{"id":"53.1.5.a","row":308,"stored":{"file":"master_catalog.json","row":308,"construction":null,"columns_sha256":null},"parameters":{"n":53,"k":1,"d":5,"N":11,"r":10,"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":[[0,2,4],[0,1,3,4],[0,1,4,5],[0,1,2,3,4,5],[0,1,3,6],[0,1,3,7],[0,1,3,4,6,7],[0,1,3,5,6,7],[0,1,4,5,6,7],[0,2,8],[0,1,2,3,8],[0,3,4,8],[0,1,4,5,8],[0,1,2,3,4,6,8],[0,1,2,4,5,6,8],[0,1,2,3,4,7,8],[0,1,2,4,5,7,8],[0,2,6,7,8],[0,2,3,6,7,8],[0,1,2,3,6,7,8],[0,1,2,5,6,7,8],[0,1,2,3,9],[0,2,5,9],[0,1,2,6,9],[0,2,5,6,9],[0,1,2,7,9],[0,2,5,7,9],[0,1,3,4,6,7,9],[0,4,5,6,7,9],[0,1,8,9],[0,1,5,8,9],[0,3,5,8,9],[0,6,8,9],[0,7,8,9],[0,1,2,4,6,7,8,9],[0,1,5,6,7,8,9],[0,2,3,4,5,6,7,8,9],[0,1,2,5,10],[0,1,2,5,6,10],[0,1,2,5,7,10],[0,1,2,5,6,7,10],[0,1,2,9,10],[0,1,2,5,9,10],[0,2,6,9,10],[0,2,5,6,9,10],[0,2,7,9,10],[0,2,5,7,9,10],[0,1,2,6,7,9,10],[0,1,2,5,6,7,9,10],[0,1,2,8,9,10],[0,1,2,6,8,9,10],[0,1,2,7,8,9,10],[0,1,2,6,7,8,9,10]]},"distance":{"d":5,"is_exact":true,"upper":5,"witness":[20,24,26,27,36],"A_d":3149,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,2],[1,2],[2,2],[6,3],[7,3],[9,2]],"CZ":[[0,6],[0,7],[1,6],[1,7],[2,6],[2,7],[6,9],[7,9]]},"rotation_powers":[[4,5],[5,5],[24,3],[26,3],[42,3],[43,3],[45,3],[48,3],[49,3],[50,3],[51,3],[52,3]]},"metrics":{"gamma":2.4668810662894636,"gamma_t":2.4668810662894636,"gamma_rho":2.4668810662894636,"gamma_claim":2.4668810662894636,"gamma_t_claim":2.4668810662894636,"gamma_rho_claim":2.4668810662894636,"v_ex":1,"rate":0.018867924528301886,"t_count":1,"poly_degree":1,"effective_width":1},"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":"gamma_frontier_release/protocols/classified_direct_protocol_03_53_1_5.json","label":"classified_direct_protocol_03_53_1_5","N":11,"d":5,"d_is_exact":true,"origin":"Product-T basis recovered from supplied generalized triorthogonal classification","provenance":"gamma frontier release record classified_direct_protocol_03_53_1_5 (source catalogue id classified_direct_protocol_03_53_1_5); construction: Product-T basis recovered from supplied generalized triorthogonal classification; matrix 9699d55ab25d526c...; the release states d=5 (exact), gamma=2.4668810662894636 (exact)","notes":"matrix gamma_frontier_release/matrices/9699d55ab25d526c3115f2d98a46c9ddbf65b6095b91bde5d8e8687c2722b783.matrix"},{"regime":"exhaustive classification n<=54 (Pareto point)","file":"classification/length54/pareto_frontier.json","label":"protocol 3","N":11,"d":5,"d_is_exact":true,"origin":"exhaustive classification of generalised triorthogonal protocols through length 54 (Wills, Jain and Singh)","provenance":"Pareto point 3 of the length-54 classification: output class Q1_0000000000000001 (T), (n, S, d_Z) = (53, 11, 5), leading error coefficient 3149; 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"}]}