 Change to browse by: hep-th math math-ph math.AG math.MP math.RT 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?)