The circuit
The 36\times511 matrix. Rows are wires and columns are rotations. A 1 means the wire takes part in the rotation, and the number beside each wire is its row weight. Scroll sideways to see every column.
| wire | 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 |
|---|---|
| out 0 (259) | 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111···························································································································································································································································· |
| out 1 (261) | 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·····························································································································1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111····························································································································· |
| out 2 (267) | 111111111111111111111111111111111111111111111111111111111111111111111111······························································11111111111111111111111111111111111111111111111111111111111111·······························································11111111111111111111111111111111111111111111111111111111111111111111···························································11111111111111111111111111111111111111111111111111111111111111111···························································· |
| out 3 (243) | 11111111111111111111111111111111111111111111111·························11111111111111111111111111111·································1111111111111111111111111·····································1111111111111111111111111······································11111111111111111111111111111·······································111111111111111111111111111································1111111111111111111111111111111··································111111111111111111111111111111······························ |
| out 4 (263) | 111111111111111111111111·······················1111111111111111·········11111111111111111············11111111111111111················11111111111··············1111111111111111111··················111111111111·············11111111111111111111··················111111111111111111···········1111111111111111111111·················1111111111111··············1111111111111···················1111111111111··················111111111111111···················1111111111111111··············11111111111111111············· |
| out 5 (253) | 1111111111··············1111111111·············111111111·······1111·····111111···········1111111·····1111111111111····1111111·········111111·····111111········1111111············111111············1111111·····111111·······11111111············11111111··········111111111111······111111·····1111111111············1111111111111····1111111······1111··········1111111······111111111··········11111111·····11111111··········1111111111·····1111111············1111111111······11111·········11111111·········111111111···· |
| out 6 (257) | 1111111···111111111·····1111······11111········11·······111········11···1111··1111111····111····111··1111111······11··1111···1111·····1111·······111···1111····111111·11111111····1111··111·········11111··11···1·····11111··1111····111111······1111····11111·····111111······1·····111···11···1111······1111111·····111111·······111·1······11111·1···111·······11111··11····1111111··1111······111111··111··11111···111111····1111111···11111111····11111·······111111····1111··1····111······11······1111·····11111····11·· |
| out 7 (253) | 111111·1··111······11···111·11····1····1·········1111···1··111·11··1·11·····1·111····11··11·11··1111·11111··111···1·1·11··1111···1111·11··1·111··11····11······11·····1111····1···111·1·111111111···111··1···1··1111··111··1·11··11········11····1111111·11111111··111···111····111··11·1··111··111·111···11111··11···111···111·········111···11···1·11····1111···11···111111··11111··1·111·1·············1··1·111111··111···1···111····111111·····111111···111····11····1···11··1·111··1··111·····111···11··11···1111·111·1·1· |
