{"id":"43.4.3.e","row":171,"stored":{"file":"master_catalog.json","row":171,"construction":null,"columns_sha256":null},"parameters":{"n":43,"k":4,"d":3,"N":11,"r":7,"level":3},"gate":{"string":"0+12+13","human":"T0\u00b7CS12\u00b7CS13","monomials":[[0],[1,2],[1,3]],"sk_key":[[0],[1,2],[1,3]],"sk_fingerprint":"4bbf112dfb4f3ecd","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[1,3,4,5],[1,3,4,6],[1,2,5,6],[1,2,7],[3,4,7],[3,5,7],[1,2,4,5,6,7],[2,4,8],[1,3,5,8],[0,4,6,8],[3,4,5,6,8],[2,4,5,7,8],[2,5,6,7,8],[3,4,9],[2,5,9],[1,3,6,9],[0,1,4,5,6,9],[1,2,4,5,7,9],[2,4,5,6,7,9],[0,1,5,8,9],[1,2,4,5,8,9],[3,4,7,8,9],[1,3,6,7,8,9],[2,3,4,10],[1,5,10],[0,2,5,6,10],[1,2,3,4,5,6,10],[1,2,3,4,5,7,10],[2,3,4,6,7,10],[8,10],[0,2,4,5,8,10],[1,4,5,7,8,10],[6,7,8,10],[0,1,2,4,9,10],[4,5,9,10],[5,7,9,10],[6,7,9,10],[0,1,2,6,8,9,10],[4,5,6,8,9,10],[7,8,9,10],[6,7,8,9,10],[4,6,7,8,9,10],[5,6,7,8,9,10]]},"distance":{"d":3,"is_exact":true,"upper":3,"witness":[0,1,2],"A_d":117,"certified":null},"clifford":{"kind":"required","correction":{"S":[[0,1],[1,3],[2,3],[3,1],[6,2],[7,2],[8,2],[9,2],[10,2]],"CZ":[[1,2],[1,4],[1,7],[1,8],[1,10],[2,8],[2,9],[3,4],[3,5],[3,6],[3,7],[4,5],[4,6],[4,7],[4,8],[4,9],[4,10],[5,6],[5,7],[5,8],[5,9],[5,10],[6,7],[6,8],[6,9],[6,10],[7,8],[7,9],[7,10],[8,9],[8,10],[9,10]]},"rotation_powers":null},"metrics":{"gamma":2.1617323773547645,"gamma_t":1.958618363779752,"gamma_rho":null,"gamma_claim":2.1617323773547645,"gamma_t_claim":1.958618363779752,"gamma_rho_claim":null,"v_ex":null,"rate":0.09302325581395349,"t_count":5,"poly_degree":2,"effective_width":4},"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":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+12+13","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+3+01+02+03+12+012+013+023+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+3+01+02+03+12+13+012+013+023","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+3+01+02+03+012+013+023","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+01+02+12+012+013+023+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+01+02+12+13+23+012+013+023+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) all-origin census witness rank7_w44_result-39 (n43-t5-07); high_tcount_subframes_REPS.json"},{"regime":"symmetry-SAT search","file":"symmetry_sat_search/catalog/factories.json","label":"T0 . CS01 . CS02 . CS12 . CS13 . CS23 . CCZ012 . CCZ013 . CCZ023 . CCZ123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"explicit circuit from ansatz-free SAT; archived result: t5_43_4_3.json"},{"regime":"AI search","file":"master_catalog/catalogue_new","label":"T0.CS01.CS02.CS12.CS13.CS23.CCZ012.CCZ013.CCZ023.CCZ123","N":11,"d":3,"d_is_exact":true,"origin":"frontier_campaign","provenance":"frontier_campaign \u00b7 catalogue.jsonl:124"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+3+01+02+12+13+012+013+023+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) all-origin census witness rank7_w44_result-246 (n43-t5-08); high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+01+02+12+13+23+012+013+023","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+01+02+03+012+013","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+1+2+01+02+03+12+012+013","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+01+02+03+23+012+013+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+2+01+02+13+012+013+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+2+3+01+02+13+23+012+013+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+3+01+02+12+23+012+013","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+2+01+02+012+013","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 3430, origin 3; high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+2+3+01+02+23+012+013","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+1+2+3+01+012+013+023","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) all-origin census witness rank7_w44_result-39 (n43-t5-09); high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+2+01+012+013+023","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+01+12+012+013+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+2+3+01+12+012+013+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) all-origin census witness rank7_w44_result-246 (n43-t5-10); high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+01+02+03+12+13+012","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) all-origin census witness rank7_w44_result-39 (n43-t5-11); high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+3+01+02+03+12+13+23+012","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) all-origin census witness rank7_w44_result-39 (n43-t5-12); high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+3+01+02+03+12+012","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+01+02+03+12+13+012","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+01+02+13+23+012+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+01+02+012+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+1+2+01+02+13+012","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+01+03+12+23+012+023","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+01+03+13+012+023+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+01+03+13+012+023+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+3+01+12+23+012+023","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+2+3+01+12+012+023","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) all-origin census witness rank7_w44_result-246 (n43-t5-13); high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+01+23+012+023","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+3+01+012+023","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 3430, origin 3; high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+2+01+03+12+13+012","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) all-origin census witness rank7_w44_result-246 (n43-t5-14); high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+3+01+03+12+13+012","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) all-origin census witness rank7_w44_result-39 (n43-t5-15); high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+01+03+13+012","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+2+01+03+012","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+3+01+03+012","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+01+12+13+012","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+01+13+012+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+01+02+03","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+3+01+02+03+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+01+02+12+13+23+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+01+02+13+23","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+01+02+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+1+01+02+23","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/census_r7.json","label":"0+1+01+23+023","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; high_tcount_subframes_REPS.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+12+13+23+012+013+023+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+1+2+012+013+023+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+12+13+012+013","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+2+3+012+013+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+12+012+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+12+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/legacy/rank7_census/catalog/sk_classes_r7.json","label":"0+12+13+23+123","N":11,"d":3,"d_is_exact":true,"origin":null,"provenance":"RM(3,7) orbit class 2936, origin 3; census_shard_06.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/length54/pareto_frontier.json","label":"protocol 22","N":11,"d":3,"d_is_exact":true,"origin":"exhaustive classification of generalised triorthogonal protocols through length 54 (Wills, Jain and Singh)","provenance":"Pareto point 22 of the length-54 classification: output class Q4_00000000000000e0 (T tensor (CS12CS13)), (n, S, d_Z) = (43, 11, 3), leading error coefficient 125; the outputs are its 4 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"}]}