[[1023, 14, 5]] magic-state factory, T0·T1·T2·T3·T4·T5·T6·T7·T8·T9·T10·T11...

The circuit

The 34\times1023 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.

output wires
check wires, measured and postselected

wire	0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740 750 760 770 780 790 800 810 820 830 840 850 860 870 880 890 900 910 920 930 940 950 960 970 980 990 1000 1010 1020 \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \| ' \|
out 0 (509)	1·····1·111··1·111·1·11··1··111··1··111···1·1···111··1·1·11111·1···11·11·11111·11·11······1·1··111·1·1111·11····1·····11···11·1·1·····1·111··1·111·1·11··1··111··1··111···1·1···111··1·1·11111··111··1··1·····1··1··111111·1·11···1·1····1··1111·11111··111··1111·11······1·1··1···11·11·11111·1·11111··111··1····1·1····1··111111·1·11··1··111·1·····1·111··1·1···11·1·1·····111·11···111·1·1·11·11······1·1··1···11·11·11111·1·11111··111··1····1·1····1··11····1·1··11·11···1·11111·1···11···111··111·1·1111··1··1···111·111·1··111··11111·1111··1····1·1·····1·1······11·11·11111·11·11···11·····1·1·11···111·1·111···11·11111··1··11·1·111·1··111·1·····11·1··111··11111·1111··1····1·1·····1·1······11·11·11111·11·11·····11111·1·1··111···1·1···111··1·1···11·11··1111·11·11·······1·11·11·1·111···11·111·····1·1·11···11·11···1·11111·1···11·11··1·1··1111··1····1·1··1·1··1111·11·11·11·····11·1·111·111·11·1·1··11·1·11·1·111···111·11··1·1111·1··1·11·11··1···11111··1··111·1·11·1·····11··111·1·1·11··1····1·1···1·····11···1··1·1··1··1···1·······
out 1 (513)	·1111···11··11·1··1·11·11··11··1·1·1·1·····111111111111·1·11·1·1·1··1·1············11111·1·1·1·11··11··1··1·11·1··11··111····111··1·11·11··11··11····111··11··1········1·1··1·11·1·1·1·····1111····11111·1·1·1··1·11·1·11111111·11··11···1111··11··11··1··1·111····11111·1·1·1··1·11·1·111111111··11··111····11··11··11·11·1··11··1·11·11··11··11····111··11··111111111·1·11·1··1·1·1·11111···11·1··1·1············11111·1·1·1···11··11·11·1··1·11··11···1111·1··1111···11··11·1··1·11·11··11·1·1·1·1··111····1··11····1111·1·1·1·11·1·111111111111·····1·1·1·111··11··1··1·11·1··11··111····11··1111···11··11·1··1·11·11··11···1·1·1·11111············1·1··1··1111·····1·1·1·11·1··1·1·········11··11···1111··11··11··1··1·1111··1·11·11··11··11····111··11···11111111·1··1111·······111··111··11·1··1··11··11··1111···11··11·11111111·1·11·1··1·1·1·11111···1····11111·1·1·11·1·11·11111·1111·11··1···1···1·1·11··11111·11···11···11111···1···11·1·1···11·1···1·1·1··1···1······11111···1····1·11·111·111···1···111111·····1·1··1·1····1·1····11·1···········
out 2 (479)	1···1··11·111·1···1···11···1······1···11···1····1···1··11·111·11·111·1111·111·1···1···1·111·111111·111·1···1····1···1····1···1··1111······1111·11·1··1·1·11·1····1·11·1·1··1·111····111111····11····111···1111·11·1··1··1··1·1111·1··1··1··1·111····111···11111··1···1··1···1···111·111···1···1·111·111···1···1··1···1··1···1····1···1·1·111·111···1······1···1·111·111111·111·11·111·1·1···1·1···1111·1····1111·11·1····1·11·1·1··1·1111·1··1·111····1·1111··1···1111··1111····1··1·11··1·1··1·1··1·11··1·11······11·1···111111····111111····1··1·11·1·1··1·11··1·11·1·1··1·111····111111····11····111···1111·11·1··1··1··1·11··1·11·11·11·1···1111···111·····1·111·11··1···1·111·111··111·111···1···11···1····1···1··11·111··1·111·1111·111·1···1···1·111·11····1···1·11····11·1·111·1111·1·····1111·1····1111·11·1····1·11·11·11·1····1·11·1···1111·1·····11111····11····1·1···1··111··11···1·11···1···1·1··111··111···11·1··1·111·1·1111····11··1·1·1·1····1··111··11······1·····1·11·111··11··11····1111·1·11111······1····11·1·1···1···1··1··1···········
out 3 (475)	·11····1111··11·11··1·11·1··11·1111··11··11····1·1··11··11··1·1·11··1·1·1·11··1··11········11···1·11··1·11··1·1····11····11·····11··1·1·1·11··1··11········11··1·1··11·1··11·1·1111··1111··111111··1111····11··1··11·1··1·11··11111··11··11····1·1··11··11··1··1111··1111··11111·1··11·1··11·1···11········11···11··1·1·1·11··1·1·11··11··11·1·····11··11··1111·11··1·11·1··11···11····1111··1··1·11··11··11·1·····11··11··11111··11·1··1·11··111··1111····11·1····11····11·····1·11··1·11····1··11········11·1·111·11·11·····1······1111·······1·1·11·1··1·1·1··111111111111···11·1·1·1·1·1··11·1·1··11··1·1·1111111··11·······11·1·1··1·1·11···111111······1··1·1·11··11·1·1·······11··111111·11·1·1··1·1·11···111111······1·······1111·······1·1·11·1··1·1··11·········1·1·1···1·11····1·11···111111······11·11·1·1··1·1·11·······11··111111·1·1·1·1···111111·111···11·111··1·1·1······11·1···1·1··111·1·1·11··111·1·111··1·1111·1111111·11111···1···11·1··11·1·11·1·1··1·111·11·1·······11···1·11·1·111···1··1···1·1·······1·111···1·1·1···1···············
out 4 (511)	1··1······1··1··111·1··11·1···1·1111·11··1····1·1···111111···1·111·11·11·11·11111·1···1·111·1····1····1·1111·11···111·11·111······111·11·111···11·1111·1····1····1·111·1···1·11111·11·11·11·111·1···111111···1·1····1··11·1111·1···1·11··1·111··1··1······1··111·111······111·11····1··11·1111·1···1·11··1·111·1·11·111111·11·1111···1··1···11111·1111·1····1····1·111·1···1·11···1··1··1··1··1111·11·11·11·111··1·111·1···1·1111·1111·1····1·····111·11·111··1·1··1······1··1·1···1·1···1·1·111····1··11·11111·1·1·11··1···1··11·1···1·111·1··111·11·11·11·111111···1··1···11111·1111·1····1··1···1·11··1·111·1·11·111111·11·1·1···111111···1··1111·11··1······1111·11··1····11·111······111·1·1··1······1··1·1···1·11··1·111111·1111·1····1·····111·11·111·1····1··1··1··11··1··1···1·1··11····1····1·1111·11···111·11·111·····111111·1·11·11111·1·111····1111·1·····1111111111·11·1·111····11·111···1·1·11··1··1·11·1·1···1·11·····11·1·11·1·1··1111·11··11·1·111····11·1111·111·····1·1·11·11·····1·11··11··········11··11··11111···11······11·1···········