| out 8 (265) | 111···1···1··11····1·11·····1111···1···1111····1···········1·······1111·111·1·1··11··111·11·····1····1111·1·11·1···1·11·1·1··11··111····1··1···1···1111····111·1·1···11111111·11111····111·1·····11·11·1·111··11·1··111··1·1·111···1·1111··111···11111···1····1··1·1··1111··11·11··1111·11111···11·111·1··1111·111·11111·11····11111···111·11·111···11111··111·11·1·111··11111·1····11··1··1·111··111···1··1··11····1111111··1···11·111·1··1··1·1··111···1··11·1···1·111··11·····111111111·11·11·1·11····1·111·1··111·11··11·11 |
| check 9 (256) | 1··11·111·11···11··111·11·····111·1·11·111111111·111·11111··1······1·1··1····1·111·1111·11111111····111··1······11111·1··111···111··1111·11·11·1·1··1··1·1·1111···1···1···1··111··111··11···1111·1·······11···11·1111····11111·1···11111·1·11·1··11·······11···1·1··1·1···11·····11··1··1111·111111·····1·11··11·1·1111·11··11·11··1··1111·11·1··111···111·11··111·111·111111··1·········1··11··1·11·1···111····1···1··11·1···1··11·1·······11·····11··1····11111··1·1··11···1111111·····11··11··11···11··11·1·11·11····1··1··1 |
| check 10 (256) | 11·1·11·111·1··1·11111·11··1··1··1··11111······111··111·····111·····1111·11·1111··11·11··111····11·1·11··1··1·1·1·11·111·11·11·1··11··1·11··1·11·1·1·1····1···1·1·11···11··1······111111111·1·····1·11········1·11····1··1111111···1·1·····1·111······11······111··11·1111111·111··111··1···1·1111111··11·1·11···11·······11·····11·1·····11··1111·1···1·1·11·11·11·1·1····11···11111·1·11···1111111····1····11··1111··1··111·11·1···1··11··1··111·11111··1·11···1·1··1··1···11111111111··11·11111····11·11·1···111·····1·1·1·· |
| check 11 (256) | ·11·1··1···1·1···1··1111·111·····11·11··11·····11·111111·11111·1·1·1·1·1·1·1··1·11·111···11·1·1···1·111···1·1··1··111·1111111·1·1111··11··1··11·111·11···1111·11·1···111··111·1·1····11·1·1··11···1·1·1···111··········1··1··1·111·1·11·11··1···1··1·11·1·11···1··11···11·1·1···11·111··11··11111···111111·1111·1111··1··1111·11···1··11··11······111··1··11··11·1···1·1·1······11····1·1·1···1·1·1·1···1·1····1·11·11·1·111·1·1···11·1·1·1··1··1··1··1···111··11·1111··11111··111111·1····11·11··11·11··11·11····1·11·1·1··11· |
| check 12 (256) | ·111·11··11·1···1··1···1·1·111·····11·1·1··111·11·111···1·11··1·1111·111·11111·1·1·11···11111·1·1······1·1··1···1111·1·1····1···111··1·11··111·1·1·1·111··111··111·1··11··11·11·11·1···111···1·····11·1··1····111·11111·1····1··1·1·11·1111·1···1····1111···1·11·11111···1·1···11····1·1111·111·1·1··1···1111·······1·1····11·11····1·1·1··1····111·1··11··1······1·1··1·1·11111111··111·111····1·11111··1·1·1·111111··11·····1111··1····11111··1·11··1·11·111·1·11·······1·11····11····11···1··1···11··11111···11111·1·111111· |
| check 13 (256) | ···1·111·1··1······111·1·1··1·111···11····1111·1111·111··1··1·1····1111··1··11··1·1·111·····1·····1·11·11·1·1···1·111·11111··11···1··1·1·1111·11····1·1111·11····1111·11··11····11·1·111·1·11·111·1···1····111·111··11······1·1·1·1··1··1·11·1··11·111·11·111···1·····11···11····111··1·111········11·1·········111·11·11··1·1111··1·11··11111111···111·1·11111·11·1···1·1·1·1111111··1···11·1·1···111···11······1··1··1·11111·1·11··11·11·111····1·11····111··11·····111·111····11·11·1·1·1·11·11··1··1··111·1·111··111··111·1 |
| check 14 (256) | 1·11·1111·11··1··1····1111··11····1··1··1·1···1111···1·111····11·····1·1··1·111······11·11·1111·11·1·111·1·1111···1···1·1····1··111111···1····1111····1·1111··111·1··1·····1·····1··1··1111····11··111111··1··11··1111·11····11·11·1··1111·11111111111·1··1·11····111·11·1·11·1··11···1··11·1111111·1··11········1···11·11111··1···11·11111··11·1···11···11··1·1·11····11·11··11··1111······11········11111111·1·1·11··11111·111··1··11·111··1··11···11111··1··1··1·111·11·111·1···1·1··11·1···1···1·····1·1·1111·1·1·1·····11· |
