{"schema":"pubphys.bundle/1","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},"content_withheld":"false","files":[],"attested":{"attestation":{"batch":null,"client_id":null,"id_token_sha256":null,"kind":"platform"},"record_hash":"97d74f7c193293b8a86ea8ae3ac37ab517dfaa9fd4c1fdc72f6b57a9afe538de","schema":"pubphys.attested/1"},"id_token":null,"id_token_withheld":"false","envelope":{"attested_hash":"608c8284cff21444c321958fc793baa26cef64a454e8b5ec7787b4dd20f69028","platform_signature":{"key_id":"c6afc19b31429869751f06879c75cd64ea92654423d15b44be775bf1310a60da","sig":"laQCLNcS4aOge1Ytn1JtSRK38HJQXhh7PnOt_OZpGtYnZuy3YroJwTRRxaAxpE5ZolJw3y9Jgxde4m3vMR6RAQ"},"schema":"pubphys.envelope/1"},"ots":{"attested":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIYIyChM_yFETDIZWPx5O6omzvZKRU6LXsd4e03SD2kCjwIGCb3dntTiLs6j8_QbdKZ_dLGLlmlWgmgzNJlJMqzljqCPEgKARpCHOzUHEc5p609h0x4-t5g2PxkZ2teIpojEMBEqsI8SBIoZSjK3Nq3HFOeINd8W9U4t8g5KkcSo1y_NWN4ZH8iQjxIJ_X56GwnzsfUUByDW93nhfOiaieNQ3a-lQNZI6wFcqCCPEgHEDaJMIpZLmckgDcuG8Y-VG27NzbOaph8h6VZtfcStgI8SCoIDI38o3m8gxPq9ipueT8_RYayaZzNWFftCYfZWwP6wjxIPz45zKf3oui_6fsPFoesAdn8c7jCNtvma5ZIef8PtuLCPEg-b8Lkm9aIcNajW2D_tonI2BpSvWgXJkN2LygUbv3ObsI8CCFjG9pSMiq3eVzHAmZsjRKj0iAZic3nIa3eyIChZggbAjwIOv_x0B6HyZ6WMHKeIf_d_EUIvwPGdZeVY-hXN3RLDM5CPEgHd9u6OIGCvUV12A1B1xdiOprSJ-4w0cqwfe7CzVX5HsI8CDQo_sIWZiWXaTGRSQBLmU-lHJKskYKqsjO1r02DhB8LAj_8Ajn9YmozHwo9QjwEJ46aM9ZDv3TdtqV3ZbRsNgI8SBOF8pILGf1kodr5MKmQs6Cd93joraGn3g6UPt_-kewPwjwIOG-gmM7bd_eR4AiDMNfluS3xvr7vicLR-sCBr5qtG2cCPAgrVNFrCoeQcCxhlmnE0BzHgR8HQpZAlfqzDDFzfWb8GUI8QRq0WWT8AjZJ2cF_c7k8ACD3-MNLvkMjiwraHR0cHM6Ly9ib2IuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZ_AI9ZMS1zxd7IEI8BDd00WgJZ0RncsC1Fyai_4aCPEgvxMFrnu7F05IW5INiraQ3OVEYMjNx6JfBxw1Q4jPLi4I8QRq0WWS8Ag2zxPAtI5JsP8Ag9_jDS75DI4uLWh0dHBzOi8vYWxpY2UuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZwjxIHAnv4KFFf4uiYEFJvLxiMxSyz8VowJP4roTfDD9h7bICPAgj5tgvbNF7mxSP4jGGE66ezUFRLnI4g1VTkvuV4VtCPAI8CCxlc1HuKORuHgn7DYIYcDhc4oo99OkTWBEvddBypoa7AjwIFcjjaF8kDNDYCZGaNISlRjvX-I7in-QNwsr4JYQbFJvCPAgnl0k4gs7LuY5i1umiHO-J7My3uBThLJUWQoY9LdVSrQI8SA6-Ft11_K_GqMNWYhkVsVqCFvXSYbp4S8JoLsLBnUzvgjxIBC8_mAE7JBP35SNxDBcG7joLJwURpMxdV2vSO64_jEOCPEge04Coj1xSi7CE8UwqiBGNhPyZ5QdD_YZwn-heSdObU4I8SBgdDSgr3VUx1j8m_pK7-OzMwwFgXeLy43ujffCv-hf5AjwIMl-PnAt4xjoBEVU4-nkyN0NKRa9AvX4kEMfllFyIJS9CPFZAQAAAAHz4rV-uwQFg02bVnEMUFgpIdGKjM4tWuID1kTVnrjisAAAAAAA_v___wJURAIAAAAAABYAFBFys2oc0-U-bYrC3VulGLzND5fbAAAAAAAAAAAiaiDwBHLTDgAICPEgMJIBYi46xOIuFDwzJ9nOx14ngW5HbglUf65ORpkoIYcICPAgHgGWEGCDHWgscwT92NOoJ7s-ezbPkyiWr2RnYZDC_dwICPAgtffMyyLu-MscJsDc0dzwGspwSrrmvVAnMNgsgWhHClgICPAgMUl1OfBDUwhp5VZjHwVA0nuz1F5HZTUJCHAnM1L1F3MICPAgStK7UWpnrUueV5EaE7uE0y6BLMp6EANmrbhrTqEBZu8ICPAg5LVW7IHtUL5mNOvPx1hOTfLMmqlQ7t9EFW-KDtrISIQICPAgABgXZ6upFU8qiDQKFtQFpqkjKwl0PsxM4JUjVLlDKbkICPAgEwb96FiOhjlbwI8GFZQ4a5GXwxs4bUdqVOMKrtBWV30ICPEgqBg-wp0c8CAPV9k6J7wIa9_GtQfpWmhpE1zqFZyi2pQICPAg2pZLTVPGJbriiz0UhMytYDVYAXOgQnWDcrMOewK9O8cICPEgxm3QwzbvT4siJzubQGnST67bWv-exYwSPP1KDi7CaUwICPEgm6udUudfjUaM0gUFTylKGDMypj7ptMTG0zmxd_T5yTcICPAgMpOInkjFCVp1pOZhfZnyPAl8hdWGmHzQcrGLgLwSKLgICAAFiJYNc9cZAQPzpjs"],"envelope":["AE9wZW5UaW1lc3RhbXBzAABQcm9vZgC_ieLohOiSlAEIzXsOvXlkJLrzUBqVFJ3Ap4BRcBRlvXkrf5ZGe26JJsvxIM11cyr8525P4lhnYmo13acI4lS0tE2p5m5v-vIDvaAMCPAg5yOyjua8_4i7oczCWQFm6UrCJ_hL4NoRTtinZN2ovz0I8CCRmeVImJfQGxhDvTOWXFMEuTYg8PaOClrrf1VFfMEpWwjwIBcfkoG7lA8AGm8Gq33RgNOChIgUNJj7GMkqOT-q5tuzCPAgSjj2ZP7Q8rAeYSFJFVFyjMLU-hn_1J-_rYmSvqDo1FkI8CC1j7CCeceErwpufSloR9lnLLVWTiU6TMroAQxKzqg7gAjwIIzg_cFUYK0fm7kzDDjEck9P8bQjDangPAxYK0BAbnx3CPEgA7tb_tb_4KngwMf2GSJCpmxJdfmzdO37wjFC3fvNh0wI8CCSqAf0sn_1-kk9EDPXEZvjRff_ciLViVfMVybjDX1OsQjxID89HsXrUag9eRv04qSQX18D2ak5pxDN4H8m3_0Qkp84CPAg3mczMYDKOd7Nhf3ojA9ZvuTsN66Y1NnyYTp9j8GKQNwI8SCvAiiN-Hrv7IEejd8wdUihoozu5KOvpoBFJ0vg3UnGAQj_8Ajn9YmozHwo9QjwEJ46aM9ZDv3TdtqV3ZbRsNgI8SBOF8pILGf1kodr5MKmQs6Cd93joraGn3g6UPt_-kewPwjwIOG-gmM7bd_eR4AiDMNfluS3xvr7vicLR-sCBr5qtG2cCPAgrVNFrCoeQcCxhlmnE0BzHgR8HQpZAlfqzDDFzfWb8GUI8QRq0WWT8AjZJ2cF_c7k8ACD3-MNLvkMjiwraHR0cHM6Ly9ib2IuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZ_AI9ZMS1zxd7IEI8BDd00WgJZ0RncsC1Fyai_4aCPEgvxMFrnu7F05IW5INiraQ3OVEYMjNx6JfBxw1Q4jPLi4I8QRq0WWS8Ag2zxPAtI5JsP8Ag9_jDS75DI4uLWh0dHBzOi8vYWxpY2UuYnRjLmNhbGVuZGFyLm9wZW50aW1lc3RhbXBzLm9yZwjxIHAnv4KFFf4uiYEFJvLxiMxSyz8VowJP4roTfDD9h7bICPAgj5tgvbNF7mxSP4jGGE66ezUFRLnI4g1VTkvuV4VtCPAI8CCxlc1HuKORuHgn7DYIYcDhc4oo99OkTWBEvddBypoa7AjwIFcjjaF8kDNDYCZGaNISlRjvX-I7in-QNwsr4JYQbFJvCPAgnl0k4gs7LuY5i1umiHO-J7My3uBThLJUWQoY9LdVSrQI8SA6-Ft11_K_GqMNWYhkVsVqCFvXSYbp4S8JoLsLBnUzvgjxIBC8_mAE7JBP35