out 5 (481)	11··11·11····1111····111··11··1·1·11·1·················1·1··1·1····1111·1·1·1·11·1·1·1·····1111··11··111··1·11··11·1··1··11··1111··11··1··1·11·1··1·11···11··11····11111·1·1·1·1·1·1·1·1111·····1·11·1·11111111·········1·11·1·1··11··111····11··1111··1··11···1·1·1·1·1111········11111·1·1·1·1··1·11···11··1111··11··1··1·11···1111··1··11··11··11··111····11·········1·11·1··1·11·1·1111111·········1·1··1·1·1·11·1·········11····111··11··1·11··11·11····1··11·1··1··11··11··11··1·1··1··111·1·1·1·····111····1···1······111111····1·1·1·1·1·1·1·1·····111111··11···11·1··1·11·1··1··11··111··11··1··1111··11····111··11··11·1··1·111111111········1·1··1···1·11·1·11111111111111111·1··1·11··11··111····1111····11·11··11111··11··1··1·11··11·1··111··111·····11111·1·111····11·1···111·····1111··1···1·1··1·1······1···111·111··111·1·1·1·111···11··1·1··111···111·····11111·1··111·11···11·1·1·1·1111111·1·1···1·11··11···1··11··11··1····1···111·111·11·1····1111·1······1·1·1··1·1····1111111···111·······111·11··11···1·1··1···1·········1···1·······
out 6 (553)	··1·111·111111··111111····1·111·1··11·111·11·1111·11·11·1··11·1···1·111·111111·1······1111·1···1·11··1···1··1··11·11·11·1··11·1··1·1·111·1111·11·1111·1··1·1·111···111·111··111111··1111···111···1·1·111·1111·1·1····1·11·1·1···111···1···11···111··1111···1111···11···1···111··111···1111··1111·1111·111·1·1····1·1·11·1····1·111··111·111···1·111···1111··1111·1111·111·1·1··11·1·1··1·1111····1··1···1··11·11·11··1·11·11·11·111111·111·1······1·1111·······11·11·111·11··1·1·11··1111·111·1·111111·11111···111·11·111·1111···1··1···1··11·1·1··11·1··1··1··1······1···1·111···1·1111······1··1··1···1··11·11·11··1·11·11·11·111111·111·1······1·1111·······111··111·111···1·111···1111··1111·1111·111·1·1··11·1·1··1·1111··111··111·111···111··111·····1·1·1111·1····1·11·1·11111111·11·11·1···1·1111111·····111·111·1··1····1·11·11111··1·1·11·1111111·1····1·1··11·11·11·1···1·····11····1·········111··1··1111111·1·111·11·111·11···1···1··1·1·1111····11·11·111··1···1111·11····1·1111·111·111·1··111111111·1··1····11·11·1111·····1····11·1···1·······
out 7 (521)	··1111··1···1···1···1·····1111····1111··1···1··1·111·11111····1··1···1·1····111·1111···11·111·1··1···1·1····1111····111··1···1··111·11111·1··1···1·11·11···1····111·11111·1··1·11·1··1··111·111·1··1·11···1···1···1···1·1··1·11·1··1·11···1···1111·111·1·11·1·111·111·11····1111····11111·111·111·111·11····111·1111·····1···1····1111·1·111·11·1···1··111····1···1111·1·111·111·111·11···111111·11·1·····1···1111·111··1··1·111·11·1·····1···1···1···11·11·1·1·111·111··1·11·1··1·11···111···1·111·111··11111·111111111·1·1··11···1···11·1··1···1·11·1·111·111·111·111··1·11·1··1·11·1·111·111·1··1·11111·111·111·111··1··1·111·11·1·····1···1111·111··1··1·1····1111·1·111·111·111·11···1111·111····1·1···1··1·111·11···111··1·11······11··111·111···1···11·1··111·····1···11··1······11··11·11··111·11···1·····11·111·111·1···11111····111·1···11·111·····11111·11·1·11·1·11·····1··1111··11111···11·111··1····11111·11·····1··1·1·11·111·1··111111·11·1·11···1···111·11···1·1··111·1··1··1·1111··1·1·11·····1·1····1···111···11·11··11·1·1·1···1···1·······
out 8 (493)	11·11·······1·11··1··1111111·1·1·1·111111···11··1·1······111··1··111··11·1·1111··111··11·1·1111·····1·11··1··11·····1·11··1··11··11·11··1·1111111··1··11·1·····1111·1·11··111······1·1··11···111··111······1·1·1··111······1·1·1·1·······11·11·1·1·······11·111··11·11··1·1111111··1··11·1·····1111·1·11··111······1·1··11···111··111······1·1·1··111······1·1·1·1·······11·11·1·1·······11·111·11·11·······1·11··1··1111111·1·1·1·111111···11··1·1······111·····111··11·1·1111··111···1·1·1··1·····1·11········1·····111·11·111··111··1111·1·1·11···11····1·1··1·11111··11·11·1·1·····11··1··111··1··1·1·1111111··1··1·1·111111111·1·1·11···111111·1·1·11···1···111··1·1·1····11···11·1·1·111111111111·1·1·11··111···1···1··11·1·1··1···········11·1·111·1···1··11··11111·1·11··11·11··1···11·11·111··1·1·1·1·111··1·1·11····111·1·1·11·11·1····1·111·1111111·1·1111···1·1···11111····1···1···11······1···11···11111···1·11·1·1······1·1···111·111·11····11111·1··11···1·111··11·1··1··1·11···11··111···1··1111·1··11·1···1·1·1··1111·1·1·1····1··1···1·······
out 9 (505)	11··1·1····11··11·······1·1·11·111111···11·1·1·1·1··11·11··11111·1··11···11··········11·11·1·1···111111·1·1·11··11··1·11111··11111111···11·1·1··1·11··1··11·····11··1·1····11····1111111·1·1··1··111111·1·1·11·1··11·1·····11··1·1··11···11····111111··1··1·1···1·11··1··11····111111···11·1·1·11·······1·1·11·1··11·1·1111··11·11··1·11111··1111······1·1·1··1111111··1··1·1·11·1··11···11···111·······1·1·11··11··1·1····11···1·11··1··11··········111··1·1··111111··1··1·1·1·1·11···11··1··1·11··1·11·11·11··1·1··1111··1·1···111111·1·1·11··11··1·11111··11·1·11··111··1111111111··1··1·1·1111111···11·1·1·1·1··11·11··11111··11·1·1111··11··1111111·1·1····1·11·11·1·1111···1111·111··1·11111·11······1···1··1·1·11·11111111····1111···11·1··1···1····11··1111·1·11·11··11··111····1·1·1··1······1·1···1·1·111······1111·1111··1··1·1·1·1··11····1·····111·1··1·111··11·11···11111·11······11···1··1·1··11·1111111··11········1111·1·1··11··1·1·1··1111·1····1·111··11····11·1·11·111······1··1·11·1···11·····11···11···1······11·111·1·1·1·1·············
