{"id":"52.1.4.a","row":291,"stored":{"file":"master_catalog.json","row":291,"construction":null,"columns_sha256":null},"parameters":{"n":52,"k":1,"d":4,"N":10,"r":9,"level":3},"gate":{"string":"0","human":"T0","monomials":[[0]],"sk_key":[[0]],"sk_fingerprint":"bc179efd84e614ba","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[0,1,2,3,4,7],[0,1,4,5,6,7],[0,1,2,3,5,8],[0,2,4,5,6,8],[0,1,2,3,6,9],[0,3,4,5,6,9],[0,1,4,7,8,9],[0,2,5,7,8,9],[0,3,6,7,8,9],[0,3,5,7],[0,1,6,8],[0,2,4,9],[2,3,5,7],[3,4,5,7],[1,3,6,8],[1,5,6,8],[1,6,7,8],[1,2,4,9],[2,4,6,9],[3,5,7,9],[2,4,8,9],[0,2,6,7],[0,3,4,8],[0,1,5,9],[0,1,2,6,7],[0,2,5,6,7],[0,2,3,4,8],[0,3,4,6,8],[0,3,4,7,8],[0,1,3,5,9],[0,1,4,5,9],[0,2,6,7,9],[0,1,5,8,9],[0,2,3,4,6,7],[0,1,3,4,5,8],[0,2,3,6,7,8],[0,2,4,6,7,8],[0,1,2,5,6,9],[0,1,2,5,7,9],[0,1,5,6,7,9],[0,1,3,4,8,9],[0,3,4,5,8,9],[1,3,5,6,7],[1,2,4,6,8],[1,3,5,7,8],[3,5,6,7,8],[2,3,4,5,9],[2,3,4,7,9],[2,4,5,7,9],[1,2,6,8,9],[1,4,6,8,9],[1,2,3,4,5,6,7,8,9]]},"distance":{"d":4,"is_exact":true,"upper":4,"witness":[0,1,6,51],"A_d":257,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[1,3],[2,3],[3,3],[4,3],[5,3],[6,3],[7,3],[8,3],[9,3]],"CZ":[[1,2],[1,3],[1,4],[1,5],[1,6],[1,7],[1,8],[1,9],[2,3],[2,4],[2,5],[2,6],[2,7],[2,8],[2,9],[3,4],[3,5],[3,6],[3,7],[3,8],[3,9],[4,5],[4,6],[4,7],[4,8],[4,9],[5,6],[5,7],[5,8],[5,9],[6,7],[6,8],[6,9],[7,8],[7,9],[8,9]]},"rotation_powers":[[4,5],[12,5],[13,5],[15,5],[24,5],[51,3]]},"metrics":{"gamma":2.8502198590705463,"gamma_t":2.8502198590705463,"gamma_rho":2.8502198590705463,"gamma_claim":2.8502198590705463,"gamma_t_claim":2.8502198590705463,"gamma_rho_claim":2.8502198590705463,"v_ex":1,"rate":0.019230769230769232,"t_count":1,"poly_degree":1,"effective_width":1},"provenance":{"regimes":["exhaustive classification n<=54 (Pareto point)","symmetry-SAT search","AI search"],"discovery":"pre-existing","strongest_claim":"a Pareto point of the exhaustive classification of generalised triorthogonal protocols through length 54 (classification/length54/): no protocol with n <= 54, the same CNOT+S output class and the same exact distance has n and S both no larger and one of them smaller, where S counts every matrix row -- the N of this table","sources":[{"regime":"symmetry-SAT search","file":"symmetry_sat_search/catalog/factories.json","label":"T0","N":10,"d":4,"d_is_exact":true,"origin":null,"provenance":"explicit circuit from ansatz-free SAT; archived result: beyond_cyclic_symmetric.json"},{"regime":"symmetry-SAT search","file":"symmetry_sat_search/catalog/factories.json","label":"T0","N":10,"d":4,"d_is_exact":true,"origin":null,"provenance":"explicit circuit from ansatz-free SAT; archived result: slot_ansatz_subgroup_extras.json"},{"regime":"AI search","file":"master_catalog/catalogue_new","label":"T0","N":10,"d":4,"d_is_exact":true,"origin":"ai_campaign/circuits","provenance":"ai_campaign/circuits \u00b7 catalogue.jsonl:253"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/length54/pareto_frontier.json","label":"protocol 1","N":10,"d":4,"d_is_exact":true,"origin":"exhaustive classification of generalised triorthogonal protocols through length 54 (Wills, Jain and Singh)","provenance":"Pareto point 1 of the length-54 classification: output class Q1_0000000000000001 (T), (n, S, d_Z) = (52, 10, 4), leading error coefficient 333; the outputs are its 1 logical rows"}]},"references":[{"key":"wills2026classification","short":"Wills et al. (2026)","full":"A. Wills, S. P. Jain, and S. Singh, \"Classification of Generalised Triorthogonal Codes through Length 54,\" arXiv:2609.30860 (2026).","url":"https://arxiv.org/abs/2609.30860","date":"2026-09-25"},{"key":"jain2026symmetry","short":"Jain et al. (2026)","full":"S. P. Jain, A. Wills, and S. Singh, \"Symmetry and AI-assisted discovery of magic-state factories,\" arXiv:2610.06535 (2026).","url":"https://arxiv.org/abs/2610.06535","date":"2026-10-05"}]}