| check 15 (256) | 1···11···1·····1··1····1··1···11111·1·1··111··1111··1··1111·11··111···1··111··1111·1·111·1·1··1··1······1···1·1·11······111·1··1··1··1111·111111··11···1··111···1··11111·1·1···1·1··11·1111··11111·1··11111·11···111·11···11111··11111····111···1··1·1111··111·11·111·11·1··11···1·1··1·1··1·111·1·1········1··1111·11··111·1111·1·111·········11····1·11·1··11111···1··11111·11····11··1·1·1·11·11·1······11·11··1·······11··1·11·11·····11·11·11··11·1·1··11·1····1·1·1··1111111···1··11··11·11··1··1·111111··1·1111·111····· |
| check 16 (256) | ·11·1·11···1·1·1·····111·1···1111····11·1·111·1111·11·1·1·1··11·1·1···1·1··11··11·1·11···1···1··1··1·1·111111······1····111111··1··1··1·11··1·····1·1·1··1··111··11···1·111··1····1···11·11··11··1111···1··1··1·111·11·11111··1··11·1··1····1·1·1···1········11··1··111··1111·111··111111··1111··1·11·11·11···1·1····11··1····11···1111111··11··1111111111·1··11·1···111··1··11·11·111··1······1111111·1·1···11···1·1111·1·11·11·1111··11·111·1·11·····1···1···11···1·1111·11··1···1··1··1·····111··11·1·11111·11··11··11·1·111 |
| check 17 (256) | ·11·11·11··11·····1····1·····1·1··1·1·1·1111111·······1·····1111··1111···111·1···1··1···111111···11·11····1·11·1·····11·1·1·1····1···111···11·1···1··1111··1·1·1··11·11····1···11111111·11····111····111·1·1·11·1111··11···11··11·11·······1··1··111····1111·1·1·1·1111····11111·1·1·1····1111··1·1··1·11·11111··1····11·1111··1···111··1·1··111··11111····11··111·1··111··1··1·1·111111·1·1··11·1111··111···1·11··1·1·1····11····1·11·11·····111····1111··1111·1···111··1··1·1·1·1·1·1·1·1·1111111·11··1·111···1··1111111··1·1 |
| check 18 (256) | ····11·11··11··1·1·1·1··11·1···1··11··1111··11111··1··1····11···11····1··1····1·1·1···11···1111111··11···111··1·111·1·1·111·11··1·1111·1·1····1·111·11·11····111··11···11····1·11·····11·1····1·1·1···111·11111111·1·111·1··11·1·1·11···1111····1·1·1·11·1···1··1··11····11·1··1·1·11··1··11111······11·111·1···1··1···1···1·1··11·1111··1·1111·1·11···11·1·1111·11··1···11·1·11··11·····1···1·1··11··11··1111111·111·····1··111·111···111·1·111··1···1·1·1·1··111·1·1·1··1··11·111·1·11·1·1···1··11111111·1·1111···11·111···1· |
| check 19 (256) | 111··11···11··1··1······111··11111··11111····1·········1·······11·1····1··1·11·11··11·1·1·11·1··11·1·11·111··11·11111···11····111111·111···11··111·1···1·1111·1··1···1··1·11···1··11111·····1·1·1·111····111··11···111····1··1··11··11·11···11·111·111·11···111··11·1111····11·11··11·1·1·11111·111·11·····111··1······1·1····11····11·1···11·11··1111··111·11···11111··1·11····1·1·1111·1·1·111·111··1··11111111111···1··1··1··1111·1··1··1··11·1111·11··1111··1·11··11···1··1··1·111·······1···1···1·11111··11·111···1··111·· |
| check 20 (256) | 111·111···1·11·1·11··11··1···1·11·1···11111·1··11····1···1··1111··11·1··1111···1·11··1··1·11111·1·····111·1111··1·1·1·11·11·1··11·1··1·1·1·1111···1·····1·1·1···1·11111·1·11·11·1·11··1··1··11·1··1·1··11·1·1·····111·1····1·1111111··1····11111····1··11·1··11·1··11111·11111111···1··1·1····1··111··11111·1·1·1·11··1··1···1······1····1·11···1·1···1··1·1··111··1111··1111······1··1····11·1·11··1·1··11·1···11·1·11·111111···11·1··111··111·1·1··11······1111·1·11111·1···1··1·1·11····1111·11·1·1··11·11··1111···1·11·11·· |
| check 21 (256) | 1·1111·111··11·11·11·11111·111·111···111·1··1·1····111·11·1····11111111·1···111·111··1······1·1··1·1·1···1··11·111·11···1··1··1·1·1··11····1·1··11···1·1···1···111111111·1·1·11··1···1·1·111·1·1··1111111·111····1··1···1·1·1·11····111····1···11·1·1·······1·11··1··1··1··111·11111·1··111·1·1··1·111··1···1111·1···1·1111····1111····1·1·1·1··1·11·11··11·1·111·11···1···1·1111··1·····1·1·····1·11···11··11··11·111111·1·1···1111····1···1·1···1·1·111··11··1·111·11···1111··111··11····1·1···11····1·111111·1·11·1·111····1 |