out 10 (473)	··1···1···1111·1···1···1····1111···1····1111······1···1111····1·1111···1···1······1111·111·111·111····1111·111·1····1111···1·····1···1·11·1··1··1···1··1·11·1···1···1···1··1·11··1···1···1·11·11·11·1··1·111·11··1·11·1··1···1·11·1··1···1···1··1··1·111·111·1····1···1···1111·1···1···1····111·111·1111····111111·111····1111··1111···1···1······1111·111·111····1111····1···1·1111····111·11···1···1·11·1··1··1···1··1·11·1··1·111·111·11·1··11·111·111·1··111·11·1··1·111·11··1·1111··1··111··1·11·111··11·11······111·1·1····1·11·1··1···1·1·11·1··1·111·11·1··1·111·111·1111·1··1···1···1··1···1··1·11·1····1···1·11·1··1···1···1···11·····11··111111···1·····1··1·11·11·1·1·111111·1···111·1·111·11·1·11······1·····1·····11··1·1···11··1·······11···111··11·11···11····1·1···11··1·11·1··11111·1·1·1·······1··11111·1·1··1··11··111·1·1·11111··1··1·1111111·11·······111·····11·1·1·11·11111··1···1··111·11·11111····1··11·1·······11111·11·11··111··1···11·1·1··11···11·11···111·1··11·1···1···1·11··11···1··1·111··11·1····1··111······1··1···········
out 11 (517)	1···111·1··1···1·11·111·1···111·1111·111···1·11·111·1···1111·1111·1···111·1111·11·1111···1·111·111·11·1···111·1···111·1···1··1·11·1···111·1111·11·1111···1·111····1··1·111···1·111···1·111·11·11·111···1·11·111·1··1···1·111····1111·111···1·11·111·1···1111·1····111·1···1··1·111·11·1···111·1··1····111·1···1··1·111···1····11···1·111····1··1····1···111·1··1·11·111·1···111·1···111·1··1··11···1·111····1··1····1···111·1···1··1···1·111···1·111···1·11·11·111···1·111·11·1···1····111··111··1····1·1·1·111···1·1··11·1·111·1···1111·11·1111·11·1111·111···1····1··1···1·111···1·11·1111·11··1·111·11··111·1····1··11·1111·11··111·1·1111·1·1·11··1·11·1·11··111·1··1·111···111··1··1··1·1······1···1····11·111···111·1111·1111·······1··11······111······1··111···1·1111111·1········111···11·11111·1·11·1111·1·1·1111···1··111···1·1·111···11···111··11·1·1·····11·111··1111111·11·11·11····11··11·11·1··1111··1·11·11·1···111111·111·11··1111·11·········1··111··11111··1·1··1··11·1····1111·1···1···1·1·111··111·1··1··1····11··11···1·1·1·1···········
out 12 (499)	··1··1··1111·11·1111·11···1··1··1···111··1·111·11·1···11·111···1····1·····1··1····1··1·1····1····1·111·1·111····1···11111·1···1·111·1··111···1·111···1··111·1····1····11·11·111·1··1···11·1111·111···1·1···1·111···1·11111···1··1··1·····1····111·1111·1·11·11··111·1·····111·1111···1·1···1·11··1····1·1··1····1··1·····1····1···111·11···1·11·111·1··111···1·1·11·111··1····1··1····11·11·11·111·11·1·1111·111····1·····1··1·1·111·····1·111···1·111·1·111··11····1··111·11·1···1·1·1·1111·11··1·111·11·····1··1·11·1·111·111··1·111·1·111···1·111·····1·111··1111·11111··1·11·111··111·111·1·111··1·11·111111·11·11111··11·11111·1·11··11·111·111·111·1··11···11111·111···1·1·1·1·11·1··11··········1···1·1·11111··11·11·1·1·11·1·1·1·1·1·1···111111·1···1····11111·11111····1·1111·11····1·11····1·1··1··11··111···1111··1···11·111·111·1·1·11·1·1·111·11·1··1··111····1··1·1····1··1··1·111··111·111··11111···111····111··1·····1·1·1·1·1·1·1111··1·1···11·1111·········1··1111····1··1···1·1·1111···1···11···1·1······11····1··1··11·1·1·1·······1·······
out 13 (509)	·111··1····1·1··1·1····1··111····111··11111·1·11·1·11111··111···1·1·····11···11··111··11111·1·1·1·1····1··111··11···11·1111·1·111···11·····1·1··1·1·····11···11··111··1····1·1··1·1····1··111··1·1·1111·11···11··111··1····1·1··1·1·····11···11··111··11111·1··11111·1···11·11··11·11···1·1111111111·1·11··1··11··1··11·1·111111··1··11·1·11111·····1·1··11·11·1··1··111·1·····11111·1···11·11111111·1·11··1··11··1··11·1·11111·····1·111··1··11··1··111·1····11··1··111·1·····111111·1··11·11··11·11·····1·1·111···1·1····1··1·1·111111··11·1·1111·1111··1··11·1111··1·····11·1··1·1··1··1··1111111····1···111··1··1111·111···11·11···111··11···11··1···1·11··11··1··1·1·11··1···1·1··1·1·11·1111·11·1···111111·1··11·1···111···1·1·1·11··1111·1···1·1·111·1··1··1··11·1·111111···1·11··111····111111·111·1·1···111·11·······1··1111··1·1·11···11·111·11··11···1··11···1··11111··1·1···1111·1·111·1·1·······111·····111·········11·1111·1·11·1····11······11··1··11···1111·11111·11····11111··1111······11·1·1·11·1111··1···1·1111····1·1·1···1·1·1···········
check 14 (512)	1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1·1
check 15 (512)	·11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11··11
check 16 (512)	···1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111····1111
check 17 (512)	·······11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111········11111111
check 18 (512)	···············1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111················1111111111111111
check 19 (512)	·······························11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111································11111111111111111111111111111111
check 20 (512)	·······························································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111
check 21 (512)	·······························································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111································································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111································································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111································································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
check 22 (512)	·······························································································································································································································································1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111································································································································································································································································1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
