{"id":"396.2.2.a","row":1016,"stored":{"file":"master_catalog.json","row":1016,"construction":null,"columns_sha256":null},"parameters":{"n":396,"k":2,"d":2,"N":13,"r":11,"level":3},"gate":{"string":"0+1","human":"T0\u00b7T1","monomials":[[0],[1]],"sk_key":[[0],[1]],"sk_fingerprint":"68f83724f5668f5b","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[0,2,3,4,5,6,7,8,9],[0,2,3,4,5,6,7,8,10],[0,2,3,4,5,6,7,8,11],[0,2,3,4,5,6,7,8,12],[0,2,3,4,5,6,7,9,10],[0,2,3,4,5,6,7,9,11],[0,2,3,4,5,6,7,9,12],[0,2,3,4,5,6,7,10,11],[0,2,3,4,5,6,7,10,12],[0,2,3,4,5,6,7,11,12],[0,2,3,4,5,6,8,9,10],[0,2,3,4,5,6,8,9,11],[0,2,3,4,5,6,8,9,12],[0,2,3,4,5,6,8,10,11],[0,2,3,4,5,6,8,10,12],[0,2,3,4,5,6,8,11,12],[0,2,3,4,5,6,9,10,11],[0,2,3,4,5,6,9,10,12],[0,2,3,4,5,6,9,11,12],[0,2,3,4,5,6,10,11,12],[0,2,3,4,5,7,8,9,10],[0,2,3,4,5,7,8,9,11],[0,2,3,4,5,7,8,9,12],[0,2,3,4,5,7,8,10,11],[0,2,3,4,5,7,8,10,12],[0,2,3,4,5,7,8,11,12],[0,2,3,4,5,7,9,10,11],[0,2,3,4,5,7,9,10,12],[0,2,3,4,5,7,9,11,12],[0,2,3,4,5,7,10,11,12],[0,2,3,4,5,8,9,10,11],[0,2,3,4,5,8,9,10,12],[0,2,3,4,5,8,9,11,12],[0,2,3,4,5,8,10,11,12],[0,2,3,4,5,9,10,11,12],[0,2,3,4,6,7,8,9,10],[0,2,3,4,6,7,8,9,11],[0,2,3,4,6,7,8,9,12],[0,2,3,4,6,7,8,10,11],[0,2,3,4,6,7,8,10,12],[0,2,3,4,6,7,8,11,12],[0,2,3,4,6,7,9,10,11],[0,2,3,4,6,7,9,10,12],[0,2,3,4,6,7,9,11,12],[0,2,3,4,6,7,10,11,12],[0,2,3,4,6,8,9,10,11],[0,2,3,4,6,8,9,10,12],[0,2,3,4,6,8,9,11,12],[0,2,3,4,6,8,10,11,12],[0,2,3,4,6,9,10,11,12],[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],[0,2,3,4,8,9,10,11,12],[0,2,3,5,6,7,8,9,10],[0,2,3,5,6,7,8,9,11],[0,2,3,5,6,7,8,9,12],[0,2,3,5,6,7,8,10,11],[0,2,3,5,6,7,8,10,12],[0,2,3,5,6,7,8,11,12],[0,2,3,5,6,7,9,10,11],[0,2,3,5,6,7,9,10,12],[0,2,3,5,6,7,9,11,12],[0,2,3,5,6,7,10,11,12],[0,2,3,5,6,8,9,10,11],[0,2,3,5,6,8,9,10,12],[0,2,3,5,6,8,9,11,12],[0,2,3,5,6,8,10,11,12],[0,2,3,5,6,9,10,11,12],[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],[0,2,3,5,8,9,10,11,12],[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],[0,2,3,6,8,9,10,11,12],[0,2,3,7,8,9,10,11,12],[0,2,4,5,6,7,8,9,10],[0,2,4,5,6,7,8,9,11],[0,2,4,5,6,7,8,9,12],[0,2,4,5,6,7,8,10,11],[0,2,4,5,6,7,8,10,12],[0,2,4,5,6,7,8,11,12],[0,2,4,5,6,7,9,10,11],[0,2,4,5,6,7,9,10,12],[0,2,4,5,6,7,9,11,12],[0,2,4,5,6,7,10,11,12],[0,2,4,5,6,8,9,10,11],[0,2,4,5,6,8,9,10,12],[0,2,4,5,6,8,9,11,12],[0,2,4,5,6,8,10,11,12],[0,2,4,5,6,9,10,11,12],[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],[0,2,4,5,8,9,10,11,12],[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],[0,2,4,6,8,9,10,11,12],[0,2,4,7,8,9,10,11,12],[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],[0,2,5,6,8,9,10,11,12],[0,2,5,7,8,9,10,11,12],[0,2,6,7,8,9,10,11,12],[0,3,4,5,6,7,8,9,10],[0,3,4,5,6,7,8,9,11],[0,3,4,5,6,7,8,9,12],[0,3,4,5,6,7,8,10,11],[0,3,4,5,6,7,8,10,12],[0,3,4,5,6,7,8,11,12],[0,3,4,5,6,7,9,10