| check 22 (256) | 11···1··1··11···1·11···1·11··11··111··1···1·1111··1·1111·1·1111·1·1···1·1·1··1···1·1····1··1·11·111·111···11·111111····11···11·111····1111··11··11··111111·1····1···1···111··1···11·11·1111·····1·11111··1·1·11111·····1··1··1·1···1···11···11····1111·111··1·111····1··11·11·1··111·11··1···1·1·1111··1··11··1·1·1111·1····1·1111··1····1·1··111·11·111·1111111····1··1·11······1···1··1·1···11·1111···111···1··11111·····11···11····11·11··111··11·1111··1··11·11··1···111·111·1·11111··11111111··11··1···1···1···1·11····111 |
| check 23 (256) | 11··111·1·····11··1··11·11··1·1··11·111·1··11···11·11·11····111··1·11·11111····11··1·11·1·111···1·1·11··1·111·11·1·11·1··1·1··1··1·111···1·1·11·11··11·1·1··11·1111111··11·1·1··111··1···1···11····1·1·1···1···11··111·11··11··11··11·1··111···1···11·1··11111·11···11·1111·1·1·11·1··1····1··1···1·······1·111··11·1·1·1·1·11·····1·1··1·1···1111··11···1··11·1····1···1·11···11···11··1···1··1··1·1111·11·11·1··111···1·1··1··1·11111···111···111·11··1··11·11·1111·1111··1111·1·1111·1····11·1111·111···111·····111·····1111 |
| check 24 (256) | ····111·1·····111111·11···1111·1·11··1······111··111····11111·1···1·111·111····11········1··111·111··1111·1·111··11111·111··1·1111111··11·11··1··1111······1·1111···1111·11·1··1·1··11111···11····11····11·1·11·11·1···11··1···11·1··1····1··11·11·111···1···1····1···1·1111·1·1111·1·11·1···1·1··111·11··1·1··1·1111······1···1···11111111···1·11111···1····1··1·111111····111····1111111·111·1·1·1····1····1·11··1·11·1·11·111111·1····1·1·1·1·11·1··1····11·1··11·11·1····111··11·1·1·····11·111111····1··11·1·1·····111···1 |
| check 25 (256) | ·1111··11··1·1······111·1···11····1·1·····1··1······11··1··1·1·····1111·11111··11·1·1···1111·····11··1·1·1111·111·1·1·11111·1111··1··1·11·11·1··1···1·11·11··1111··1·111··11··11·1·11··1·1·11······11··11·11···1·1···1·1·11·1··11111··111·1····1·11··1111·1111····1·1·1··1··11····1·11111111···11···1··111·1··111····11·111·11·11····11·11·1··11111·1·11·111··11···111·1111·1···111·11··11··1·1·11···1···1·····1111111····1···1·11·1···111·····1··1·1····1111·1·11···1·111·1····111··1·1·1·1·11·1··1·11···11·11····111·11·11·11 |
| check 26 (256) | ···1··1111111··1···1·11···11··11111·111··111···1··11·11··1·1··1·1··111···11····1··1·11·11··········1·······1·1·1·11··11·1···111111·1·11·1·11··11··1111·1·1··1·1·11··1··1·1·1··1·11111·11·11·1·1··1··1··11·11·1·11·111·····1···11·11·1·····11111·····1······1·1·11··1·1·····1·1··1····11·1111111··1·····1·111··1111····1··1111··111·111111·11·11·····11··11··111·1·1··1·1··1·11·1·1······1·1···1·11·1··111·11·11·1··11···1·1·1···111111·1··1··1·111·1·111·1111111·····1·11···1111·1·1111··1····11·1···11·111·11111·11·111·111·1· |
| check 27 (256) | ·11·1·11····1··11···1·1··11··111··1··1111111·1·1·1···1··11·1·1·········1·1111······11···111··111·1·1··111·11·1·1··11········111·111·1·11111···11··1··1·1··1·11·1·11·1·1·1111·1·1·1·1·111·11·11·1······1·111·11·1··111··1·11··1····111·1·····1·111·111111···1···11·1111111··11··········11111····1···1·····1·····1··1·1··1····1·11···1111·1·11·11····1111·111·111·11·111·11·1·11··1···11···11111···1111···11111·1·····1·11·11··1··11·11··11·1·11·1111111·11···1····1···11111·1·1···1·1·1··1···111·11·11···1111111111···1·11···1· |
| check 28 (256) | 1····11111··1·111·1·1···1····1111··111·1111···11·1·11111111·1111·1·111·1···1···1··1·1·11·11·11···11·1··1·1111·1···11·11···1111···1·1·1111···11·1·11···1·111·1·····11111···1·1···1·1·1·1·1·11···1·1·1111·11·1···11·1·1···1····1····1·11111·11···11·11··11111111··1·11·1··1·1···111·111·111··1····11·1·11···1111111··111·····111·111··1···1·111··11·1··1···1···1·1·1····1·1·1111···1··11··········1·11···1·1·1111··1····1111·1··11···1··111··1·1····11·11·1····111·11···1··1··1···111··1·11·111·1·1···1·1111·····111·1·111···1··1 |
| check 29 (256) | 1···111111·111·1·1111·11···11·111···11··111···1·1·1·1·1·1111··11·1··11···111111··11·11·11····11···111·1··11··1·1·1·11··11···1111111·1·11·111···1···1·1·····11··1·1·1···1·111·1·1····1·1·11·1·····1·····11·1·11·11111·1·11·······1111·1···11111·1···1··11111···11··11···1·1·1·11·1···111111··11·1·1···1···1···1·1·111·1·11·1··11··1··1·111·11·111·11··1··1··1·1·1·11····1·1·111··111·1···1·11·1············1·111··111·1··11·····11111·1··1·1··1·1····11···111··111·11·1·1··11111·11··1·11·111··1111··1·1······1·11···1·····11111 |
