{"id":"553.7.2.a","row":1125,"stored":{"file":"master_catalog.json","row":1125,"construction":null,"columns_sha256":null},"parameters":{"n":553,"k":7,"d":2,"N":13,"r":6,"level":3},"gate":{"string":"0+1+2+3+4+5+6","human":"T0\u00b7T1\u00b7T2\u00b7T3\u00b7T4\u00b7T5\u00b7T6","monomials":[[0],[1],[2],[3],[4],[5],[6]],"sk_key":[[0],[1],[2],[3],[4],[5],[6]],"sk_fingerprint":"4227720e4bdc2b66","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[0,1,2,3,4,5,6,7,8],[0,1,2,3,4,5,6,7,8,9,10,11,12],[0,1,2,3,4,5,6,7,9],[0,1,2,3,4,5,6,7,10],[0,1,2,3,4,5,6,7,11],[0,1,2,3,4,5,6,7,12],[0,1,2,3,4,5,6,8,9],[0,1,2,3,4,5,6,8,10],[0,1,2,3,4,5,6,8,11],[0,1,2,3,4,5,6,8,12],[0,1,2,3,4,5,6,9,10],[0,1,2,3,4,5,6,9,11],[0,1,2,3,4,5,6,9,12],[0,1,2,3,4,5,6,10,11],[0,1,2,3,4,5,6,10,12],[0,1,2,3,4,5,6,11,12],[0,1,2,3,7],[0,1,2,3,7,8,9,10,11],[0,1,2,3,7,8,9,10,12],[0,1,2,3,7,8,9,11,12],[0,1,2,3,7,8,10,11,12],[0,1,2,3,7,9,10,11,12],[0,1,2,3,8],[0,1,2,3,8,9,10,11,12],[0,1,2,3,9],[0,1,2,3,10],[0,1,2,3,11],[0,1,2,3,12],[0,1,2,4,7],[0,1,2,4,7,8,9,10,11],[0,1,2,4,7,8,9,10,12],[0,1,2,4,7,8,9,11,12],[0,1,2,4,7,8,10,11,12],[0,1,2,4,7,9,10,11,12],[0,1,2,4,8],[0,1,2,4,8,9,10,11,12],[0,1,2,4,9],[0,1,2,4,10],[0,1,2,4,11],[0,1,2,4,12],[0,1,2,5,7],[0,1,2,5,7,8,9,10,11],[0,1,2,5,7,8,9,10,12],[0,1,2,5,7,8,9,11,12],[0,1,2,5,7,8,10,11,12],[0,1,2,5,7,9,10,11,12],[0,1,2,5,8],[0,1,2,5,8,9,10,11,12],[0,1,2,5,9],[0,1,2,5,10],[0,1,2,5,11],[0,1,2,5,12],[0,1,2,6,7],[0,1,2,6,7,8,9,10,11],[0,1,2,6,7,8,9,10,12],[0,1,2,6,7,8,9,11,12],[0,1,2,6,7,8,10,11,12],[0,1,2,6,7,9,10,11,12],[0,1,2,6,8],[0,1,2,6,8,9,10,11,12],[0,1,2,6,9],[0,1,2,6,10],[0,1,2,6,11],[0,1,2,6,12],[0,1,3,4,7],[0,1,3,4,7,8,9,10,11],[0,1,3,4,7,8,9,10,12],[0,1,3,4,7,8,9,11,12],[0,1,3,4,7,8,10,11,12],[0,1,3,4,7,9,10,11,12],[0,1,3,4,8],[0,1,3,4,8,9,10,11,12],[0,1,3,4,9],[0,1,3,4,10],[0,1,3,4,11],[0,1,3,4,12],[0,1,3,5,7],[0,1,3,5,7,8,9,10,11],[0,1,3,5,7,8,9,10,12],[0,1,3,5,7,8,9,11,12],[0,1,3,5,7,8,10,11,12],[0,1,3,5,7,9,10,11,12],[0,1,3,5,8],[0,1,3,5,8,9,10,11,12],[0,1,3,5,9],[0,1,3,5,10],[0,1,3,5,11],[0,1,3,5,12],[0,1,3,6,7],[0,1,3,6,7,8,9,10,11],[0,1,3,6,7,8,9,10,12],[0,1,3,6,7,8,9,11,12],[0,1,3,6,7,8,10,11,12],[0,1,3,6,7,9,10,11,12],[0,1,3,6,8],[0,1,3,6,8,9,10,11,12],[0,1,3,6,9],[0,1,3,6,10],[0,1,3,6,11],[0,1,3,6,12],[0,1,4,5,7],[0,1,