{"id":"51.6.3.a","row":288,"stored":{"file":"master_catalog.json","row":288,"construction":null,"columns_sha256":null},"parameters":{"n":51,"k":6,"d":3,"N":13,"r":7,"level":3},"gate":{"string":"0+12+34+012+345","human":"T0\u00b7CS12\u00b7CS34\u00b7CCZ012\u00b7CCZ345","monomials":[[0],[1,2],[3,4],[0,1,2],[3,4,5]],"sk_key":[[0],[0,1,2],[1,2],[3,4],[3,4,5]],"sk_fingerprint":"07f39887fa4bc120","sk_canonical_frame":true,"relabelled_into_canonical_frame":true},"circuit":{"columns":[[1,3,5,7],[0,1,3,6,8],[0,2,3,4,6,7,8],[0,1,4,6,9],[0,2,3,6,7,9],[0,1,3,4,8,9],[0,2,6,7,8,9],[1,2,3,4,5,10],[0,2,4,6,7,10],[1,2,4,5,6,8,10],[4,5,7,8,10],[1,2,3,5,6,9,10],[3,4,5,7,9,10],[1,2,5,8,9,10],[5,7,8,9,10],[1,3,5,6,11],[0,1,2,6,7,11],[3,4,5,6,9,11],[5,6,7,9,11],[3,4,5,7,8,9,11],[4,5,6,7,8,9,11],[3,4,5,6,10,11],[0,2,7,10,11],[0,2,4,7,8,10,11],[0,2,4,6,7,8,10,11],[3,5,6,9,10,11],[3,5,7,9,10,11],[0,1,2,5,6,12],[1,3,6,7,12],[0,2,3,5,6,8,12],[0,2,3,4,5,6,7,8,12],[0,2,4,5,7,8,9,12],[0,2,4,5,6,7,8,9,12],[0,2,3,5,6,10,12],[3,4,7,10,12],[0,2,5,6,8,10,12],[0,2,3,4,5,7,8,10,12],[4,7,9,10,12],[6,7,9,10,12],[1,3,11,12],[0,1,2,5,7,11,12],[4,8,9,11,12],[6,7,8,9,11,12],[6,10,11,12],[0,2,5,7,10,11,12],[3,4,8,10,11,12],[6,8,10,11,12],[9,10,11,12],[6,9,10,11,12],[6,8,9,10,11,12],[6,7,8,9,10,11,12]]},"distance":{"d":3,"is_exact":true,"upper":3,"witness":[0,1,2],"A_d":173,"certified":null},"clifford":{"kind":"required","correction":{"S":[[0,2],[1,1],[2,1],[3,1],[4,1],[5,3],[6,1],[7,2],[10,2]],"CZ":[[0,1],[0,2],[0,4],[0,5],[0,6],[0,7],[0,9],[0,11],[0,12],[2,4],[2,5],[2,6],[2,7],[2,9],[2,11],[2,12],[3,5],[3,7],[3,10],[4,6],[4,7],[4,8],[4,9],[5,6],[5,7],[5,10],[5,11],[5,12],[6,7],[6,8],[6,9],[6,10],[6,11],[6,12],[7,8],[7,9],[7,10],[8,11],[9,12],[10,11],[10,12]]},"rotation_powers":null},"metrics":{"gamma":1.9479721695911083,"gamma_t":1.8076581740011435,"gamma_rho":null,"gamma_claim":1.9479721695911083,"gamma_t_claim":1.8076581740011435,"gamma_rho_claim":null,"v_ex":null,"rate":0.11764705882352941,"t_count":7,"poly_degree":2,"effective_width":6},"provenance":{"regimes":["exhaustive classification n<=54 (Pareto point)"],"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/length54/pareto_frontier.json","label":"protocol 65","N":13,"d":3,"d_is_exact":true,"origin":"exhaustive classification of generalised triorthogonal protocols through length 54 (Wills, Jain and Singh)","provenance":"Pareto point 65 of the length-54 classification: output class Q6_0000004002090400 (T^(tensor 3) tensor (CS12CS13)), (n, S, d_Z) = (51, 13, 3), leading error coefficient 173; 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"}]}