check 23 (512)	·······························································································································································································································································································································································································································································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
check 24 (480)	1·1·1·1·1·1·1·1·1·11·1·1·1··1·1····················11111111·····1·11·1··1·11·1··1·1·1·11·1·1·1·1111····1111····11111111·········1·1·1·11·1·1·1··1·11·1··1·11·1·11111111········1111····1111····1·1··1·1·1·11·1·1·1·1·1·1·1·1·1·1111········111111111111111111111·1·1·1·1·1·1·1··1·11·1·1·1··1·11111111111111111····11111111·····1·11·1··1·11·1·1·1·1·1··1·1·1·11111····1111············111111111·1·1·1··1·1·1·1·1·11·1··1·11·1·········111111111111····1111····1·1··1·1·1·11·1··1·1·1·1·1·1·1·11111········1111·················1·11·1··1·11·1·1·1·1·1··1·1·1·1····1111····111111111111········1·1·1·1·1·1·1·1··1·11·1·1·1··1·1················1111········1111·1·11·1·1·1··1·11·1·1·1·1·1·1·1·1111········1111·················1·1·1·11·1·1·1·1·1··1·11·1··1·1········111111111111····1111····1·1··1·11·1··1·11·1·1·1··1·1·1·11111····1111····11111111········1·1·1·1·1·1·1·1·1·1··1·1·1·11·1·····················11111111····1·1··1·1·1·11·1·1·1·1·1·1·1·1·1·····11111111·····················1·1·1·11·1·1·1··1·11·1··1·11·1·········11111111····1111····1111
check 25 (528)	··1···1···1···1···1···1111·111···1···1·11·111·1··1···1···1···1·111·111·111·111·111·111····1···111·111·1··1···1·11·111·111·111·1111····1111····1···1111·111····111·1··1···1·11·1··1·11·1··1·11·1···1111····1111·111····1···1111···1·11·111·1··1·11·1··1·11·1··1····1111····1111····1111·111····1··1·11·111·1··1···1·11·1··1·11·1111····1111····1111····1···1111·11·1··1···1·11·111·1··1·11·1··1·111·111·111·111····1···1111·111·11·111·1··1···1···1···1···1···1····1···1···1···1111·111····1···1··1···1·11·111·111·111·111·111·1·111·1111···1····111·111·111·111·1···1···1···1···1···1··1·111·11·111·1111···1····111·111·111·111·1···1···1···1···1···1··1·111·11·1111···1····1111····1111····111·1··1·11·1··1·111·11·1···1··1·11·1111···1····1111····1111····111·1··1·11·1··1·111·11·1···1··1·111····111·1111···1····1111····1111·11·1··1·11·1··1·11·1···1··1·111····111·1111···1····1111····1111·11·1··1·11·1··1·11·1···1··1·111···1····111·111·111·111·111·1111·111·111·111·11·1···1··1·111·111···1····111·111·111·111·111·1111·111·111·111·11·1···1··1·111·11
check 26 (528)	··1·1·····1·1··111·1·11111·1·11···11·11···11·11111··1··111··1···1·····11·11111·1·11111··1·····1·1··111·1·11···11·11···1·1··111····1·1·····1·1··111·1·11111·1·11111··1··111··1·····11·11···11·111·11111··1·····1·1·····11·11111··1··111·1·11···11·11···1·1··111····11·11111··1··111··1·····11·11···1·1··111·1·11111·1·11···1·1··1·11···11·11···1·1··111··1··111·1·11111·1·11111··1·····1·1·····1111··1·····11·11···11·11111··1·····1·1··111·1·11111·1·11···1·1··1·11···11·11···1·1··111··1··111··1·····1·1·····11·11111·1·11111···1··111··1··111··1··111··1··11111·1·11111·1·11111·1·11111·1·111·111··1·1···11·1·111··1·1···11·11·····1··11111·11·····1··11111·1··1··111··1··111··1··111··1··111··1·1·····1·1·····1·1·····1·1···1···11·1·111··1·1···11·1·111··1·1·····1··11111·11·····1··11111·111·1·111··1·1···11·1·111··1·1·····1··11111·11·····1··11111·11····11111·1·11111·1·11111·1·11111·11···11·11···11·11···11·11···11·1··1·1···11·1·111··1·1···11·1·111··1··11111·11·····1··11111·11····11111·1·11111·1·11111·1·11111·1·111··1··111··1··111··1··111··1·
check 27 (528)	·11··11·11··11···1111··1··1·11··11··11·11··11··1··1·11·11····11····1111·1·11·1·········1·1·1·1·1·1··1·1····1111·1·1·1·1··········11··11·11··11···1111··1··1·11·1··11··1··11··11·11·1··1··1111······1111·1·11·1·········1·1·1·1··1·11·1·1111····1·1·1·1·1111111111··11··1··11··111····11·11·1··11··11··1··11··11·11·1··1··1111··1111····1·1··1·111111111·1·1·1·1·1·11·1·1111····1·1·1·1·11111111··11··11·11··11···1111··1··1·11·1··11··1··11··11·11·1··1··1111······1111·1·11·1·········1·1·1·1··1·11·1·1111····1·1·1·1·1111111111····111··1·11···11··111··11··1·11·1··111····11·11··11···11··11·········1·1·1·11111·····1·11·1··1·1·1·111111111·1·11·1·····1111··1111···11·1··111··11···11··11··11·1··111····11·11··11···11··11111111111·1·1·1·····11111·1··1·1·1·1·1·111111111·1·11·1·····1111··1111···11·1··111··11···11··11·1··1·11···1111··1··11··111··11··111111111·1·1·1·····11111·1··1·11·1·1·1·········1·1··1·11111······1111···11·1··111··11···11··11··11·1··111····11·11··11···11··11111111111·1·1·1·····11111·1··1·1·1·1·1·111111111·1·11·1·····1111
check 28 (480)	····11······11··1·1··111·1·11····11·1·111··1·1·1··111111··111111111·11·1111·11··1·111··1·1···1111···1·1··111·1·1··1····1··1·····1·111···1·111··1111·11·····1··1·11·11111··1······111·1···111·1·1·1·11··1·1·11··11111··1·····11·1··11111·11·······11·1·1··11·1·11111·11·····1··11·1···111·1···11··111·1···111·1·1··1·····11·1111·····11·11111··11·1·11··1·1·11··11··1·1·11··1·1·1··11111·11·····1·1·11···1·1··11·····11······11··11······11·······11·1·111··1·1··1·111··1·1···11····1··1····1··11··1····1··1······111·1·11···1·1·11······11·······11·1·111··1·1··1·1··111·1·11··11111··111111··1·11·1111·11·111111···1·1··111·1··1·111··1·1···11····1··1····1··111···1·111···1·1·11·11111··1····1111·11·····1··11·1···111·1···1111··1·1·11··1·1·1··11111·11·····11111··1·····11··1·1··11·1·1··11·11·11111··1······111·1···111·1·1·1···111·1···11····1··11111·11··11·····1··1111111··1·1·11··1·1·1·1·11··1·1·11··11111··1·····11·11··1·1···11·1·1·11······11··········11······11··1·1··111·1·11··11···1·1··111·1·1··1····1··1········1··1····1··11·1···11·1·111··
