{"record":{"author":{"account_ref":null,"orcid":null},"builds_on":[],"content_schema":"pubphys.content.revision/1","content_sha256":"98405dacf250df5eee667d352953c0d02a6f469aa897fa9f9ba0a752cbd35d39","created":"2026-10-15T23:43:22Z","files":[],"origin":{"assisted_by":[],"kind":"seed"},"parents":["72724ff413f94e79cfc1b07e426114fe56fb14bc05e9f8af22b052ea33fe0499","ce59fb6cd812fc05906d0c60a71c75ea22bc686bcf38e7c3ef7010834d98ee7b"],"salt":"f5c83391088fb740db8dc94b0f4263523b64975e197d78a969776f6ff56dcbff","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":"2","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":"72724ff413f94e79cfc1b07e426114fe56fb14bc05e9f8af22b052ea33fe0499","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 bound?","topic_ref":null},"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"97d74f7c193293b8a86ea8ae3ac37ab517dfaa9fd4c1fdc72f6b57a9afe538de","schema":"pubphys.attested/1"},"envelope":{"attested_hash":"608c8284cff21444c321958fc793baa26cef64a454e8b5ec7787b4dd20f69028","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"laQCLNcS4aOge1Ytn1JtSRK38HJQXhh7PnOt_OZpGtYnZuy3YroJwTRRxaAxpE5ZolJw3y9Jgxde4m3vMR6RAQ"},"schema":"pubphys.envelope/1"},"record_hash":"97d74f7c193293b8a86ea8ae3ac37ab517dfaa9fd4c1fdc72f6b57a9afe538de","leaf_index":3712}