{"id":"47.6.3.a","row":243,"stored":{"file":"master_catalog.json","row":243,"construction":null,"columns_sha256":null},"parameters":{"n":47,"k":6,"d":3,"N":12,"r":6,"level":3},"gate":{"string":"0+1+2+3+01+02+03+12+13+23+012+013+014+015+023+024+025+034+035+045+123+124+125+145","human":"T0\u00b7T1\u00b7T2\u00b7T3\u00b7CS01\u00b7CS02\u00b7CS03\u00b7CS12\u00b7CS13\u00b7CS23\u00b7CCZ012\u00b7CCZ013\u00b7CCZ014\u00b7CCZ015\u00b7CCZ023\u00b7CCZ024\u00b7CCZ025\u00b7CCZ034\u00b7CCZ035\u00b7CCZ045\u00b7CCZ123\u00b7CCZ124\u00b7CCZ125\u00b7CCZ145","monomials":[[0],[1],[2],[3],[0,1],[0,2],[0,3],[1,2],[1,3],[2,3],[0,1,2],[0,1,3],[0,1,4],[0,1,5],[0,2,3],[0,2,4],[0,2,5],[0,3,4],[0,3,5],[0,4,5],[1,2,3],[1,2,4],[1,2,5],[1,4,5]],"sk_key":[[0],[0,1],[0,1,2],[0,1,3],[0,1,4],[0,1,5],[0,2],[0,2,3],[0,2,4],[0,2,5],[0,3],[0,3,4],[0,3,5],[0,4,5],[1],[1,2],[1,2,3],[1,2,4],[1,2,5],[1,3],[1,4,5],[2],[2,3],[3]],"sk_fingerprint":"bd58d2f16e42bf1f","sk_canonical_frame":true,"relabelled_into_canonical_frame":true},"circuit":{"columns":[[0,3,5,10],[1,2,11],[0,2,4,5,10,11],[0,4,8],[1,3,4,9],[0,1,2,8,10,11],[2,3,9,10,11],[0,1,2,4,5,8,9],[4,8,9,10],[0,1,2,3,8,9,11],[0,2,3,4,5,8,9,10,11],[0,1,2,3,4,5,7],[2,7,10],[2,4,5,7,11],[0,4,7,10,11],[0,1,3,5,7,8],[0,1,3,4,7,9],[1,2,4,5,7,8,10,11],[1,2,7,9,10,11],[0,1,4,7,8,9],[1,4,7,8,9,10],[1,4,7,8,9,11],[0,2,3,7,8,9,10,11],[1,4,5,6],[0,1,3,6,10],[2,5,6,11],[3,4,5,6,10,11],[0,1,2,4,6,8],[2,3,5,6,9],[0,4,5,6,8,10,11],[1,3,6,9,10,11],[1,3,4,5,6,8,9],[1,4,5,6,8,9,10],[0,2,3,5,6,8,9,11],[6,8,9,10,11],[1,4,5,6,7],[1,2,4,6,7,10],[1,2,5,6,7,11],[2,3,5,6,7,10,11],[0,2,3,5,6,7,8],[0,2,3,5,6,7,9],[6,7,8,10,11],[6,7,9,10,11],[1,2,3,5,6,7,8,9],[6,7,8,9,10],[6,7,8,9,11],[6,7,8,9,10,11]]},"distance":{"d":3,"is_exact":true,"upper":3,"witness":[0,1,2],"A_d":275,"certified":null},"clifford":{"kind":"required","correction":{"S":[[0,3],[1,1],[2,1],[3,3],[4,1],[5,1]],"CZ":[[0,1],[0,4],[0,5],[0,6],[1,3],[1,4],[1,5],[1,6],[2,5],[2,6],[2,8],[2,9],[2,10],[2,11],[3,4],[3,6],[4,7],[4,9],[4,10],[5,6],[5,7],[5,8],[5,11]]},"rotation_powers":null},"metrics":{"gamma":1.8736256218082787,"gamma_t":1.3218970367355982,"gamma_rho":null,"gamma_claim":1.8736256218082787,"gamma_t_claim":1.3218970367355982,"gamma_rho_claim":null,"v_ex":null,"rate":0.1276595744680851,"t_count":11,"poly_degree":3,"effective_width":6},"provenance":{"regimes":["exhaustive classification n<=54 (Pareto point)","symmetry-SAT 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 . T2 . T3 . T4 . CS02 . CS03 . CS04 . CS23 . CS24 . CS34 . CCZ012 . CCZ014 . CCZ015 . CCZ023 . CCZ024 . CCZ025 . CCZ034 . CCZ045 . CCZ123 . CCZ124 . CCZ125 . CCZ234 . CCZ235 . CCZ245","N":12,"d":3,"d_is_exact":true,"origin":null,"provenance":"explicit circuit from ansatz-free SAT; archived result: ccz47_relabel_k6_t11.json"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/length54/pareto_frontier.json","label":"protocol 69","N":12,"d":3,"d_is_exact":true,"origin":"exhaustive classification of generalised triorthogonal protocols through length 54 (Wills, Jain and Singh)","provenance":"Pareto point 69 of the length-54 classification: output class Q6_0000000002600000 (T tensor (CCZ145CCZ235)), (n, S, d_Z) = (47, 12, 3), leading error coefficient 275; the outputs are its 6 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"}]}