check 29 (528)	·····11··11····111111··11··11111·1··11·1··1·1·1·1·11··1·11·1·1··1·1·11··11··1·11·1·1··11··11·1·1111··1111··········11····111111··11··········11··11··········11·11·1·1··1·11··1·11·1·1··1·11··11··11·1·1·1·1··11··11·1·1·1·1··111······1111··1111······1111··11····11··11······1111··11··1111111·1·1··1·11··1·1·1·1·11·1··11·1··1·11··11··1·1·11·1··11··11·1·1·111111····11··········1111··1111··1111111111··11··1111111111··11·11··1·11·1·1··1·11··1·11·1·1··11··1·1·1·1·11··11··1·1·1·1·11··111··1111······1111··1111······11··11····111111··11··1111······11·11·1·1·1·1··11·1··1·1·1·1·11··1·11··1·11·1·1··11··11·1··1·1·11···1111111111··1111··········11··111111····11····111111····11·····1·11··11··1·1·1·1·11··11··1·1·1·1·1·11·1··11·1··1·1·11·1··11·1·1111··11··1111111111··11··111111··111111····11··11······1111··11·11··1·1·1·1·11·1··11·1·1·1·1··1·11·1·1··1·11··11··1·1·11·1··11···11··········1111··1111111111··1111··1111······1111··1111·······1·1·11··11··1·1·1·1·11··11··1·1·1·11··1·11·1·1··1·11··1·11·1·1·111111··11··1111111111··11··1111
check 30 (528)	···1111·1·11·1···1111···11·1··1····11111·1··1·111····11·11·1··1·11·1··1··1111··1·1··1·11111····1··1·11···1111··1·1··1·1····1111········1·1·1·1···11··111··11··1·········1·1·1·111··11··1··11··11··11··1··11··11·1·1·1·111111111·11··11···11··11·1·1·1·1·········1·11·1·····1111·11·1··1··1111···1·11·1·1111····1··1·11···1111··11····111··1·11·····1111·1·11·1···1111··1··1·11·····11111·1··1·1·1·1·1·111111111·11··11·11··11···1·1·1·1········1··11··111··11····11··111··11··111111111·1·1·1·111··11··1··11··1111111111·1·1·1·1111····1·1··1·111····111··1·11·1111·····1·11·1···1111··1··1·11·1··1·11·11····11·1·11·1·····1111·11·1··111····11·1·11·1·1111····11111111·1·1·1·111··11···11··11·111111111·1·1·1···11··11·11··11··11··11·11··11··1·1·1·1·········1··11··111··11··1·1·1·1·11111111·1·11·1·····1111·11·1··1··1111···1·11·1·1111····1··1·11···1111··11····111··1·11·····1111·1·11·1···1111··1··1·11·····11111·1··1·1·1·1·1·111111111·11··11·11··11···1·1·1·1········1··11··111··11····11··111··11··111111111·1·1·1·111··11··1··11··1111111111·1·1·1·
check 31 (480)	·········1111············1111····11··11····111111··11··1111········1111··11··111111····11··11··11····111111111111····111111111111111111··1111··11111111··1111····11··111111····11··11······11111111······11··11····111111··11··11····11········11····11··········11··111111······11··111111····11111111··1111··········11····11··1111··1111111111····11············111111··11······111111··11··11··11··1111····11··11··1111····1111111111····11··········1111··11····1111111111··1111··············1111··11··11····1111··11··1111111111··1111··········11····1111··11······111111··11······1111····111111··11······111111··11··11····11··········1111··11111111··········1111··1111111111····1111··11··1111····11··11··1111····1111····11··11··1111····11··11··11····1111111111··1111··········11··11··1111······11··11····1111··········1111············1111····1111············1111··············1111··11··111111····11··11····11··111111····11··11······1111········11····11········11····1111····11········11····11············111111··11··1111······11··11
check 32 (528)	···········1111·1·11·1··1·1·1·1111111111111····1·1··1·11·1·1·1·11··11····1111··1··1·11··11··11···11··1111····11·11·1··11··11··1··11··11··1111···11·1··1·11··11·11··11··11····111··1·11·1··11··111111111····11111·1··1·1·1·1·1·1········1111·····1·11·1·1·1·1·1··1·11·1··1·1·1·1············1111·1·11·1··1·1·1·1············1111·11·1··11··11··1··11··1111····11·11·1··11··11··1··11··1111····111··1·11·1··11··111··11··11····111··1·11·1··11··111··11··11····111·1··1·1·1·1·1·111111111····11111·1··1·1·1·1·1·111111111····1111·1·1·1·11·1··1·11111············1·1·1·1··1·11·1·····1111111111111··11··11··1·11···1111····11··11·11··11··11·1··111····1111··11···11··11·1··1·11·11····11··11··111··11··1·11·1··1··1111··11··11··1·1·1·1·1·1··1·1····1111·········1·1·1·1·1·11·1·1111····11111111····1111111111111·1·1·1··1·11·1·····1111111111111·1·1·1··1·11·1···1111····11··111··11··11··1·11···1111····11··111··11··11··1·11·11····11··11··11·11··11·1··1·11·11····11··11··11·11··11·1··1·11·1111····11111111·1·1·1·1·1·11·1·1111····11111111·1·1·1·1·1·11·1·
check 33 (528)	·······11111111··11··1111··11······11111111······1111··11····11········1111111111··11····11··111111········1111··1111··11····11··1111··11····11····11111111····11··11····11··1111111111········11····11··1111······11111111····11··11····11··11········11111111·1·1·1·1·1·1·1·11··11··11··11··1·1·11·1··1·11·1·1··1·11·1··1·11··1·1·1·1·1·1·1·1·11··11··11··11·1·1··1·11·1··1·11··1·11·1··1·11·1··1·11·1··1·11··1·11·1··1·11·1··11··11··11··11·1·1·1·1·1·1·1·1··11·1··1·11·1··1·1·11·1··1·11·1··11··11··11··11··1·1·1·1·1·1·1·1·11··11·1··11··1·1·1·1·11·1·1·1··11·1··11··1·11··1·11·1·1·1··1·1·11··11·1··11··11·1·1·1··1·1·1·11··1·11··11·1··1·1·11·1·1·1··1·11·1··1·1·1·11·1·1··1·11··11·1··1·1·1·1·11·1·1·1··11··11·1··11··1·1·11·1·1·1··1·11··1·11··11·1··1·1·1·1·11·1·1·1·1··11··1·11··11···11··11··11··111111111111111111··1111····1111··1111····1111······11··11··11··11················11····1111····111111····1111········1111····111111····1111····111111111111111111··11··11··11··111111····1111····11····1111····11111111111111111111··11··11··11··