| check 30 (256) | ··1·11·····11··1··11·1·1111··1··111·111··111···11111·1··11··11··1··1···11··1··11111···11··1111··1111·1··111··1······1···1·····11·111·1·11·111··111····1··111111··1·1··111111···1······1··11·1···111111·1·11111··1··1··1··1·1·11····1111···11111111·1·1·1··1111··11·1···1···1···11·11·11···1·1··111··11··11··11·111··11··1··1··1··1··111···1····1···1··1·111·11···1·1··11·····1·11·1111·1·1·111111·1··11···111111···1111·111·1··1··1··11···11·11111··1···11·1··11····111··11·1··1··1111·1···1···1··1·1111··11··1··1111··1····1·· |
| check 31 (256) | ··111111···11··1····11·1·11111·1··1··1·11·11····111111·1··11·11·111·11·1·11·1··11··1·11··111······1···1··11··1··1·1·111·1·1·······11··1·1··11··1······1111·1····11·11··11··1··1·1·11·1·11·1···111111·11·11···1·····111·11·1·1··11·11···1··1····11·11··1··111·1··1·111·1·1·111111·1···1···1·1·11·11·11111·1·1··11111·1·1·1·11·1·11··111··111··11··1·111···1·111111·11····11······1····1·11·11··111····11·1···111111·111··1·1····111·1111··11·····111··1···111··11·1··1··11··111·11··11····1111·11····1····11··11·1····111·1····1 |
| check 32 (256) | ·11·1··111··1·1·11·11111··1·1··11··1·111····11·11··1·111··11·1·1·1···11·····1·111·11··11·11·····1·11··1·1····111·1·1·1·1··1111·1111·11·11··11·11···1··11···11·1·11··1····1··1111111···1···11·1·111··1·11···11·11···1111··1·111·11·11·1·111··1·1··1······11111111·11·111··1··111·1·1·111··11······11··1111···1···11·11111·11·1·1·1·11··1··11111·1···1·1····1111·······11·1····11·1·1··1····1·1·11····1·1··1·1·1·11·····11111·1·11·11111····1·111··1··1·111·1··1····1·····11·11··1111···1·1111··11·1··1·1···1···1111·111·1····1·· |
| check 33 (256) | 1··1·1·1111···1111·1······1111··1111·1··1···1·1·11·1··1··11··1···1··1··111··11111111····1···1·····1···1····11111·11·11···1··1··111··111111·1·11·11··1···111·1··1··11·1··1·11·1··1·11·1····1·111·1·1·1111··1·1·1·1··111··11·111·1··11·111·1·11·····111111··11111·11··11111·11·······11··111111··1··11···111···1····1·11····11·1··11·1111·1·11····11··11·1·····1··11111·1·11····1·11····1··1·11111·11·11··11··111···111·11···11···111··11·1·111·11···1····1·11···11111···1111·1··11··111··1·11····1·1···1···1···111·1······1·1·11 |
| check 34 (256) | ··11·····1··111·1··11···1··111···1··11·····1·1··11·1·1·11·1·1·11111·111·1··111·11····11111·······1·····1··1·····1111··1·11····1·1·1·11·11·1·1·1··1··1··11·1··11·1·1··1·1··11··111····11···1··11111·1·11111·1···11111····11··1·1·1·1·11·11·····11·11·1·1···1·1·11··1··111·1·1··11···1111·11··1····11·1·1111··11·111···1·1111·111·1···1111·11·1·111111111111111·11·111·1·1···11··11·······1···111···11·1······1111·1111···1··1111···1··11111··1··1··1·1···11····11·11·1··111·1··1··11···1111··11·11·11··1·1····111·11·1·1·1··1·1· |
| check 35 (256) | 11·11·11··1111·1·1·1·1···111·11111··1·1·1··1·········11·1···11·1111·11···11···11·1·1·····11·11··11111··1··111·1····1·11·1·11··1·1····1·1·1·11·1111·111·····111····111···1··1·11······1··1·1·····111··11·11··1·1···········1··1111111·1·1··11····111··1·11·····111111·111·11··1·11···1·1··1·1·11······111··1111···1111·1··111111······11·11111··11··1···1··1·····111111·11··111··11111··11·111·1·1·1·1··1··1111·1·111·111····1····1·111·111·11·11·1··1··111111111·111·11···1·1·111·······11·····111·1·1·1·1··11···111··1111···1· |
The gate is read off the output rows. An output row of odd weight gives a T, a pair of output rows with odd overlap gives a CS, and an odd triple overlap gives a CCZ. The circuit is a valid factory because every such parity that touches a check row is even.
Figures of merit
inputs n511Noisy T states per run.
outputs k9
distance d≥ 6A lower bound proved here.
error coefficient A_{6}not definedUndefined while the distance is only a lower bound.
wires N36
checks r27
V_{\mathrm{ex}}9T states per run. CCZ counts as two.