4,5,7,8,9,10,11],[0,1,4,5,7,8,9,10,12],[0,1,4,5,7,8,9,11,12],[0,1,4,5,7,8,10,11,12],[0,1,4,5,7,9,10,11,12],[0,1,4,5,8],[0,1,4,5,8,9,10,11,12],[0,1,4,5,9],[0,1,4,5,10],[0,1,4,5,11],[0,1,4,5,12],[0,1,4,6,7],[0,1,4,6,7,8,9,10,11],[0,1,4,6,7,8,9,10,12],[0,1,4,6,7,8,9,11,12],[0,1,4,6,7,8,10,11,12],[0,1,4,6,7,9,10,11,12],[0,1,4,6,8],[0,1,4,6,8,9,10,11,12],[0,1,4,6,9],[0,1,4,6,10],[0,1,4,6,11],[0,1,4,6,12],[0,1,5,6,7],[0,1,5,6,7,8,9,10,11],[0,1,5,6,7,8,9,10,12],[0,1,5,6,7,8,9,11,12],[0,1,5,6,7,8,10,11,12],[0,1,5,6,7,9,10,11,12],[0,1,5,6,8],[0,1,5,6,8,9,10,11,12],[0,1,5,6,9],[0,1,5,6,10],[0,1,5,6,11],[0,1,5,6,12],[0,2,3,4,7],[0,2,3,4,7,8,9,10,11],[0,2,3,4,7,8,9,10,12],[0,2,3,4,7,8,9,11,12],[0,2,3,4,7,8,10,11,12],[0,2,3,4,7,9,10,11,12],[0,2,3,4,8],[0,2,3,4,8,9,10,11,12],[0,2,3,4,9],[0,2,3,4,10],[0,2,3,4,11],[0,2,3,4,12],[0,2,3,5,7],[0,2,3,5,7,8,9,10,11],[0,2,3,5,7,8,9,10,12],[0,2,3,5,7,8,9,11,12],[0,2,3,5,7,8,10,11,12],[0,2,3,5,7,9,10,11,12],[0,2,3,5,8],[0,2,3,5,8,9,10,11,12],[0,2,3,5,9],[0,2,3,5,10],[0,2,3,5,11],[0,2,3,5,12],[0,2,3,6,7],[0,2,3,6,7,8,9,10,11],[0,2,3,6,7,8,9,10,12],[0,2,3,6,7,8,9,11,12],[0,2,3,6,7,8,10,11,12],[0,2,3,6,7,9,10,11,12],[0,2,3,6,8],[0,2,3,6,8,9,10,11,12],[0,2,3,6,9],[0,2,3,6,10],[0,2,3,6,11],[0,2,3,6,12],[0,2,4,5,7],[0,2,4,5,7,8,9,10,11],[0,2,4,5,7,8,9,10,12],[0,2,4,5,7,8,9,11,12],[0,2,4,5,7,8,10,11,12],[0,2,4,5,7,9,10,11,12],[0,2,4,5,8],[0,2,4,5,8,9,10,11,12],[0,2,4,5,9],[0,2,4,5,10],[0,2,4,5,11],[0,2,4,5,12],[0,2,4,6,7],[0,2,4,6,7,8,9,10,11],[0,2,4,6,7,8,9,10,12],[0,2,4,6,7,8,9,11,12],[0,2,4,6,7,8,10,11,12],[0,2,4,6,7,9,10,11,12],[0,2,4,6,8],[0,2,4,6,8,9,10,11,12],[0,2,4,6,9],[0,2,4,6,10],[0,2,4,6,11],[0,2,4,6,12],[0,2,5,6,7],[0,2,5,6,7,8,9,10,11],[0,2,5,6,7,8,9,10,12],[0,2,5,6,7,8,9,11,12],[0,2,5,6,7,8,10,11,12],[0,2,5,6,7,9,10,11,12],[0,2,5,6,8],[0,2,5,6,8,9,10,11,12],[0,2,5,6,9],[0,2,5,6,10],[0,2,5,6,11],[0,2,5,6,12],[0,3,4,5,7],[0,3,4,5,7,8,9,10,11],[0,3,4,5,7,8,9,10,12],[0,3,4,5,7,8,9,11,12],[0,3,4,5,7,8,10,11,12],[0,3,4,5,7,9,10,11,12],[0,3,4,5,8],[0,3,4,5,8,9,10,11,12],[0,3,4,5,9],[0,3,4,5,10],[0,3,4,5,11],[0,3,4,5,12],[0,3,4,6,7],[0,3,4,6,7,8,9,10,11],[0,3,4,6,7,8,9,10,12],[0,3,4,6,7,8,9,11,12],[0,3,4,6,7,8,10,11,12],[0,3,4,6,7,9,10,11,12],[0,3,4,6,8],[0,3,4,6,8,9,10,11,12],[0,3,4,6,9],[0,3,4,6,10],[0,3,4,6,11