,11],[0,3,4,5,6,7,9,10,12],[0,3,4,5,6,7,9,11,12],[0,3,4,5,6,7,10,11,12],[0,3,4,5,6,8,9,10,11],[0,3,4,5,6,8,9,10,12],[0,3,4,5,6,8,9,11,12],[0,3,4,5,6,8,10,11,12],[0,3,4,5,6,9,10,11,12],[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],[0,3,4,5,8,9,10,11,12],[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],[0,3,4,6,8,9,10,11,12],[0,3,4,7,8,9,10,11,12],[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],[0,3,5,6,8,9,10,11,12],[0,3,5,7,8,9,10,11,12],[0,3,6,7,8,9,10,11,12],[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],[0,4,5,6,8,9,10,11,12],[0,4,5,7,8,9,10,11,12],[0,4,6,7,8,9,10,11,12],[0,5,6,7,8,9,10,11,12],[1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,10],[1,2,3,4,5,6,7,8,11],[1,2,3,4,5,6,7,8,12],[1,2,3,4,5,6,7,9,10],[1,2,3,4,5,6,7,9,11],[1,2,3,4,5,6,7,9,12],[1,2,3,4,5,6,7,10,11],[1,2,3,4,5,6,7,10,12],[1,2,3,4,5,6,7,11,12],[1,2,3,4,5,6,8,9,10],[1,2,3,4,5,6,8,9,11],[1,2,3,4,5,6,8,9,12],[1,2,3,4,5,6,8,10,11],[1,2,3,4,5,6,8,10,12],[1,2,3,4,5,6,8,11,12],[1,2,3,4,5,6,9,10,11],[1,2,3,4,5,6,9,10,12],[1,2,3,4,5,6,9,11,12],[1,2,3,4,5,6,10,11,12],[1,2,3,4,5,7,8,9,10],[1,2,3,4,5,7,8,9,11],[1,2,3,4,5,7,8,9,12],[1,2,3,4,5,7,8,10,11],[1,2,3,4,5,7,8,10,12],[1,2,3,4,5,7,8,11,12],[1,2,3,4,5,7,9,10,11],[1,2,3,4,5,7,9,10,12],[1,2,3,4,5,7,9,11,12],[1,2,3,4,5,7,10,11,12],[1,2,3,4,5,8,9,10,11],[1,2,3,4,5,8,9,10,12],[1,2,3,4,5,8,9,11,12],[1,2,3,4,5,8,10,11,12],[1,2,3,4,5,9,10,11,12],[1,2,3,4,6,7,8,9,10],[1,2,3,4,6,7,8,9,11],[1,2,3,4,6,7,8,9,12],[1,2,3,4,6,7,8,10,11],[1,2,3,4,6,7,8,10,12],[1,2,3,4,6,7,8,11,12],[1,2,3,4,6,7,9,10,11],[1,2,3,4,6,7,9,10,12],[1,2,3,4,6,7,9,11,12],[1,2,3,4,6,7,10,11,12],[1,2,3,4,6,8,9,10,11],[1,2,3,4,6,8,9,10,12],[1,2,3,4,6,8,9,11,12],[1,2,3,4,6,8,10,11,12],[1,2,3,4,6,9,10,11,12],[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],[1,2,3,4,8,9,10,11,12],[1,2,3,5,6,7,8,9,10],[1,2,3,5,6,7,8,9,11],[1,2,3,5,6,7,8,9,12],[1,2,3,5,6,7,8,10,11],[1,2,3,5,6,7,8,10,12],[1,2,3,5,6,7,8,11,12],[1,2,3,5,6,7,9,10,11],[1,2,3,5,6,7,9,10,12],[1,2,3,5,6,7,9,11,12],[1,2,3,5,6,7,10,11,12],[1,2,3,5,6,8,9,10,11],[1,2,3,5,6,8,9,10,12],[1,2,3,5,6,8,9,11,12],[1,2,3,5,6,8,10,11,12],[1,2,3,5,6,9,10,11,12],[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],[1,2,3,5,8,9,10,11,12],[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],[1,2,3,6,8,9,10,11,12],[1,2,3,7,8,9,10,11,12],[1,2,4,5,6,7,8,9,10],[1,2,4,5,6,7,8,9,11],[1,2,4,5,6,7,8,9,12],[1,2,4,5,6,7,8,10,11],[1,2,4,5,6,7,8,10,12],[1,2,4,5,6,7,8,11,12],[1,2,4,5,6,7,9,10,11],[1,2,4,5,6,7,9,10,12],[1,2,4,5,6,7,9,11,12],[1,2,4,5,6,7,10,11,12],[1,2,4,5,6,8,9,10,11],[1,2,4,5,6,8,9,10,12],[1,2,4,5,6,8,9,11,12],[1,2,4,5,6,8,10,11,12],[1,2,4,5,6,9,10,11,12],[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],[1,2,4,5,8,9,10,11,12],[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],[1,2,4,6,8,9,10,