SNxDBcG7joLJwURpMxdV2vSO64_jEOCPEge04Coj1xSi7CE8UwqiBGNhPyZ5QdD_YZwn-heSdObU4I8SBgdDSgr3VUx1j8m_pK7-OzMwwFgXeLy43ujffCv-hf5AjwIMl-PnAt4xjoBEVU4-nkyN0NKRa9AvX4kEMfllFyIJS9CPFZAQAAAAHz4rV-uwQFg02bVnEMUFgpIdGKjM4tWuID1kTVnrjisAAAAAAA_v___wJURAIAAAAAABYAFBFys2oc0-U-bYrC3VulGLzND5fbAAAAAAAAAAAiaiDwBHLTDgAICPEgMJIBYi46xOIuFDwzJ9nOx14ngW5HbglUf65ORpkoIYcICPAgHgGWEGCDHWgscwT92NOoJ7s-ezbPkyiWr2RnYZDC_dwICPAgtffMyyLu-MscJsDc0dzwGspwSrrmvVAnMNgsgWhHClgICPAgMUl1OfBDUwhp5VZjHwVA0nuz1F5HZTUJCHAnM1L1F3MICPAgStK7UWpnrUueV5EaE7uE0y6BLMp6EANmrbhrTqEBZu8ICPAg5LVW7IHtUL5mNOvPx1hOTfLMmqlQ7t9EFW-KDtrISIQICPAgABgXZ6upFU8qiDQKFtQFpqkjKwl0PsxM4JUjVLlDKbkICPAgEwb96FiOhjlbwI8GFZQ4a5GXwxs4bUdqVOMKrtBWV30ICPEgqBg-wp0c8CAPV9k6J7wIa9_GtQfpWmhpE1zqFZyi2pQICPAg2pZLTVPGJbriiz0UhMytYDVYAXOgQnWDcrMOewK9O8cICPEgxm3QwzbvT4siJzubQGnST67bWv-exYwSPP1KDi7CaUwICPEgm6udUudfjUaM0gUFTylKGDMypj7ptMTG0zmxd_T5yTcICPAgMpOInkjFCVp1pOZhfZnyPAl8hdWGmHzQcrGLgLwSKLgICAAFiJYNc9cZAQPzpjs"]},"log":{"leaf_index":"3712","tree_size":"3804","proof":["556b289206a74c15f77e0f9f39fb54e7e8c94ffc488cec6865d1070fbe2ce384","923a8f090ccc7484695895299d6d352973a4f0cd670b1422d715cfab038b0d1a","618dbdc890a1ba61891cd532a5dff368b50da0fe6baf12b5b1c66d5e14f3d11b","b498cb8603cfe55a6aa7be7194f48f884727040713ae8e26019eff22a984dce3","74290274b1cf51c84f29057d384a458fefc3897feab52e42febdc6651ee3591a","9a081e67452f79119d54e86bb56291e213ce50cf95c39fed99560bbf0ef2fb1e","17c91a7812aa72cd79b13c02f19c3226fff2624c8ff753f84916b2fa345baaf3","0455d4faa8c1028929d27c21062fe4bb47127ac22b5867e956f5c4f72c680cf1","e55158332fc5066d8199fad50d3929c561e805c1a4826619f59cd2254d322eb8","af34d80d0065c0ab502282775e865492e2cb49c9493f4594d63ba5577922854a","3c578dccc15502c5f3efe228cadc99fee76ab536ed01f66b14d316fdcc157429"],"checkpoint":"pubphys.com/log/v1\n3804\nAK6t4qgu96+M9wScTrIlsJ10d/zhKVEtYguyM3aeJvo=\n\n— pubphys.com/log/v1 JFPwIPP8EtoxgcsQg7V7CwIdL4UKPqJtpv+eB8sFNJjyQHIdyA1VabmBsPyLqq2bLxJPn57rA6fQi8y0QDDjRXHvlQg=\n","promise":"pubphys.com/log/v1 promise\nd2b32270b78a3a3baa9a8c5d21ae8f7371b79c779ad7586b44586e2f095f3d73\n3712\n2026-10-15T23:43:57Z\n2026-10-16T01:43:57Z\n\n— pubphys.com/log/v1 JFPwIPOKdBx2u9uRS8KPmZgNYmRtYsJi1CuKjPCuobyvCqzcGeXlHVj506PuVEDtVWoB573WDHEw9VkWrqv49tCe2A4=\n"},"orcid_key_evidence":[],"attesting":null}