Magic State Factory Catalog

[\![208,\,44,\,3]\!] factory

\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}\cdot \mathrm{T}_{14}\cdot \mathrm{T}_{15}\cdot \mathrm{T}_{16}\cdot \mathrm{T}_{17}\cdot \mathrm{T}_{18}\cdot \mathrm{T}_{19}\cdot \mathrm{T}_{20}\cdot \mathrm{T}_{21}\cdot \mathrm{T}_{22}\cdot \mathrm{T}_{23}\cdot \mathrm{T}_{}\cdots

distance, proved here
d=3
error coefficient
A_{3}=1{,}398
\gamma_\rho
1.4139
V_{\mathrm{ex}}
44
wires
N=54
Clifford correction
none with T powers
found by
AI search pure T

The circuit

The 54\times208 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
out 0 (85)11··1·1·······1·11··1··11·11···11·11····111·····1·1111······11·1··1······1·1·1···1·11··1·11·1111··1·111····1·11·······1···1···1··11·11····11·1·1····111·1··1111·····11·1·1····1···1··1····1····11·1·11·1·111··1·
out 1 (123)1111111·1·····11·111··11···1111·11·1··11·1······11··11·····11111111··111·111·111·1111·11·11·1111111·111··111111·1·····11··1·11··11···111·1·11··111···1··11···11···1·111111111·1··11··111··1·1··1111·11·1·111111·
out 2 (77)1···1·1··111··1·····1····111···11·11·····1······1·1·11········11··1···1·11·11·1··1·1···1·11···11··1·11·····1·11··111··1·······1····111····11·111·····1··1··1·11·······11·1·1·11···1·1·1···1····1··1·11····11··1·
out 3 (77)1111·1··1·····11··111·111···1111·11···1···1······1·········1··1·11·1·1·1··1·1·1···1···1····11···11···11··11·1···1·····11····111·111···11·11·11··11····1··1········1···1·1·1·1····1··1·1·····1····1····11····11··
out 4 (85)1··1···1·1····11··········1111·1··1···1·111··1·····1111·1··11··1111··11·····111··111···1·111·1··1····1····1····1·1····11············1111··1··1··11··1111··1·11111·1·1··11111········111···1·1··1··1·111··1··1···
out 5 (75)·1···111·11······11·11···11···1111····1···1·1·····1·1·······11·1111····1·11111·········1·1····1·1···1·1·····1111·11·······1·1·11···11····1111···11····1····1·1······11·111··1·1··11·11············1·1·····1·1···
out 6 (95)·1·1···11·····11···1·1·111·111·1···1····111··1··11·1·11·1··1111······11···1·111··1111·1···11·1··111···1···1····11·····11·····1·1·111·111··1····1····111111··1·111·1·111···11·····1·1111···1·1··111···11··1··111·
out 7 (75)·1···111·11······11·11·11·····1111····1····1·1····1·1·······11·1111····1·111··11·······1·1···1·1····1·1·····1111·11·······1·1·11·11······1111···11·····1···1·1······11·111··1·1··11···11··········1·1····1·1····
out 8 (77)····111·111····1···111··11··1·1·1··1··1··1······1·1·1·1·····1···1·1···11·11···11·1·1····11·1·1·1·1·········11·1·111····1·····111··11··1··1·1···111···1··1··1·1·1····1····1·11·1··1····11··1····1····1·1··1·1·1··
out 9 (55)1···1··1··1·········11··111··1·1··1·······1··1····1·····1··1·1············11·111··11·········1··1····1·····1···1··1···········11··111··1··1··1········11···1····1·1··1···········11··111·1··1··1·········1··1···
out 10 (85)·1··1·11·11···1··1··11·1·1·1····1·11·····1···11·1··111·····1···1··1···1··1····11·1111··1·11···11111·1·1····1·111·11···1···1···11·1·1·1·····1·1·1·····1·11···111···1····1·1·1··1·······11··1·1··11·1·11····11111·
out 11 (85)1·11··1111·1·····1·1·11··11·11··1·1··········1···11·1······11··1··1·11···1·11·1···1···11·1·11·1111··11···11··11111·1······1··1·11··11·11···1·1·········1·1·1·1····1·1··1·11···1···1·1·1·····1····11·1·11··1111··
out 12 (77)···1·1·11······1·1111·1·1·1·111··11···1·1·1··1···111··1·······1·11···1·11·1···11·1····1····1·····1········1·1··11······1··1·111·1·1·1·1111··11··11··1·11·1·11··1······1·1·1·11···1····11··1······1····1······1··
out 13 (75)1···111··········11·1··1·11···11111···1·1·1········11···1··1···11·1····1·11··1·····1···1·1··1·111···11·····1111·····1·····1·1·1··1·11····11111··11··1·1·····11··1·1····1·1··1·1··1···1·········1··1·1··1··111···
out 14 (77)11··1·1··1····1·····1··11111···11··1·····1······1·11·1·······1·1··1···1·11·1·11··1··1··1·11···11··1·1111···1·11··1····1·······1··11111····11···1·····1··1··11·1······1·1·1·1·11···1··11···1·····1·1·11····11··1·
out 15 (93)111·1·11·11···1······1···1······1·11·····11··1··1·11111····1···1··1···1·111·1·1111111··1·11··1111·1·111··1·1·111·11···1········1···1·······1·1·1·····1111··11111··1····1·1·1·11··1··1·11··1·1··11·1·11···1111·1·