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 n1023Noisy T states per run.

outputs k14

distance d5Exact, proved and witnessed here.

error coefficient A_{5}8,368,120Faults of weight 5 in the circuit shown that pass every check and damage the output. The output error is A_{5}\,p^{5} to leading order.

wires N34

checks r20

V_{\mathrm{ex}}14T states per run. CCZ counts as two.

\gamma_\rho = \log(n/V_{\mathrm{ex}})/\log d2.6664Overhead per extractable T state. Lower is better.

\gamma = \log(n/k)/\log d2.6664Overhead 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 width14Rank of the output rows modulo the check rows.

Output gate

14 T0 CS0 CCZon 14 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}\cdot \mathrm{T}_{9}\cdot \mathrm{T}_{10}\cdot \mathrm{T}_{11}\cdot \mathrm{T}_{12}\cdot \mathrm{T}_{13}

As monomials: 0+1+2+3+4+5+6+7+8+9+10+11+12+13 (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 8 single-wire gates and 192 \mathrm{CZ} gates: \mathrm{Z} on output 0, \mathrm{S} on output 2, \mathrm{S}^\dagger on output 3, \mathrm{S} on output 4, \mathrm{Z} on output 8, \mathrm{Z} on output 11, \mathrm{S}^\dagger on output 12, \mathrm{Z} on output 13, \mathrm{CZ} on output 0, output 1, \mathrm{CZ} on output 0, output 3, \mathrm{CZ} on output 0, output 4, \mathrm{CZ} on output 0, output 5, \mathrm{CZ} on output 0, output 6, \mathrm{CZ} on output 0, output 7, \mathrm{CZ} on output 0, output 10, \mathrm{CZ} on output 0, output 11, \mathrm{CZ} on output 0, output 12, \mathrm{CZ} on output 0, output 13, \mathrm{CZ} on output 0, check 16, \mathrm{CZ} on output 0, check 17, \mathrm{CZ} on output 0, check 18, \mathrm{CZ} on output 0, check 20, \mathrm{CZ} on output 0, check 21, \mathrm{CZ} on output 0, check 24 and 176 more (listed below). The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.

