A diagrammatic field theory of quantum error correction

• URL: http://arxiv.org/abs/2607.08911v1
• Date: Thu, 09 Jul 2026 20:10:13 GMT
• Title: A diagrammatic field theory of quantum error correction
• Abstract summary: We develop a framework for quantum error correction centred on fusionspace codes in unitary fusion categories.<n>We compute geometry-dependent Ising four-point weights for growing code families.<n>We also formulate conformal-block likelihood data and compute geometry-dependent Ising four-point weights.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $σ$ punctures show that pair-charge footprints can be complementary logical diagnostics and realize an exact one-qubit Clifford shadow. A proper six-$σ$ 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 $\Pr_L(\mathrm{fail})\le C|Ω_L|e^{-cL}$ 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.

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