],[0,3,4,6,12],[0,3,5,6,7],[0,3,5,6,7,8,9,10,11],[0,3,5,6,7,8,9,10,12],[0,3,5,6,7,8,9,11,12],[0,3,5,6,7,8,10,11,12],[0,3,5,6,7,9,10,11,12],[0,3,5,6,8],[0,3,5,6,8,9,10,11,12],[0,3,5,6,9],[0,3,5,6,10],[0,3,5,6,11],[0,3,5,6,12],[0,4,5,6,7],[0,4,5,6,7,8,9,10,11],[0,4,5,6,7,8,9,10,12],[0,4,5,6,7,8,9,11,12],[0,4,5,6,7,8,10,11,12],[0,4,5,6,7,9,10,11,12],[0,4,5,6,8],[0,4,5,6,8,9,10,11,12],[0,4,5,6,9],[0,4,5,6,10],[0,4,5,6,11],[0,4,5,6,12],[0,7,8,9,10],[0,7,8,9,11],[0,7,8,9,12],[0,7,8,10,11],[0,7,8,10,12],[0,7,8,11,12],[0,7,9,10,11],[0,7,9,10,12],[0,7,9,11,12],[0,7,10,11,12],[0,8,9,10,11],[0,8,9,10,12],[0,8,9,11,12],[0,8,10,11,12],[0,9,10,11,12],[1,2,3,4,7],[1,2,3,4,7,8,9,10,11],[1,2,3,4,7,8,9,10,12],[1,2,3,4,7,8,9,11,12],[1,2,3,4,7,8,10,11,12],[1,2,3,4,7,9,10,11,12],[1,2,3,4,8],[1,2,3,4,8,9,10,11,12],[1,2,3,4,9],[1,2,3,4,10],[1,2,3,4,11],[1,2,3,4,12],[1,2,3,5,7],[1,2,3,5,7,8,9,10,11],[1,2,3,5,7,8,9,10,12],[1,2,3,5,7,8,9,11,12],[1,2,3,5,7,8,10,11,12],[1,2,3,5,7,9,10,11,12],[1,2,3,5,8],[1,2,3,5,8,9,10,11,12],[1,2,3,5,9],[1,2,3,5,10],[1,2,3,5,11],[1,2,3,5,12],[1,2,3,6,7],[1,2,3,6,7,8,9,10,11],[1,2,3,6,7,8,9,10,12],[1,2,3,6,7,8,9,11,12],[1,2,3,6,7,8,10,11,12],[1,2,3,6,7,9,10,11,12],[1,2,3,6,8],[1,2,3,6,8,9,10,11,12],[1,2,3,6,9],[1,2,3,6,10],[1,2,3,6,11],[1,2,3,6,12],[1,2,4,5,7],[1,2,4,5,7,8,9,10,11],[1,2,4,5,7,8,9,10,12],[1,2,4,5,7,8,9,11,12],[1,2,4,5,7,8,10,11,12],[1,2,4,5,7,9,10,11,12],[1,2,4,5,8],[1,2,4,5,8,9,10,11,12],[1,2,4,5,9],[1,2,4,5,10],[1,2,4,5,11],[1,2,4,5,12],[1,2,4,6,7],[1,2,4,6,7,8,9,10,11],[1,2,4,6,7,8,9,10,12],[1,2,4,6,7,8,9,11,12],[1,2,4,6,7,8,10,11,12],[1,2,4,6,7,9,10,11,12],[1,2,4,6,8],[1,2,4,6,8,9,10,11,12],[1,2,4,6,9],[1,2,4,6,10],[1,2,4,6,11],[1,2,4,6,12],[1,2,5,6,7],[1,2,5,6,7,8,9,10,11],[1,2,5,6,7,8,9,10,12],[1,2,5,6,7,8,9,11,12],[1,2,5,6,7,8,10,11,12],[1,2,5,6,7,9,10,11,12],[1,2,5,6,8],[1,2,5,6,8,9,10,11,12],[1,2,5,6,9],[1,2,5,6,10],[1,2,5,6,11],[1,2,5,6,12],[1,3,4,5,7],[1,3,4,5,7,8,9,10,11],[1,3,4,5,7,8,9,10,12],[1,3,4,5,7,8,9,11,12],[1,3,4,5,7,8,10,11,12],[1,3,4,5,7,9,10,11,12],[1,3,4,5,8],[1,3,4,5,8,9,10,11,12],[1,3,4,5,9],[1,3,4,5,10],[1,3,4,5,11],[1,3,4,5,12],[1,3,4,6,7],[1,3,4,6,7,8,9,10,11],[1,3,4,6,7,8,9,10,12],[1,3,4,6,7,8,9,11,12],[1,3,4,6,7,8,10,11,12],[1,3,4,6,7,9,10,11,12],[1,3,4,6,8],[1,3,4,6,8,9,10,11,12],[1,3,