{"id":"406.2.2.a","row":1028,"stored":{"file":"master_catalog.json","row":1028,"construction":null,"columns_sha256":null},"parameters":{"n":406,"k":2,"d":2,"N":11,"r":9,"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":[[2],[3],[4],[5],[6],[7],[8],[9],[10],[2,3,4,5,6],[2,3,4,5,7],[2,3,4,5,8],[2,3,4,5,9],[2,3,4,5,10],[2,3,4,6,7],[2,3,4,6,8],[2,3,4,6,9],[2,3,4,6,10],[2,3,4,7,8],[2,3,4,7,9],[2,3,4,7,10],[2,3,4,8,9],[2,3,4,8,10],[2,3,4,9,10],[2,3,5,6,7],[2,3,5,6,8],[2,3,5,6,9],[2,3,5,6,10],[2,3,5,7,8],[2,3,5,7,9],[2,3,5,7,10],[2,3,5,8,9],[2,3,5,8,10],[2,3,5,9,10],[2,3,6,7,8],[2,3,6,7,9],[2,3,6,7,10],[2,3,6,8,9],[2,3,6,8,10],[2,3,6,9,10],[2,3,7,8,9],[2,3,7,8,10],[2,3,7,9,10],[2,3,8,9,10],[2,4,5,6,7],[2,4,5,6,8],[2,4,5,6,9],[2,4,5,6,10],[2,4,5,7,8],[2,4,5,7,9],[2,4,5,7,10],[2,4,5,8,9],[2,4,5,8,10],[2,4,5,9,10],[2,4,6,7,8],[2,4,6,7,9],[2,4,6,7,10],[2,4,6,8,9],[2,4,6,8,10],[2,4,6,9,10],[2,4,7,8,9],[2,4,7,8,10],[2,4,7,9,10],[2,4,8,9,10],[2,5,6,7,8],[2,5,6,7,9],[2,5,6,7,10],[2,5,6,8,9],[2,5,6,8,10],[2,5,6,9,10],[2,5,7,8,9],[2,5,7,8,10],[2,5,7,9,10],[2,5,8,9,10],[2,6,7,8,9],[2,6,7,8,10],[2,6,7,9,10],[2,6,8,9,10],[2,7,8,9,10],[3,4,5,6,7],[3,4,5,6,8],[3,4,5,6,9],[3,4,5,6,10],[3,4,5,7,8],[3,4,5,7,9],[3,4,5,7,10],[3,4,5,8,9],[3,4,5,8,10],[3,4,5,9,10],[3,4,6,7,8],[3,4,6,7,9],[3,4,6,7,10],[3,4,6,8,9],[3,4,6,8,10],[3,4,6,9,10],[3,4,7,8,9],[3,4,7,8,10],[3,4,7,9,10],[3,4,8,9,10],[3,5,6,7,8],[3,5,6,7,9],[3,5,6,7,10],[3,5,6,8,9],[3,5,6,8,10],[3,5,6,9,10],[3,5,7,8,9],[3,5,7,8,10],[3,5,7,9,10],[3,5,8,9,10],[3,6,7,8,9],[3,6,7,8,10],[3,6,7,9,10],[3,6,8,9,10],[3,7,8,9,10],[4,5,6,7,8],[4,5,6,7,9],[4,5,6,7,10],[4,5,6,8,9],[4,5,6,8,10],[4,5,6,9,10],[4,5,7,8,9],[4,5,7,8,10],[4,5,7,9,10],[4,5,8,9,10],[4,6,7,8,9],[4,6,7,8,10],[4,6,7,9,10],[4,6,8,9,10],[4,7,8,9,10],[5,6,7,8,9],[5,6,7,8,10],[5,6,7,9,10],[5,6,8,9,10],[5,7,8,9,10],[6,7,8,9,10],[2,3,4,5,6,7,8,9,10],[0,2,3,4,5],[0,2,3,4,6],[0,2,3,4,7],[0,2,3,4,8],[0,2,3,4,9],[0,2,3,4,10],[0,2,3,5,6],[0,2,3,5,7],[0,2,3,5,8],[0,2,3,5,9],[0,2,3,5,10],[0,2,3,6,7],[0,2,3,6,8],[0,2,3,6,9],[0,2,3,6,10],[0,2,3,7,8],[0,2,3,7,9],[0,2,3,7,10],[0,2,3,8,9],[0,2,3,8,10],[0,2,3,9,10],[0,2,4,5,6],[0,2,4,5,7],[0,2,4,5,8],[0,2,4,5,9],[0,2,4,5,10],[0,2,4,6,7],[0,2,4,6,8],[0,2,4,6,9],[0,2,4,6,10],[0,2,4,7,8],[0,2,4,7,9],[0,2,4,7,10],[0,2,4,8,9],[