\gamma_\rho = \log(n/V_{\mathrm{ex}})/\log d2.2543Overhead per extractable T state. Lower is better.
\gamma = \log(n/k)/\log d2.2543Overhead per output qubit.
\gamma_T = \log(n/T)/\log dnot computedOverhead per unit of T-count.
T-countnot computedMinimal T-count of the output gate. Not computable at this width.
phase-polynomial degreenot computedMinimised over changes of output basis.
effective width9Rank of the output rows modulo the check rows.
Output gate
9 T0 CS0 CCZon 9 output wires
As gates: \mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3}\cdot \mathrm{T}_{4}\cdot \mathrm{T}_{5}\cdot \mathrm{T}_{6}\cdot \mathrm{T}_{7}\cdot \mathrm{T}_{8}
As monomials: 0+1+2+3+4+5+6+7+8 (output wires per term, separated by +)
This row stands for every gate that a CNOT circuit and S, Z and CZ gates on the outputs can reach from this one. They all prepare the same magic state. The wires are shown in the S_k-canonical frame.
Clifford correction
With every rotation a \mathrm{T}=\mathrm{diag}(1,e^{i\pi/4}), the rotations deposit the gate above times a diagonal Clifford. Undo it after the rotations and before the checks are measured, with 7 single-wire gates and 142 \mathrm{CZ} gates: \mathrm{S}^\dagger on output 0, \mathrm{Z} on output 1, \mathrm{S}^\dagger on output 2, \mathrm{S}^\dagger on output 3, \mathrm{S} on output 4, \mathrm{Z} on output 5, \mathrm{Z} on output 7, \mathrm{CZ} on output 0, output 1, \mathrm{CZ} on output 0, output 2, \mathrm{CZ} on output 0, output 3, \mathrm{CZ} on output 0, output 6, \mathrm{CZ} on output 0, output 7, \mathrm{CZ} on output 0, check 11, \mathrm{CZ} on output 0, check 14, \mathrm{CZ} on output 0, check 15, \mathrm{CZ} on output 0, check 16, \mathrm{CZ} on output 0, check 18, \mathrm{CZ} on output 0, check 19, \mathrm{CZ} on output 0, check 20, \mathrm{CZ} on output 0, check 21, \mathrm{CZ} on output 0, check 24, \mathrm{CZ} on output 0, check 26, \mathrm{CZ} on output 0, check 28, \mathrm{CZ} on output 0, check 29 and 125 more (listed below). The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.
All 149 correction gates
\mathrm{S}^\dagger on output 0, \mathrm{Z} on output 1, \mathrm{S}^\dagger on output 2, \mathrm{S}^\dagger on output 3, \mathrm{S} on output 4, \mathrm{Z} on output 5, \mathrm{Z} on output 7, \mathrm{CZ} on output 0, output 1, \mathrm{CZ} on output 0, output 2, \mathrm{CZ} on output 0, output 3, \mathrm{CZ} on output 0, output 6, \mathrm{CZ} on output 0, output 7, \mathrm{CZ} on output 0, check 11, \mathrm{CZ} on output 0, check 14, \mathrm{CZ} on output 0, check 15, \mathrm{CZ} on output 0, check 16, \mathrm{CZ} on output 0, check 18, \mathrm{CZ} on output 0, check 19, \mathrm{CZ} on output 0, check 20, \mathrm{CZ} on output 0, check 21, \mathrm{CZ} on output 0, check 24, \mathrm{CZ} on output 0, check 26, \mathrm{CZ} on output 0, check 28, \mathrm{CZ} on output 0, check 29, \mathrm{CZ} on output 0, check 30, \mathrm{CZ} on output 0, check 31, \mathrm{CZ} on output 0, check 33, \mathrm{CZ} on output 0, check 34, \mathrm{CZ} on output 0, check 35, \mathrm{CZ} on output 1, output 5, \mathrm{CZ} on output 1, output 6, \mathrm{CZ} on output 1, output 7, \mathrm{CZ} on output 1, output 8, \mathrm{CZ} on output 1, check 9, \mathrm{CZ} on output 1, check 11, \mathrm{CZ} on output 1, check 12, \mathrm{CZ} on output 1, check 15, \mathrm{CZ} on output 1, check 18, \mathrm{CZ} on output 1, check 19, \mathrm{CZ} on output 1, check 20, \mathrm{CZ} on output 1, check 21, \mathrm{CZ} on output 1, check 23, \mathrm{CZ} on output 1, check 25, \mathrm{CZ} on output 1, check 27, \mathrm{CZ} on output 1, check 29, \mathrm{CZ} on output 1, check 30, \mathrm{CZ} on output 1, check 32, \mathrm{CZ} on output 1, check 33, \mathrm{CZ} on output 1, check 35, \mathrm{CZ} on