{"id":"24.3.2.a","row":29,"stored":{"file":"master_catalog.json","row":29,"construction":null,"columns_sha256":null},"parameters":{"n":24,"k":3,"d":2,"N":5,"r":2,"level":3},"gate":{"string":"012","human":"CCZ012","monomials":[[0,1,2]],"sk_key":[[0,1,2]],"sk_fingerprint":"d0c21becf7c06e56","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[3],[4],[3,4],[0,3],[0,4],[1,3],[1,4],[2,3],[2,4],[0,3,4],[1,3,4],[2,3,4],[0,1,3],[0,1,4],[0,2,3],[0,2,4],[1,2,3],[1,2,4],[0,1,3,4],[0,2,3,4],[1,2,3,4],[0,1,2,3],[0,1,2,4],[0,1,2,3,4]]},"distance":{"d":2,"is_exact":true,"upper":2,"witness":[0,3],"A_d":84,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,2],[1,2],[2,2]],"CZ":[[0,1],[0,2],[1,2]]},"rotation_powers":[[3,3],[5,3],[7,3],[21,3]]},"metrics":{"gamma":3.0,"gamma_t":1.7776075786635521,"gamma_rho":3.5849625007211565,"gamma_claim":3.0,"gamma_t_claim":1.7776075786635521,"gamma_rho_claim":3.5849625007211565,"v_ex":2,"rate":0.125,"t_count":7,"poly_degree":3,"effective_width":3},"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-row052-two-group","N":5,"d":2,"d_is_exact":true,"origin":"https://github.com/shraggy/Magic_state_factory_search at cae4982, outputs/factory_catalogue_l3.csv row 52","provenance":"two-group l=3 n=5 k=3 s_total=1 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"}]}