0,2,4,8,10],[0,2,4,9,10],[0,2,5,6,7],[0,2,5,6,8],[0,2,5,6,9],[0,2,5,6,10],[0,2,5,7,8],[0,2,5,7,9],[0,2,5,7,10],[0,2,5,8,9],[0,2,5,8,10],[0,2,5,9,10],[0,2,6,7,8],[0,2,6,7,9],[0,2,6,7,10],[0,2,6,8,9],[0,2,6,8,10],[0,2,6,9,10],[0,2,7,8,9],[0,2,7,8,10],[0,2,7,9,10],[0,2,8,9,10],[0,3,4,5,6],[0,3,4,5,7],[0,3,4,5,8],[0,3,4,5,9],[0,3,4,5,10],[0,3,4,6,7],[0,3,4,6,8],[0,3,4,6,9],[0,3,4,6,10],[0,3,4,7,8],[0,3,4,7,9],[0,3,4,7,10],[0,3,4,8,9],[0,3,4,8,10],[0,3,4,9,10],[0,3,5,6,7],[0,3,5,6,8],[0,3,5,6,9],[0,3,5,6,10],[0,3,5,7,8],[0,3,5,7,9],[0,3,5,7,10],[0,3,5,8,9],[0,3,5,8,10],[0,3,5,9,10],[0,3,6,7,8],[0,3,6,7,9],[0,3,6,7,10],[0,3,6,8,9],[0,3,6,8,10],[0,3,6,9,10],[0,3,7,8,9],[0,3,7,8,10],[0,3,7,9,10],[0,3,8,9,10],[0,4,5,6,7],[0,4,5,6,8],[0,4,5,6,9],[0,4,5,6,10],[0,4,5,7,8],[0,4,5,7,9],[0,4,5,7,10],[0,4,5,8,9],[0,4,5,8,10],[0,4,5,9,10],[0,4,6,7,8],[0,4,6,7,9],[0,4,6,7,10],[0,4,6,8,9],[0,4,6,8,10],[0,4,6,9,10],[0,4,7,8,9],[0,4,7,8,10],[0,4,7,9,10],[0,4,8,9,10],[0,5,6,7,8],[0,5,6,7,9],[0,5,6,7,10],[0,5,6,8,9],[0,5,6,8,10],[0,5,6,9,10],[0,5,7,8,9],[0,5,7,8,10],[0,5,7,9,10],[0,5,8,9,10],[0,6,7,8,9],[0,6,7,8,10],[0,6,7,9,10],[0,6,8,9,10],[0,7,8,9,10],[1,2,3,4,5],[1,2,3,4,6],[1,2,3,4,7],[1,2,3,4,8],[1,2,3,4,9],[1,2,3,4,10],[1,2,3,5,6],[1,2,3,5,7],[1,2,3,5,8],[1,2,3,5,9],[1,2,3,5,10],[1,2,3,6,7],[1,2,3,6,8],[1,2,3,6,9],[1,2,3,6,10],[1,2,3,7,8],[1,2,3,7,9],[1,2,3,7,10],[1,2,3,8,9],[1,2,3,8,10],[1,2,3,9,10],[1,2,4,5,6],[1,2,4,5,7],[1,2,4,5,8],[1,2,4,5,9],[1,2,4,5,10],[1,2,4,6,7],[1,2,4,6,8],[1,2,4,6,9],[1,2,4,6,10],[1,2,4,7,8],[1,2,4,7,9],[1,2,4,7,10],[1,2,4,8,9],[1,2,4,8,10],[1,2,4,9,10],[1,2,5,6,7],[1,2,5,6,8],[1,2,5,6,9],[1,2,5,6,10],[1,2,5,7,8],[1,2,5,7,9],[1,2,5,7,10],[1,2,5,8,9],[1,2,5,8,10],[1,2,5,9,10],[1,2,6,7,8],[1,2,6,7,9],[1,2,6,7,10],[1,2,6,8,9],[1,2,6,8,10],[1,2,6,9,10],[1,2,7,8,9],[1,2,7,8,10],[1,2,7,9,10],[1,2,8,9,10],[1,3,4,5,6],[1,3,4,5,7],[1,3,4,5,8],[1,3,4,5,9],[1,3,4,5,10],[1,3,4,6,7],[1,3,4,6,8],[1,3,4,6,9],[1,3,4,6,10],[1,3,4,7,8],[1,3,4,7,9],[1,3,4,7,10],[1,3,4,8,9],[1,3,4,8,10],[1,3,4,9,10],[1,3,5,6,7],[1,3,5,6,8],[1,3,5,6,9],[1,3,5,6,10],[1,3,5,7,8],[1,3,5,7,9],[1,3,5,7,10],[1,3,5,8,9],[1,3,5,8,10],[1,3,5,9,10],[1,3,6,7,8],[1,3,6,7,9],[1,3,6,7,10],[1,3,6,8,9],[1,3,6,8,10],[1,3,6,9,10],[1,3,7,8,9],[1,3,7,8,10],[1,3,7,9,10],[1,3,8,9,10],[1,4,5