All 200 correction gates

\mathrm{Z} on output 0, \mathrm{S} on output 2, \mathrm{S}^\dagger on output 3, \mathrm{S} on output 4, \mathrm{Z} on output 8, \mathrm{Z} on output 11, \mathrm{S}^\dagger on output 12, \mathrm{Z} on output 13, \mathrm{CZ} on output 0, output 1, \mathrm{CZ} on output 0, output 3, \mathrm{CZ} on output 0, output 4, \mathrm{CZ} on output 0, output 5, \mathrm{CZ} on output 0, output 6, \mathrm{CZ} on output 0, output 7, \mathrm{CZ} on output 0, output 10, \mathrm{CZ} on output 0, output 11, \mathrm{CZ} on output 0, output 12, \mathrm{CZ} on output 0, output 13, \mathrm{CZ} on output 0, check 16, \mathrm{CZ} on output 0, check 17, \mathrm{CZ} on output 0, check 18, \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 27, \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 1, output 4, \mathrm{CZ} on output 1, output 5, \mathrm{CZ} on output 1, output 8, \mathrm{CZ} on output 1, output 9, \mathrm{CZ} on output 1, output 10, \mathrm{CZ} on output 1, output 13, \mathrm{CZ} on output 1, check 14, \mathrm{CZ} on output 1, check 15, \mathrm{CZ} on output 1, check 16, \mathrm{CZ} on output 1, check 18, \mathrm{CZ} on output 1, check 22, \mathrm{CZ} on output 1, check 23, \mathrm{CZ} on output 1, check 24, \mathrm{CZ} on output 1, check 25, \mathrm{CZ} on output 1, check 26, \mathrm{CZ} on output 1, check 30, \mathrm{CZ} on output 1, check 31, \mathrm{CZ} on output 1, check 33, \mathrm{CZ} on output 2, output 4, \mathrm{CZ} on output 2, output 9, \mathrm{CZ} on output 2, check 15, \mathrm{CZ} on output 2, check 17, \mathrm{CZ} on output 2, check 19, \mathrm{CZ} on output 2, check 22, \mathrm{CZ} on output 2, check 23, \mathrm{CZ} on output 2, check 24, \mathrm{CZ} on output 2, check 25, \mathrm{CZ} on output 2, check 26, \mathrm{CZ} on output 2, check 30, \mathrm{CZ} on output 2, check 32, \mathrm{CZ} on output 2, check 33, \mathrm{CZ} on output 3, output 4, \mathrm{CZ} on output 3, output 5, \mathrm{CZ} on output 3, output 8, \mathrm{CZ} on output 3, output 9, \mathrm{CZ} on output 3, output 12, \mathrm{CZ} on output 3, check 14, \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 20, \mathrm{CZ} on output 3, check 25, \mathrm{CZ} on output 3, check 26, \mathrm{CZ} on output 3, check 27, \mathrm{CZ} on output 3, check 28, \mathrm{CZ} on output 3, check 30, \mathrm{CZ} on output 4, output 5, \mathrm{CZ} on output 4, output 6, \mathrm{CZ} on output 4, output 8, \mathrm{CZ} on output 4, output 13, \mathrm{CZ} on output 4, check 14, \mathrm{CZ} on output 4, check 17, \mathrm{CZ} on output 4, check 20, \mathrm{CZ} on output 4, check 22, \mathrm{CZ} on output 4, check 23, \mathrm{CZ} on output 4, check 24, \mathrm{CZ} on output 4, check 27, \mathrm{CZ} on output 4, check 28, \mathrm{CZ} on output 4, check 29, \mathrm{CZ} on output 4, check 32, \mathrm{CZ} on output 4, check 33, \mathrm{CZ} on output 5, output 7, \mathrm{CZ} on output 5, output 9, \mathrm{CZ} on output 5, output 11, \mathrm{CZ} on output 5, output 13, \mathrm{CZ} on output 5, check 14, \mathrm{CZ} on output 5, check 16, \mathrm{CZ} on output 5, check 17, \mathrm{CZ} on output 5, check 18, \mathrm{CZ} on output 5, check 20, \mathrm{CZ} on output 5, check 21, \mathrm{CZ} on output 5, check 22, \mathrm{CZ} on output 5, check 23, \mathrm{CZ} on output 5, check 24, \mathrm{CZ} on output 5, check 27, \mathrm{CZ} on output 5, check 28, \mathrm{CZ} on output 5, check 29, \mathrm{CZ} on output 5, check 31, \mathrm{CZ} on output 6, output 8, \mathrm{CZ} on output 6, output 10, \mathrm{CZ} on output 6, output 13, \mathrm{CZ} on output 6, check 14, \mathrm{CZ} on output 6, check 18, \mathrm{CZ} on output 6, check 19, \mathrm{CZ} on output 6, check 21, \mathrm{CZ} on output 6, check 23, \mathrm{CZ} on output 6, check 24, \mathrm{CZ} on output 6, check 30, \mathrm{CZ} on output 6, check 33, \mathrm{CZ} on output 7, output 8, \mathrm{CZ} on output 7, output 10, \mathrm{CZ} on output 7, output 11, \mathrm{CZ} on output 7, check 14, \mathrm{CZ} on output 7, check 16, \mathrm{CZ} on output 7, check 17, \mathrm{CZ} on output 7, check 18, \mathrm{CZ} on output 7, check 20, \mathrm{CZ} on output 7, check 21, \mathrm{CZ} on output 7, check 23, \mathrm{CZ} on output 7, check 25, \mathrm{CZ} on output 7, check 29, \mathrm{CZ} on output 7, check 31, \mathrm{CZ} on output 8, output 9, \mathrm{CZ} on output 8, output 10, \mathrm{CZ} on output 8, output 11, \mathrm{CZ} on output 8, output 12, \mathrm{CZ} on output 8, check 15, \mathrm{CZ} on output 8, check 16, \mathrm{CZ} on output 8, check 17, \mathrm{CZ} on output 8, check 19, \mathrm{CZ} on output 8, check 22, \mathrm{CZ} on output 8, check 24, \mathrm{CZ} on output 8, check 25, \mathrm{CZ} on output 8, check 26, \mathrm{CZ} on output 8, check 27, \mathrm{CZ} on output 8, check 28, \mathrm{CZ} on output 8, check 30, \mathrm{CZ} on output 8, check 32, \mathrm{CZ} on output 9, output 13, \mathrm{CZ} on output 9, check 15, \mathrm{CZ} on output 9, check 18, \mathrm{CZ} on output 9, check 19, \mathrm{CZ} on output 9, check 21, \mathrm{CZ} on output 9, check 28, \mathrm{CZ} on output 9, check 29, \mathrm{CZ} on output 9, check 30, \mathrm{CZ} on output 10, output 12, \mathrm{CZ} on output 10, output 13, \mathrm{CZ} on output 10, check 15, \mathrm{CZ} on output 10, check 16, \mathrm{CZ} on output 10, check 17, \mathrm{CZ} on output 10, check 21, \mathrm{CZ} on output 10, check 22, \mathrm{CZ} on output 10, check 24, \mathrm{CZ} on output 10, check 25, \mathrm{CZ} on output 10, check 29, \mathrm{CZ} on output 10, check 32, \mathrm{CZ} on output 11, output 12, \mathrm{CZ} on output 11, output 13, \mathrm{CZ} on output 11, check 14, \mathrm{CZ} on output 11, check 15, \mathrm{CZ} on output 11, check 16, \mathrm{CZ} on output 11, check 21, \mathrm{CZ} on output 11, check 22, \mathrm{CZ} on output 11, check 23, \mathrm{CZ} on output 11, check 24, \mathrm{CZ} on output 11, check 26, \mathrm{CZ} on output 11, check 31, \mathrm{CZ} on output 12, output 13, \mathrm{CZ} on output 12, check 15, \mathrm{CZ} on output 12, check 18, \mathrm{CZ} on output 12, check 21, \mathrm{CZ} on output 12, check 22, \mathrm{CZ} on output 12, check 23, \mathrm{CZ} on output 12, check 24, \mathrm{CZ} on output 12, check 27, \mathrm{CZ} on output 12, check 28, \mathrm{CZ} on output 12, check 31, \mathrm{CZ} on output 13, check 14, \mathrm{CZ} on output 13, check 15, \mathrm{CZ} on output 13, check 16, \mathrm{CZ} on output 13, check 17, \mathrm{CZ} on output 13, check 18, \mathrm{CZ} on output 13, check 23, \mathrm{CZ} on output 13, check 24, \mathrm{CZ} on output 13, check 25, \mathrm{CZ} on output 13, check 28, \mathrm{CZ} on output 13, check 29, \mathrm{CZ} on output 13, check 31, \mathrm{CZ} on output 13, check 32, \mathrm{CZ} on output 13, check 33