11,12],[1,2,4,7,8,9,10,11,12],[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],[1,2,5,6,8,9,10,11,12],[1,2,5,7,8,9,10,11,12],[1,2,6,7,8,9,10,11,12],[1,3,4,5,6,7,8,9,10],[1,3,4,5,6,7,8,9,11],[1,3,4,5,6,7,8,9,12],[1,3,4,5,6,7,8,10,11],[1,3,4,5,6,7,8,10,12],[1,3,4,5,6,7,8,11,12],[1,3,4,5,6,7,9,10,11],[1,3,4,5,6,7,9,10,12],[1,3,4,5,6,7,9,11,12],[1,3,4,5,6,7,10,11,12],[1,3,4,5,6,8,9,10,11],[1,3,4,5,6,8,9,10,12],[1,3,4,5,6,8,9,11,12],[1,3,4,5,6,8,10,11,12],[1,3,4,5,6,9,10,11,12],[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],[1,3,4,5,8,9,10,11,12],[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],[1,3,4,6,8,9,10,11,12],[1,3,4,7,8,9,10,11,12],[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],[1,3,5,6,8,9,10,11,12],[1,3,5,7,8,9,10,11,12],[1,3,6,7,8,9,10,11,12],[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],[1,4,5,6,8,9,10,11,12],[1,4,5,7,8,9,10,11,12],[1,4,6,7,8,9,10,11,12],[1,5,6,7,8,9,10,11,12],[2],[2,3,4,5,6,7,8,9,10],[2,3,4,5,6,7,8,9,11],[2,3,4,5,6,7,8,9,12],[2,3,4,5,6,7,8,10,11],[2,3,4,5,6,7,8,10,12],[2,3,4,5,6,7,8,11,12],[2,3,4,5,6,7,9,10,11],[2,3,4,5,6,7,9,10,12],[2,3,4,5,6,7,9,11,12],[2,3,4,5,6,7,10,11,12],[2,3,4,5,6,8,9,10,11],[2,3,4,5,6,8,9,10,12],[2,3,4,5,6,8,9,11,12],[2,3,4,5,6,8,10,11,12],[2,3,4,5,6,9,10,11,12],[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],[2,3,4,5,8,9,10,11,12],[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],[2,3,4,6,8,9,10,11,12],[2,3,4,7,8,9,10,11,12],[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],[2,3,5,6,8,9,10,11,12],[2,3,5,7,8,9,10,11,12],[2,3,6,7,8,9,10,11,12],[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],[2,4,5,6,8,9,10,11,12],[2,4,5,7,8,9,10,11,12],[2,4,6,7,8,9,10,11,12],[2,5,6,7,8,9,10,11,12],[3],[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],[3,4,5,6,8,9,10,11,12],[3,4,5,7,8,9,10,11,12],[3,4,6,7,8,9,10,11,12],[3,5,6,7,8,9,10,11,12],[4],[4,5,6,7,8,9,10,11,12],[5],[6],[7],[8],[9],[10],[11],[12]]},"distance":{"d":2,"is_exact":true,"upper":2,"witness":[0,165],"A_d":165,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,2],[1,2],[2,1],[3,1],[4,1],[5,1],[6,1],[7,1],[8,1],[9,1],[10,1],[11,1],[12,1]],"CZ":[]},"rotation_powers":[[0,5],[165,5],[330,3],[377,3],[378,3],[379,3],[380,3],[381,3],[382,3],[383,3],[384,3],[385,3],[387,3]]},"metrics":{"gamma":7.6293566200796095,"gamma_t":7.6293566200796095,"gamma_rho":7.6293566200796095,"gamma_claim":7.6293566200796095,"gamma_t_claim":7.6293566200796095,"gamma_rho_claim":7.6293566200796095,"v_ex":2,"rate":0.005050505050505051,"t_count":2,"poly_degree":1,"effective_width":2},"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-0082","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 82","provenance":"larger unified search l=3, two-group s_total=8,s_O=2"}]},"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"}]}