Local decoders for fault-tolerant quantum computation and translation-invariant stabilizer codes

This paper presents the first fully spatially local fault-tolerant quantum-computing architecture based on topological codes in 2D, using geometrically local quantum and classical operations, bounded-speed communication, and constant resource density. Its core is a time-translation-invariant cellular-automaton decoder that drives defect clusters toward designated corners, where they annihilate, while linear defect and message erosion yield nonzero thresholds, stretched-exponential memory lifetimes, and polylogarithmic average decoding time. Translation-invariant streaming decoders reduce the classical overhead to poly(log log L) bits per site, while hierarchical coarse-grained decoders achieve constant density for toric and surface codes, including decoding during state preparation, state injection, lattice surgery, fold-transversal Hadamards, and readout. Numerically, the code-capacity toric decoder reaches a 6.3% threshold and extrapolates to 8.2% as its clock period grows, while the surface-code streaming decoder reaches approximately 1.1% under phenomenological noise; the repetition-code decoder is consistent with a 50% threshold. The architecture implements universal Clifford+T computation using lattice surgery, magic-state distillation, and Y-state distillation, with one constant-bandwidth input/output wire per logical qubit. A general proof based on Haah’s polynomial formalism and Gröbner-basis division establishes local decodability for every translation-invariant topological Pauli stabilizer code, including fracton models. The paper also reports using GPT 5.6 Sol, GPT 5.6, GPT 6, and Claude Fable 5 for literature review, revision, figures, and simulations.