{"schema":"pubphys.bundle/1","record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"2b5d4ad924cec8a2006b84405f338d85183a4c79b3639dfb6bec1a5eaed80dbf","created":"2026-10-03T07:18:09Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["5475cf5dc5b1a72283d26dc95dc6e00d7ea4e00096a7c6492e409a1816eeb011"],"salt":"eb4ee98321192bfaa8a5e297a7983d2fe685f98d56b6f7a672f1d12be802308f","schema":"pubphys.record/2","site":"pubphys.com","target":null,"type":"revision"},"content":{"answer_type":"value","assisted_by":[],"external_id":"qi.fault-tolerance-noise.surface-code-hashing-bound","kind":"well-posed","literature_status":"open","n":"1","parents":[],"plain":"With a perfect decoder (the classical routine that infers which errors occurred), the toric code tolerates a certain depolarizing error rate, and numerics put it at about 18.9%. Whether this equals exactly the hashing bound, a limit from information theory with the same value to three digits, is unknown.","posed_since":"2012","precise":"For the toric code under independent depolarizing noise of strength p with perfect syndrome measurement and maximum-likelihood decoding, the threshold $p_c$ equals the Nishimori-line critical point of a disordered eight-vertex (coupled Ising) model. Decide whether $p_c$ equals the hashing value $p_h \\sim 0.1893$ solving $1 - H(p_h) - p_h \\operatorname{log2} 3 = 0$ (H the binary entropy), or compute p_c to a precision that separates them; Bombin et al. found $p_c = 0.189(3)$ (Phys. Rev. X 2, 021004, 2012).","problem_ref":null,"references":"","settled_by":"High-precision Monte Carlo or tensor-network evaluation of the Nishimori critical point with error below $10^{-4}$, or an exact duality argument fixing $p_{c}$.","status_note":"","title":"Does the optimal toric-code depolarizing threshold equal the hashing 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