Magic State Factory Catalog

[\![2295,\,7,\,2]\!] factory

\mathrm{CS}_{01}\cdot \mathrm{CS}_{02}\cdot \mathrm{CS}_{03}\cdot \mathrm{CS}_{04}\cdot \mathrm{CS}_{05}\cdot \mathrm{CS}_{06}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{13}\cdot \mathrm{CS}_{14}\cdot \mathrm{CS}_{15}\cdot \mathrm{CS}_{16}\cdot \mathrm{CS}_{23}\cdot \mathrm{CS}_{24}\cdot \mathrm{CS}_{25}\cdot \mathrm{CS}_{26}\cdot \mathrm{CS}_{34}\cdot \mathrm{CS}_{35}\cdot \mathrm{CS}_{}\cdots

distance, proved here
d=2
error coefficient
A_{2}=9{,}180
\gamma_\rho
—
V_{\mathrm{ex}}
—
wires
N=15
Clifford correction
none with T powers
found by
pre-existing

The circuit

The 15\times2295 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 1030 1040 1050 1060 1070 1080 1090 1100 1110 1120 1130 1140 1150 1160 1170 1180 1190 1200 1210 1220 1230 1240 1250 1260 1270 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440 1450 1460 1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570 1580 1590 1600 1610 1620 1630 1640 1650 1660 1670 1680 1690 1700 1710 1720 1730 1740 1750 1760 1770 1780 1790 1800 1810 1820 1830 1840 1850 1860 1870 1880 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2110 2120 2130 2140 2150 2160 2170 2180 2190 2200 2210 2220 2230 2240 2250 2260 2270 2280 2290
out 0 (510)111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·························································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································
out 1 (510)111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·······························································································································································································································································111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111··························································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································
out 2 (510)111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111······························································································································································································································································································································································································································································································································111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111···························································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································
out 3 (510)111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·····························································································································································································································································································································································································································································································································································································································································································································································································111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111····························································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································
out 4 (510)111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111····························································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·····························································································································································································································································································································································································································································································································································································································································································································································································
out 5 (510)111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111···························································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111······························································································································································································································································································································································································································································································································
out 6 (510)111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111··························································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·······························································································································································································································································
check 7 (1152)11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·······························································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·······························································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·······························································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·······························································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·······························································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·······························································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·······························································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·······························································································································11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111·······························································································································
check 8 (1152)·1111111111111111111111111111111111111111111111111111111111111111·······························································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111·······························································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111·······························································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111·······························································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111·······························································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111·······························································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111·······························································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111·······························································1111111111111111111111111111111111111111111111111111111111111111································································1111111111111111111111111111111111111111111111111111111111111111·······························································1111111111111111111111111111111111111111111111111111111111111111·······························································
check 9 (1152)··11111111111111111111111111111111·······························11111111111111111111111111111111································11111111111111111111111111111111·······························11111111111111111111111111111111·································11111111111111111111111111111111·······························11111111111111111111111111111111································11111111111111111111111111111111·······························11111111111111111111111111111111·································11111111111111111111111111111111·······························11111111111111111111111111111111································11111111111111111111111111111111·······························11111111111111111111111111111111·································11111111111111111111111111111111·······························11111111111111111111111111111111································11111111111111111111111111111111·······························11111111111111111111111111111111·································11111111111111111111111111111111·······························11111111111111111111111111111111································11111111111111111111111111111111·······························11111111111111111111111111111111·································11111111111111111111111111111111·······························11111111111111111111111111111111································11111111111111111111111111111111·······························11111111111111111111111111111111·································11111111111111111111111111111111·······························11111111111111111111111111111111································11111111111111111111111111111111·······························11111111111111111111111111111111·································11111111111111111111111111111111·······························11111111111111111111111111111111································11111111111111111111111111111111·······························11111111111111111111111111111111·································11111111111111111111111111111111·······························11111111111111111111111111111111································11111111111111111111111111111111·······························11111111111111111111111111111111·······························
check 10 (1152)···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··················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··················1111111111111111···············1111111111111111················1111111111111111···············1111111111111111·················1111111111111111···············1111111111111111················1111111111111111···············1111111111111111···············
check 11 (1152)····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···········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···········11111111·······11111111········11111111·······11111111·········11111111·······11111111········11111111·······11111111··········11111111·······11111111········11111111·······11111111·········11111111·······11111111········11111111·······11111111·······
check 12 (1152)·····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········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········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 13 (1152)······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·······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·······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 14 (1152)·······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·······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·······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

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 n2295Noisy T states per run.
outputs k7
distance d2Exact, proved and witnessed here.
error coefficient A_{2}9,180Faults of weight 2 in the circuit shown that pass every check and damage the output. The output error is A_{2}\,p^{2} to leading order.
wires N15
checks r8
V_{\mathrm{ex}}not definedUndefined, because the gate's terms overlap.
\gamma_\rho = \log(n/V_{\mathrm{ex}})/\log dnot definedOverhead per extractable T state. Lower is better.
\gamma = \log(n/k)/\log d8.3569Overhead 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 width7Rank of the output rows modulo the check rows.