output 2, output 4, \mathrm{CZ} on output 2, output 7, \mathrm{CZ} on output 2, output 8, \mathrm{CZ} on output 2, check 10, \mathrm{CZ} on output 2, check 12, \mathrm{CZ} on output 2, check 13, \mathrm{CZ} on output 2, check 14, \mathrm{CZ} on output 2, check 17, \mathrm{CZ} on output 2, check 18, \mathrm{CZ} on output 2, check 20, \mathrm{CZ} on output 2, check 23, \mathrm{CZ} on output 2, check 26, \mathrm{CZ} on output 2, check 28, \mathrm{CZ} on output 2, check 30, \mathrm{CZ} on output 2, check 31, \mathrm{CZ} on output 2, check 33, \mathrm{CZ} on output 2, check 34, \mathrm{CZ} on output 2, check 35, \mathrm{CZ} on output 3, output 5, \mathrm{CZ} on output 3, output 6, \mathrm{CZ} on output 3, output 8, \mathrm{CZ} on output 3, check 10, \mathrm{CZ} on output 3, check 11, \mathrm{CZ} on output 3, check 15, \mathrm{CZ} on output 3, check 16, \mathrm{CZ} on output 3, check 17, \mathrm{CZ} on output 3, check 18, \mathrm{CZ} on output 3, check 19, \mathrm{CZ} on output 3, check 21, \mathrm{CZ} on output 3, check 26, \mathrm{CZ} on output 3, check 28, \mathrm{CZ} on output 3, check 29, \mathrm{CZ} on output 3, check 30, \mathrm{CZ} on output 3, check 31, \mathrm{CZ} on output 4, output 5, \mathrm{CZ} on output 4, output 8, \mathrm{CZ} on output 4, check 10, \mathrm{CZ} on output 4, check 11, \mathrm{CZ} on output 4, check 16, \mathrm{CZ} on output 4, check 18, \mathrm{CZ} on output 4, check 19, \mathrm{CZ} on output 4, check 25, \mathrm{CZ} on output 4, check 28, \mathrm{CZ} on output 4, check 29, \mathrm{CZ} on output 4, check 31, \mathrm{CZ} on output 4, check 32, \mathrm{CZ} on output 5, output 6, \mathrm{CZ} on output 5, output 7, \mathrm{CZ} on output 5, output 8, \mathrm{CZ} on output 5, check 10, \mathrm{CZ} on output 5, check 12, \mathrm{CZ} on output 5, check 13, \mathrm{CZ} on output 5, check 14, \mathrm{CZ} on output 5, check 16, \mathrm{CZ} on output 5, check 18, \mathrm{CZ} on output 5, check 19, \mathrm{CZ} on output 5, check 20, \mathrm{CZ} on output 5, check 26, \mathrm{CZ} on output 5, check 30, \mathrm{CZ} on output 5, check 32, \mathrm{CZ} on output 5, check 34, \mathrm{CZ} on output 6, output 7, \mathrm{CZ} on output 6, check 12, \mathrm{CZ} on output 6, check 13, \mathrm{CZ} on output 6, check 14, \mathrm{CZ} on output 6, check 15, \mathrm{CZ} on output 6, check 17, \mathrm{CZ} on output 6, check 20, \mathrm{CZ} on output 6, check 23, \mathrm{CZ} on output 6, check 25, \mathrm{CZ} on output 6, check 26, \mathrm{CZ} on output 6, check 28, \mathrm{CZ} on output 6, check 30, \mathrm{CZ} on output 6, check 31, \mathrm{CZ} on output 6, check 32, \mathrm{CZ} on output 6, check 34, \mathrm{CZ} on output 7, output 8, \mathrm{CZ} on output 7, check 9, \mathrm{CZ} on output 7, check 10, \mathrm{CZ} on output 7, check 13, \mathrm{CZ} on output 7, check 14, \mathrm{CZ} on output 7, check 21, \mathrm{CZ} on output 7, check 22, \mathrm{CZ} on output 7, check 25, \mathrm{CZ} on output 7, check 29, \mathrm{CZ} on output 7, check 30, \mathrm{CZ} on output 7, check 31, \mathrm{CZ} on output 7, check 32, \mathrm{CZ} on output 7, check 34, \mathrm{CZ} on output 8, check 13, \mathrm{CZ} on output 8, check 14, \mathrm{CZ} on output 8, check 16, \mathrm{CZ} on output 8, check 20, \mathrm{CZ} on output 8, check 22, \mathrm{CZ} on output 8, check 24, \mathrm{CZ} on output 8, check 28, \mathrm{CZ} on output 8, check 30, \mathrm{CZ} on output 8, check 31, \mathrm{CZ} on output 8, check 34, \mathrm{CZ} on output 8, check 35
These Clifford gates cannot be avoided. No choice of \mathrm{T}, \mathrm{T}^3, \mathrm{T}^5 or \mathrm{T}^\dagger for each rotation removes them; the catalogue proves this by showing that the linear system over \mathbb{Z}_4 the powers would have to solve has no solution.