Or no Clifford gate at all: run \mathrm{T}^3 on rotations 1, 2, 7, 8, 17, 19, 35, 63, 255, 469, 499, 507, 511, 514, 519, 520, 521, 523, 525, 527, 530, 531, 535, 536 and 236 more (listed below); \mathrm{T}^5 on rotations 0, 9, 23, 31, 254, 468, 470, 487, 488, 493, 496; \mathrm{T}^\dagger on rotations 15, 127, 477, 497, and \mathrm{T} on the rest. A power of \mathrm{T} costs one magic state, like \mathrm{T} itself. For a code this is a transversal \mathrm{T} gate, \mathrm{T}^j on each qubit.

All 260 rotations at T³

1, 2, 7, 8, 17, 19, 35, 63, 255, 469, 499, 507, 511, 514, 519, 520, 521, 523, 525, 527, 530, 531, 535, 536, 537, 538, 539, 540, 541, 542, 544, 548, 550, 551, 552, 557, 558, 563, 564, 565, 566, 567, 568, 571, 575, 576, 577, 582, 583, 584, 587, 588, 590, 593, 600, 601, 602, 610, 611, 612, 614, 615, 616, 617, 618, 621, 622, 624, 625, 626, 627, 639, 642, 645, 648, 650, 651, 652, 657, 658, 663, 666, 668, 669, 673, 675, 677, 678, 681, 682, 683, 684, 685, 687, 688, 690, 691, 694, 695, 696, 699, 701, 708, 709, 710, 712, 715, 716, 718, 719, 722, 723, 727, 728, 730, 736, 737, 744, 746, 747, 748, 750, 758, 759, 763, 765, 767, 769, 770, 771, 773, 774, 776, 777, 781, 782, 783, 786, 795, 797, 799, 800, 801, 802, 804, 805, 806, 807, 808, 810, 811, 812, 818, 819, 823, 824, 825, 829, 830, 832, 837, 839, 840, 842, 843, 844, 847, 848, 850, 853, 854, 855, 856, 858, 860, 861, 862, 868, 869, 870, 872, 874, 876, 877, 880, 881, 882, 883, 889, 890, 892, 893, 894, 896, 897, 899, 902, 904, 906, 907, 908, 909, 912, 914, 915, 921, 922, 924, 925, 928, 929, 937, 938, 940, 942, 943, 944, 947, 950, 952, 953, 960, 961, 963, 965, 966, 967, 968, 970, 973, 976, 980, 982, 984, 986, 987, 988, 989, 993, 995, 997, 998, 1000, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1010, 1011, 1014, 1015, 1016, 1017, 1019, 1020, 1021, 1022

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/1023.14.5.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-1023.14.5.a.txt", "w") as out:
    out.write("\n".join(lines) + "\n")

Distance

d=5, exact. No fault of weight below 5 is both undetectable and damaging. One of weight 5 is.

Exactly 8,368,120 faults of weight 5 are undetectable and damaging, so A_{5}=8{,}368{,}120. If each rotation fails independently with probability p, an accepted run has a damaged output with probability A_{5}\,p^{5}+O(p^{6}).

A damaging fault of weight 5 sits on columns [95,197,281,549,921].

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_catalog/catalogue_new",
  "label": "T0.T1.T2.T3.T4.T5.T6.T7.T8.T9.T10.T11.T12.T13",
  "N": 34,
  "d": 5,
  "d_is_exact": true,
  "origin": "frontier_campaign",
  "provenance": "frontier_campaign \u00b7 catalogue.jsonl:411"
 }
]

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/1023.14.5.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[1333].

[\![1023,\,14,\,5]\!] factory