{"id":"20.4.2.a","row":21,"stored":{"file":"master_catalog.json","row":21,"construction":null,"columns_sha256":null},"parameters":{"n":20,"k":4,"d":2,"N":7,"r":3,"level":3},"gate":{"string":"0+1+2+3","human":"T0\u00b7T1\u00b7T2\u00b7T3","monomials":[[0],[1],[2],[3]],"sk_key":[[0],[1],[2],[3]],"sk_fingerprint":"473e0903fa3e9dc7","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],[0,1,2,3,4],[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":22,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,1],[1,1],[2,1],[3,1],[4,2],[5,2],[6,2]],"CZ":[[4,5],[4,6],[5,6]]},"rotation_powers":[[0,3],[1,3],[4,7],[7,7],[10,7],[13,7],[18,3],[19,3]]},"metrics":{"gamma":2.321928094887362,"gamma_t":2.321928094887362,"gamma_rho":2.321928094887362,"gamma_claim":2.321928094887362,"gamma_t_claim":2.321928094887362,"gamma_rho_claim":2.321928094887362,"v_ex":4,"rate":0.2,"t_count":4,"poly_degree":1,"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-row086-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 86","provenance":"two-group l=3 n=7 k=4 s_total=2 s_O=3 s_S=1","notes":"earlier publication: the k = 4 member of the Bravyi-Haah [[3k+8,k,2]] family (arXiv:2606.28518, App. F.3a)"}]},"references":[{"key":"bravyi2012magic","short":"Bravyi & Haah (2012)","full":"S. Bravyi and J. Haah, \"Magic-state distillation with low overhead,\" Phys. Rev. A 86, 052329 (2012).","url":"https://doi.org/10.1103/PhysRevA.86.052329","date":"2012-11-27"}]}