4,6,9],[1,3,4,6,10],[1,3,4,6,11],[1,3,4,6,12],[1,3,5,6,7],[1,3,5,6,7,8,9,10,11],[1,3,5,6,7,8,9,10,12],[1,3,5,6,7,8,9,11,12],[1,3,5,6,7,8,10,11,12],[1,3,5,6,7,9,10,11,12],[1,3,5,6,8],[1,3,5,6,8,9,10,11,12],[1,3,5,6,9],[1,3,5,6,10],[1,3,5,6,11],[1,3,5,6,12],[1,4,5,6,7],[1,4,5,6,7,8,9,10,11],[1,4,5,6,7,8,9,10,12],[1,4,5,6,7,8,9,11,12],[1,4,5,6,7,8,10,11,12],[1,4,5,6,7,9,10,11,12],[1,4,5,6,8],[1,4,5,6,8,9,10,11,12],[1,4,5,6,9],[1,4,5,6,10],[1,4,5,6,11],[1,4,5,6,12],[1,7,8,9,10],[1,7,8,9,11],[1,7,8,9,12],[1,7,8,10,11],[1,7,8,10,12],[1,7,8,11,12],[1,7,9,10,11],[1,7,9,10,12],[1,7,9,11,12],[1,7,10,11,12],[1,8,9,10,11],[1,8,9,10,12],[1,8,9,11,12],[1,8,10,11,12],[1,9,10,11,12],[2,3,4,5,7],[2,3,4,5,7,8,9,10,11],[2,3,4,5,7,8,9,10,12],[2,3,4,5,7,8,9,11,12],[2,3,4,5,7,8,10,11,12],[2,3,4,5,7,9,10,11,12],[2,3,4,5,8],[2,3,4,5,8,9,10,11,12],[2,3,4,5,9],[2,3,4,5,10],[2,3,4,5,11],[2,3,4,5,12],[2,3,4,6,7],[2,3,4,6,7,8,9,10,11],[2,3,4,6,7,8,9,10,12],[2,3,4,6,7,8,9,11,12],[2,3,4,6,7,8,10,11,12],[2,3,4,6,7,9,10,11,12],[2,3,4,6,8],[2,3,4,6,8,9,10,11,12],[2,3,4,6,9],[2,3,4,6,10],[2,3,4,6,11],[2,3,4,6,12],[2,3,5,6,7],[2,3,5,6,7,8,9,10,11],[2,3,5,6,7,8,9,10,12],[2,3,5,6,7,8,9,11,12],[2,3,5,6,7,8,10,11,12],[2,3,5,6,7,9,10,11,12],[2,3,5,6,8],[2,3,5,6,8,9,10,11,12],[2,3,5,6,9],[2,3,5,6,10],[2,3,5,6,11],[2,3,5,6,12],[2,4,5,6,7],[2,4,5,6,7,8,9,10,11],[2,4,5,6,7,8,9,10,12],[2,4,5,6,7,8,9,11,12],[2,4,5,6,7,8,10,11,12],[2,4,5,6,7,9,10,11,12],[2,4,5,6,8],[2,4,5,6,8,9,10,11,12],[2,4,5,6,9],[2,4,5,6,10],[2,4,5,6,11],[2,4,5,6,12],[2,7,8,9,10],[2,7,8,9,11],[2,7,8,9,12],[2,7,8,10,11],[2,7,8,10,12],[2,7,8,11,12],[2,7,9,10,11],[2,7,9,10,12],[2,7,9,11,12],[2,7,10,11,12],[2,8,9,10,11],[2,8,9,10,12],[2,8,9,11,12],[2,8,10,11,12],[2,9,10,11,12],[3,4,5,6,7],[3,4,5,6,7,8,9,10,11],[3,4,5,6,7,8,9,10,12],[3,4,5,6,7,8,9,11,12],[3,4,5,6,7,8,10,11,12],[3,4,5,6,7,9,10,11,12],[3,4,5,6,8],[3,4,5,6,8,9,10,11,12],[3,4,5,6,9],[3,4,5,6,10],[3,4,5,6,11],[3,4,5,6,12],[3,7,8,9,10],[3,7,8,9,11],[3,7,8,9,12],[3,7,8,10,11],[3,7,8,10,12],[3,7,8,11,12],[3,7,9,10,11],[3,7,9,10,12],[3,7,9,11,12],[3,7,10,11,12],[3,8,9,10,11],[3,8,9,10,12],[3,8,9,11,12],[3,8,10,11,12],[3,9,10,11,12],[4,7,8,9,10],[4,7,8,9,11],[4,7,8,9,12],[4,7,8,10,11],[4,7,8,10,12],[4,7,8,11,12],[4,7,9,10,11],[4,7,9,10,12],[4,7,9,11,12],[4,7,10,11