,6,7],[1,4,5,6,8],[1,4,5,6,9],[1,4,5,6,10],[1,4,5,7,8],[1,4,5,7,9],[1,4,5,7,10],[1,4,5,8,9],[1,4,5,8,10],[1,4,5,9,10],[1,4,6,7,8],[1,4,6,7,9],[1,4,6,7,10],[1,4,6,8,9],[1,4,6,8,10],[1,4,6,9,10],[1,4,7,8,9],[1,4,7,8,10],[1,4,7,9,10],[1,4,8,9,10],[1,5,6,7,8],[1,5,6,7,9],[1,5,6,7,10],[1,5,6,8,9],[1,5,6,8,10],[1,5,6,9,10],[1,5,7,8,9],[1,5,7,8,10],[1,5,7,9,10],[1,5,8,9,10],[1,6,7,8,9],[1,6,7,8,10],[1,6,7,9,10],[1,6,8,9,10],[1,7,8,9,10],[0,2,3,4,5,6,7,8,9],[0,2,3,4,5,6,7,8,10],[0,2,3,4,5,6,7,9,10],[0,2,3,4,5,6,8,9,10],[0,2,3,4,5,7,8,9,10],[0,2,3,4,6,7,8,9,10],[0,2,3,5,6,7,8,9,10],[0,2,4,5,6,7,8,9,10],[0,3,4,5,6,7,8,9,10],[1,2,3,4,5,6,7,8,9],[1,2,3,4,5,6,7,8,10],[1,2,3,4,5,6,7,9,10],[1,2,3,4,5,6,8,9,10],[1,2,3,4,5,7,8,9,10],[1,2,3,4,6,7,8,9,10],[1,2,3,5,6,7,8,9,10],[1,2,4,5,6,7,8,9,10],[1,3,4,5,6,7,8,9,10]]},"distance":{"d":2,"is_exact":true,"upper":2,"witness":[136,262],"A_d":135,"certified":null},"clifford":{"kind":"powers","correction":{"S":[[0,1],[1,1]],"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],[29,7],[30,7],[31,7],[32,7],[33,7],[34,7],[35,7],[36,7],[37,7],[38,7],[39,7],[40,7],[41,7],[42,7],[43,7],[44,7],[45,7],[46,7],[47,7],[48,7],[49,7],[50,7],[51,7],[52,7],[53,7],[54,7],[55,7],[56,7],[57,7],[58,7],[59,7],[60,7],[61,7],[62,7],[63,7],[64,7],[65,7],[66,7],[67,7],[68,7],[69,7],[70,7],[71,7],[72,7],[73,7],[74,7],[75,7],[76,7],[77,7],[78,7],[79,7],[80,7],[81,7],[82,7],[83,7],[84,7],[85,7],[86,7],[87,7],[88,7],[89,7],[90,7],[91,7],[92,7],[93,7],[94,7],[95,7],[96,7],[97,7],[98,7],[99,7],[100,7],[101,7],[102,7],[103,7],[104,7],[105,7],[106,7],[107,7],[108,7],[109,7],[110,7],[111,7],[112,7],[113,7],[114,7],[115,7],[116,7],[117,7],[118,7],[119,7],[120,7],[121,7],[122,7],[123,7],[124,7],[125,7],[126,7],[127,7],[128,7],[129,7],[130,7],[131,7],[132,7],[133,7],[134,7],[135,7],[136,7],[137,7],[138,7],[139,7],[140,7],[141,7],[142,7],[143,7],[144,7],[145,7],[146,7],[147,7],[148,7],[149,7],[150,7],[151,7],[152,7],[153,7],[154,7],[155,7],[156,7],[157,7],[158,7],[159,7],[160,7],[161,7],[162,7],[163,7],[164,7],[165,7],[166,7],[167,7],[168,7],[169,7],[170,7],[171,7],[172,7],[173,7],[174,7],[175,7],[176,7],[177,7],[178,7],[179,7],[180,7],[181,7],[182,7],[183,7],[184,7],[185,7],[186,7],[187,7],[188,7],[189,7],[190,7],[191,7],[192,7],[193,7],[194,7],[195,7],