Output gate

0 T21 CS35 CCZon 7 output wires

As gates: \mathrm{CS}_{01}\cdot \mathrm{CS}_{02}\cdot \mathrm{CS}_{03}\cdot \mathrm{CS}_{04}\cdot \mathrm{CS}_{05}\cdot \mathrm{CS}_{06}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{13}\cdot \mathrm{CS}_{14}\cdot \mathrm{CS}_{15}\cdot \mathrm{CS}_{16}\cdot \mathrm{CS}_{23}\cdot \mathrm{CS}_{24}\cdot \mathrm{CS}_{25}\cdot \mathrm{CS}_{26}\cdot \mathrm{CS}_{34}\cdot \mathrm{CS}_{35}\cdot \mathrm{CS}_{36}\cdot \mathrm{CS}_{45}\cdot \mathrm{CS}_{46}\cdot \mathrm{CS}_{56}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{013}\cdot \mathrm{CCZ}_{014}\cdot \mathrm{CCZ}_{015}\cdot \mathrm{CCZ}_{016}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{024}\cdot \mathrm{CCZ}_{025}\cdot \mathrm{CCZ}_{026}\cdot \mathrm{CCZ}_{034}\cdot \mathrm{CCZ}_{035}\cdot \mathrm{CCZ}_{036}\cdot \mathrm{CCZ}_{045}\cdot \mathrm{CCZ}_{046}\cdot \mathrm{CCZ}_{056}\cdot \mathrm{CCZ}_{123}\cdot \mathrm{CCZ}_{124}\cdot \mathrm{CCZ}_{125}\cdot \mathrm{CCZ}_{126}\cdot \mathrm{CCZ}_{134}\cdot \mathrm{CCZ}_{135}\cdot \mathrm{CCZ}_{136}\cdot \mathrm{CCZ}_{145}\cdot \mathrm{CCZ}_{146}\cdot \mathrm{CCZ}_{156}\cdot \mathrm{CCZ}_{234}\cdot \mathrm{CCZ}_{235}\cdot \mathrm{CCZ}_{236}\cdot \mathrm{CCZ}_{245}\cdot \mathrm{CCZ}_{246}\cdot \mathrm{CCZ}_{256}\cdot \mathrm{CCZ}_{345}\cdot \mathrm{CCZ}_{346}\cdot \mathrm{CCZ}_{356}\cdot \mathrm{CCZ}_{456}

As monomials: 01+02+03+04+05+06+12+13+14+15+16+23+24+25+26+34+35+36+45+46+56+012+013+014+015+016+023+024+025+026+034+035+036+045+046+056+123+124+125+126+134+135+136+145+146+156+234+235+236+245+246+256+345+346+356+456 (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: \mathrm{S} on output 0, \mathrm{S} on output 1, \mathrm{S} on output 2, \mathrm{S} on output 3, \mathrm{S} on output 4, \mathrm{S} on output 5, \mathrm{S} on output 6. The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.

Or no Clifford gate at all: run \mathrm{T}^3 on rotations 1, 128, 1912, 2037, 2038, 2104, 2165, 2166, 2167, 2231; \mathrm{T}^5 on rotations 6, 7; \mathrm{T}^\dagger on rotations 255, 510, 765, 1020, 1275, 1530, 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.

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/2295.7.2.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-2295.7.2.a.txt", "w") as out:
    out.write("\n".join(lines) + "\n")

Distance

d=2, exact. No fault of weight below 2 is both undetectable and damaging. One of weight 2 is.

Exactly 9,180 faults of weight 2 are undetectable and damaging, so A_{2}=9{,}180. If each rotation fails independently with probability p, an accepted run has a damaged output with probability A_{2}\,p^{2}+O(p^{3}).

A damaging fault of weight 2 sits on columns [0,255].

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. Singh, C. Gidney, and C. Jones, "Borrowed Identities: Malleable Distillation Factories and a Unified Numerical Search," arXiv:2606.28518 (2026). arxiv.org/abs/2606.28518

Provenance

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

Found by: borrowed-identity search: two-group

Discovery: pre-existing

1 recorded source
[
 {
  "regime": "borrowed-identity search: two-group",
  "file": "borrowed_identities/circuits/unified_l3.json",
  "label": "ls-0952",
  "N": 15,
  "d": 2,
  "d_is_exact": true,
  "origin": "https://github.com/shraggy/Magic_state_factory_search at 4627014, outputs/large_sweep.csv line 952",
  "provenance": "unified search l=3, two-group s_total=1,s_O=6"
 }
]

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/2295.7.2.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[1439].