{"id":"29.3.2.a","row":58,"stored":{"file":"master_catalog.json","row":58,"construction":null,"columns_sha256":null},"parameters":{"n":29,"k":3,"d":2,"N":7,"r":4,"level":3},"gate":{"string":"0+1+2","human":"T0\u00b7T1\u00b7T2","monomials":[[0],[1],[2]],"sk_key":[[0],[1],[2]],"sk_fingerprint":"b0744d1878b2f54b","sk_canonical_frame":true,"relabelled_into_canonical_frame":false},"circuit":{"columns":[[3],[4],[5],[6],[3,4,5],[3,4,6],[3,5,6],[4,5,6],[0,3,4],[0,3,5],[0,3,6],[0,4,5],[0,4,6],[0,5,6],[1,3,4],[1,3,5],[1,3,6],[1,4,5],[1,4,6],[1,5,6],[2,3,4],[2,3,5],[2,3,6],[2,4,5],[2,4,6],[2,5,6],[0,3,4,5,6],[1,3,4,5,6],[2,3,4,5,6]]},"distance":{"d":2,"is_exact":true,"upper":2,"witness":[8,14],"A_d":21,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,1],[1,1],[2,1]],"CZ":[]},"rotation_powers":[[0,7],[1,7],[2,7],[3,7],[4,7],[5,7],[6,7],[7,7],[8,7],[9,7],[10,7],[11,7],[12,7],[13,7],[14,7],[15,7],[16,7],[17,7],[18,7],[19,7],[20,7],[21,7],[22,7],[23,7],[24,7],[25,7],[26,7],[27,7],[28,7]]},"metrics":{"gamma":3.273018494406416,"gamma_t":3.273018494406416,"gamma_rho":3.273018494406416,"gamma_claim":3.273018494406416,"gamma_t_claim":3.273018494406416,"gamma_rho_claim":3.273018494406416,"v_ex":3,"rate":0.10344827586206896,"t_count":3,"poly_degree":1,"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-row054-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 54","provenance":"two-group l=3 n=7 k=3 s_total=2 s_O=3 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"}]}