{"id":"56.4.2.a","row":329,"stored":{"file":"master_catalog.json","row":329,"construction":null,"columns_sha256":null},"parameters":{"n":56,"k":4,"d":2,"N":7,"r":3,"level":3},"gate":{"string":"012+013+023+123","human":"CCZ012\u00b7CCZ013\u00b7CCZ023\u00b7CCZ123","monomials":[[0,1,2],[0,1,3],[0,2,3],[1,2,3]],"sk_key":[[0,1,2],[0,1,3],[0,2,3],[1,2,3]],"sk_fingerprint":"43e6f7db38f09da5","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[4],[5],[6],[4,5,6],[0,4,5],[0,4,6],[0,5,6],[1,4,5],[1,4,6],[1,5,6],[2,4,5],[2,4,6],[2,5,6],[3,4,5],[3,4,6],[3,5,6],[0,1,4],[0,1,5],[0,1,6],[0,2,4],[0,2,5],[0,2,6],[0,3,4],[0,3,5],[0,3,6],[1,2,4],[1,2,5],[1,2,6],[1,3,4],[1,3,5],[1,3,6],[2,3,4],[2,3,5],[2,3,6],[0,1,4,5,6],[0,2,4,5,6],[0,3,4,5,6],[1,2,4,5,6],[1,3,4,5,6],[2,3,4,5,6],[0,1,2,4,5],[0,1,2,4,6],[0,1,2,5,6],[0,1,3,4,5],[0,1,3,4,6],[0,1,3,5,6],[0,2,3,4,5],[0,2,3,4,6],[0,2,3,5,6],[1,2,3,4,5],[1,2,3,4,6],[1,2,3,5,6],[0,1,2,3,4],[0,1,2,3,5],[0,1,2,3,6],[0,1,2,3,4,5,6]]},"distance":{"d":2,"is_exact":true,"upper":2,"witness":[0,16],"A_d":196,"certified":null},"clifford":{"kind":"required","correction":{"S":[[0,2],[1,2],[2,2],[3,2]],"CZ":[[0,1],[0,2],[0,3],[1,2],[1,3],[2,3]]},"rotation_powers":null},"metrics":{"gamma":3.8073549220576037,"gamma_t":3.0,"gamma_rho":null,"gamma_claim":3.8073549220576037,"gamma_t_claim":3.0,"gamma_rho_claim":null,"v_ex":null,"rate":0.07142857142857142,"t_count":7,"poly_degree":3,"effective_width":4},"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/circuits_l3.json","label":"l3-row093-two-group","N":7,"d":2,"d_is_exact":true,"origin":"https://github.com/shraggy/Magic_state_factory_search at cae4982, outputs/factory_catalogue_l3.csv row 93","provenance":"two-group l=3 n=7 k=4 s_total=2 s_O=1 s_S=1"}]},"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"}]}