,12],[4,8,9,10,11],[4,8,9,10,12],[4,8,9,11,12],[4,8,10,11,12],[4,9,10,11,12],[5,7,8,9,10],[5,7,8,9,11],[5,7,8,9,12],[5,7,8,10,11],[5,7,8,10,12],[5,7,8,11,12],[5,7,9,10,11],[5,7,9,10,12],[5,7,9,11,12],[5,7,10,11,12],[5,8,9,10,11],[5,8,9,10,12],[5,8,9,11,12],[5,8,10,11,12],[5,9,10,11,12],[6,7,8,9,10],[6,7,8,9,11],[6,7,8,9,12],[6,7,8,10,11],[6,7,8,10,12],[6,7,8,11,12],[6,7,9,10,11],[6,7,9,10,12],[6,7,9,11,12],[6,7,10,11,12],[6,8,9,10,11],[6,8,9,10,12],[6,8,9,11,12],[6,8,10,11,12],[6,9,10,11,12],[7],[7,8,9,10,11],[7,8,9,10,12],[7,8,9,11,12],[7,8,10,11,12],[7,9,10,11,12],[8],[8,9,10,11,12],[9],[10],[11],[12]]},"distance":{"d":2,"is_exact":true,"upper":2,"witness":[16,28],"A_d":7875,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,1],[1,1],[2,1],[3,1],[4,1],[5,1],[6,1],[7,2],[8,2],[9,2],[10,2],[11,2],[12,2]],"CZ":[]},"rotation_powers":[[0,3],[1,7],[2,5],[6,5],[270,3],[405,3],[468,3],[495,3],[510,3],[525,3],[540,3],[542,3],[543,3],[544,3],[545,3],[549,3],[550,3],[551,3],[552,3]]},"metrics":{"gamma":6.303780748177103,"gamma_t":null,"gamma_rho":6.303780748177103,"gamma_claim":6.303780748177103,"gamma_t_claim":null,"gamma_rho_claim":6.303780748177103,"v_ex":7,"rate":0.012658227848101266,"t_count":null,"poly_degree":null,"effective_width":7},"provenance":{"regimes":["borrowed-identity search: two-group"],"discovery":"pre-existing","strongest_claim":"found by the two-group borrowed-identity search of Singh, Gidney and Jones (arXiv:2606.28518; borrowed_identities/): an identity circuit symmetric within its output and check blocks, with its output-only gates removed; a verified witness, not a maximum","sources":[{"regime":"borrowed-identity search: two-group","file":"borrowed_identities/circuits/large_unified_l3.json","label":"lu-0752","N":13,"d":2,"d_is_exact":true,"origin":"https://github.com/shraggy/Magic_state_factory_search at ae9fbf8, outputs/large_unified_factories_l3.csv line 752","provenance":"larger unified search l=3, two-group s_total=4,s_O=3"}]},"references":[{"key":"singh2026borrowed","short":"Singh et al. (2026)","full":"S. Singh, C. Gidney, and C. Jones, \"Borrowed Identities: Malleable Distillation Factories and a Unified Numerical Search,\" arXiv:2606.28518 (2026).","url":"https://arxiv.org/abs/2606.28518","date":"2026-06-26"}]}