Rotations are numbered by matrix column, from 0. The correction comes from the columns by the weight expansion of Bravyi and Haah (2012). verify_catalog.py re-derives it, checks the corrected circuit's phase input by input against the gate, and on small circuits simulates it gate by gate.
Convert the output with CNOT and S
A CNOT circuit on the outputs, followed by S, Z and CZ gates, turns this factory's gate into other gates that prepare the same magic state. Type a target gate to check whether it is reachable. If it is, you get the circuit, checked on every basis state.
Or list every gate it reaches. The list is built in your browser and saved as a text file, one gate per line.
Python code that writes the same list
import json, urllib.request
NAMES = {1: "T", 2: "CS", 3: "CCZ"}
def equivalent_gates(k, terms, limit=1_000_000):
"""Every gate a CNOT circuit and S, Z, CZ gates reach from this one.
A gate is its set of terms mod Clifford: {a} is T, {a,b} is CS and
{a,b,c} is CCZ on those outputs. CNOT c->t sends x_t to
x_c + x_t - 2 x_c x_t, so a term on t but not c adds the term with t
replaced by c, and, below CCZ, the term with c added.
"""
start = frozenset(frozenset(m) for m in terms)
seen, queue = {start}, [start]
for gate in queue:
for c in range(k):
for t in range(k):
if c == t:
continue
new = set(gate)
for m in gate:
if t in m and c not in m:
new ^= {(m - {t}) | {c}}
if len(m) < 3:
new ^= {m | {c}}
new = frozenset(new)
if new not in seen:
seen.add(new)
queue.append(new)
if len(seen) > limit:
raise RuntimeError(f"more than {limit} equivalent gates")
return seen
def name(gate, k):
sep = "," if k > 10 else ""
terms = sorted((sorted(m) for m in gate), key=lambda m: (len(m), m))
return "·".join(NAMES[len(m)] + sep.join(map(str, m)) for m in terms) or "I"
url = "https://shubhamj810.github.io/magic-state-factory-repo/data/factories/511.9.6.a.json"
record = json.load(urllib.request.urlopen(url))
k = record["parameters"]["k"]
gates = equivalent_gates(k, record["gate"]["monomials"])
lines = sorted(name(g, k) for g in gates)
print(len(lines), "equivalent gates")
with open("equivalent-gates-511.9.6.a.txt", "w") as out:
out.write("\n".join(lines) + "\n")
Distance
6\le d\le 7. No fault of weight below 6 is both undetectable and damaging. None of weight 6 has been found, and one of weight 7 is, so the distance lies between the two.
A damaging fault of weight 7 sits on columns [34,149,303,331,365,369,381].
A fault is a set of faulty columns. It is undetectable when their check parts cancel, and damaging when their output parts do not.
Original source
S. P. Jain, A. Wills, and S. Singh, "Symmetry and AI-assisted discovery of magic-state factories," arXiv:2610.06535 (2026). arxiv.org/abs/2610.06535
Provenance
found by an AI search campaign; a verified witness, not a maximum
Found by: AI search
Discovery: AI search
1 recorded source
[
{
"regime": "AI search",
"file": "master_catalogue/circuits/n511_k9_d6_8e0696e597b38902.json",
"label": "T0.T1.T2.T3.T4.T5.T6.T7.T8",
"N": 36,
"d": 6,
"d_is_exact": false,
"origin": "magic-state-factories for AI",
"provenance": "magic-state-factories for AI/master_catalogue/circuits/n511_k9_d6_8e0696e597b38902.json (restored 2026-09-15: a higher-distance circuit for a class the S_k-era key held only at a lower distance)"
}
]Use it in code
Load this factory and re-derive every number from its circuit with the catalogue's own verifier.
# git clone https://github.com/shubhamj810/magic-state-factory-repo
# cd magic-state-factory-repo/master_catalog
import json, urllib.request
import verify_catalog as VC
url = "https://shubhamj810.github.io/magic-state-factory-repo/data/factories/511.9.6.a.json"
record = json.load(urllib.request.urlopen(url))
columns = record["circuit"]["columns"]
k, N = record["parameters"]["k"], record["parameters"]["N"]
facts, problems = VC.derive(columns, k, N) # gate, T-count, degree, ...
distance = VC.measure_distance(columns, k, N) # exhaustive fault search
print(facts["gate_human"], distance["d_exact"] or distance["d_at_least"],
problems or "verified")
Raw record
Show the JSON this page is built from
Source: master_catalog.json, factories[1098].