[196,7],[197,7],[198,7],[199,7],[200,7],[201,7],[202,7],[203,7],[204,7],[205,7],[206,7],[207,7],[208,7],[209,7],[210,7],[211,7],[212,7],[213,7],[214,7],[215,7],[216,7],[217,7],[218,7],[219,7],[220,7],[221,7],[222,7],[223,7],[224,7],[225,7],[226,7],[227,7],[228,7],[229,7],[230,7],[231,7],[232,7],[233,7],[234,7],[235,7],[236,7],[237,7],[238,7],[239,7],[240,7],[241,7],[242,7],[243,7],[244,7],[245,7],[246,7],[247,7],[248,7],[249,7],[250,7],[251,7],[252,7],[253,7],[254,7],[255,7],[256,7],[257,7],[258,7],[259,7],[260,7],[261,7],[262,7],[263,7],[264,7],[265,7],[266,7],[267,7],[268,7],[269,7],[270,7],[271,7],[272,7],[273,7],[274,7],[275,7],[276,7],[277,7],[278,7],[279,7],[280,7],[281,7],[282,7],[283,7],[284,7],[285,7],[286,7],[287,7],[288,7],[289,7],[290,7],[291,7],[292,7],[293,7],[294,7],[295,7],[296,7],[297,7],[298,7],[299,7],[300,7],[301,7],[302,7],[303,7],[304,7],[305,7],[306,7],[307,7],[308,7],[309,7],[310,7],[311,7],[312,7],[313,7],[314,7],[315,7],[316,7],[317,7],[318,7],[319,7],[320,7],[321,7],[322,7],[323,7],[324,7],[325,7],[326,7],[327,7],[328,7],[329,7],[330,7],[331,7],[332,7],[333,7],[334,7],[335,7],[336,7],[337,7],[338,7],[339,7],[340,7],[341,7],[342,7],[343,7],[344,7],[345,7],[346,7],[347,7],[348,7],[349,7],[350,7],[351,7],[352,7],[353,7],[354,7],[355,7],[356,7],[357,7],[358,7],[359,7],[360,7],[361,7],[362,7],[363,7],[364,7],[365,7],[366,7],[367,7],[368,7],[369,7],[370,7],[371,7],[372,7],[373,7],[374,7],[375,7],[376,7],[377,7],[378,7],[379,7],[380,7],[381,7],[382,7],[383,7],[384,7],[385,7],[386,7],[387,7],[388,7],[389,7],[390,7],[391,7],[392,7],[393,7],[394,7],[395,7],[396,7],[397,7],[398,7],[399,7],[400,7],[401,7],[402,7],[403,7],[404,7],[405,7]]},"metrics":{"gamma":7.6653359171851765,"gamma_t":7.6653359171851765,"gamma_rho":7.6653359171851765,"gamma_claim":7.6653359171851765,"gamma_t_claim":7.6653359171851765,"gamma_rho_claim":7.6653359171851765,"v_ex":2,"rate":0.0049261083743842365,"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/circuits_l3.json","label":"l3-row039-two-group","N":11,"d":2,"d_is_exact":true,"origin":"https://github.com/shraggy/Magic_state_factory_search at cae4982, outputs/factory_catalogue_l3.csv row 39","provenance":"two-group l=3 n=11 k=2 s_total=4 s_O=2 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"}]}