out 16 (77)11··11·1···1··1··11··1·11··1··1··1·1··1·1·1··1··1····1······111·11····111·1·1·1··1··1·····1·1·····1··11····11··1···1··1···1·1··1·11··1···1··1··111··1·111·····1·····111·1··111···1··1·1···1·····1····1·11·····1·
out 17 (61)····11·1··1···1······1··11·1··1··1·1··1··1······1·11·1······1···11····11··1······1··1·····1··1··1·1········11··1··1···1········1··11·1···1··1··111···1··1··11·1·····1···1··11··1·1········1·····1····1···1··1·1·
out 18 (61)1··1·1·····1·1···1···11······111···1··1·111·····11······1····1·1·····111····11···1····11······1·····11····1·11·····1······1····11······1·11····11···111·11······1····1·1··111·······11····1······11·······1·····
out 19 (77)·1··1·1····1··1·····1···1·11···11·1······1······1··111······1111··1···1·11·111···1·111·1·11···11··1·1·1····1·11····1··1·······1···1·11····11·1·······1··1···111·····1111·1·1·11···1·11····1····11·1·11····11··1·
out 20 (85)1···11·1·1·1··1···1··1·11·11··1··111··1·11···1··1··1·1·······11·11···111··1·1·11·1·11·····1·11····1··1·····11··1·1·1··1·1···1··1·11·11···1··11·111··11·11···1·1······11·1·111····1··1·11··1····11····1·1·1····1·
out 21 (31)········1111···········1111·································111·············1111································1111·············1111······························1111·············1111························
out 22 (85)1··1·1·11··11··1·111111·1·1·1111·11···1······1···1·1··1········111···1·11·11··11··1···1····111···1···1····1·1··11··1···1··1·11111·1·1·11·11·11··11·····1·1··1··1·······11·1·11···11···11····1····1····11·1···1··
out 23 (31)11···············1·····1··1···············1··1····11····················11··1··1·············1··1····11···················11·····1··1·················11···11················11·····1··1·················1··1···
out 24 (85)1··111111··1···1·111·11···1·1111·1····1·1········1·1··1·1·1·····11···1·11··1·111··11··1····1·1··11···1····111·111··1···1··1·11·11···1·11·11·1···11··1····1··1··11·······1·1·11····1··111····1··1·1····1··1··11··
out 25 (85)1····1·1·1····1···1····1··11··11·111·11·111··1··1··1·1··1····1··11····11··111111·1111··1··1·1·····1··1······1··1·1····1·····1····1··11···11·11·111··11111···1·1·1····1··1··11····11·1111··1·1··11·1··1·1······1·
out 26 (79)11···11·1111·····111··11··1···111111············1111·····1·1···········1·1111··1····1·1······111111··11·····1·1·1111······1·11··11··1····11111·1········11·11·····1·········1·1··11·1··1········11·······111111·
out 27 (123)···1·11·1·11··11·111··1··1·111111··1··1·111··1··1111111·1··11··1111··111·11111·1·1·11·11·111··11·11·1·····1·111·1·11··11··1·11··1··1·111·111···111··111111·111111·1·1··111111·1··11·11·1··1···11111·111···11·11·
out 28 (77)1····1·1··1···1···1·11·····1··11·111··1·111··1··1··1·1··1···1···11····11·····11··1111····11·1·····1··1······1··1··1···1·····1·11·····1···11·11·1111·11111···1·1·1···1···1··11········11···1·1··11···11·1······1·
out 29 (85)··11··111··1···1···1·111·11·11··1·········1··1···1··1······1···1··1··1··11·1··1···11··11·111111111··1····11··1111··1···1·····1·111·11·11···1·······1··11·1···1····1····1·11··11···1···1·····1··1·11·1111·11111··
out 30 (123)··11·11·111···11·111111···11111·11·1··1·11······11111···1··1··11111··111·1··111··1111·11·111·111111·1···111·111·111···11··1·11111···1111·1·11··111··11··11·111··1·1···1111111·1·····111···1·1··1111·111··111111·
out 31 (47)11···1·1·11·········11··11················1··1····11·······111············11·11··············1··1····11·····1··1·11···········11··11··················11···11····11·11···········11··11··················1··1···
out 32 (123)·111·11111····11·111··111111·1111111··1·111··1··111·11··1····111111··111·111·····1··1·11·111··11·11·1·1··11·111111····11··1·11··111111·1·11111·111··111111·1·11·1····11111111·1··11·····1·1·····111·111···11·11·
out 33 (85)·1·1·1·11111···1·111··11·1··1111·11···1······1···11···1·····11··111··1·11···11····1···1····111···1····1···1·1··11111·1·1··1·11··11·1··11·11·11··11·····1·1·1···1····11··111·11······11······1····1····11·1···1··
out 34 (47)·········11·········11·1111····11·········1··1··············11············111111··1··········1··1················11···········11·1111·····11··········11············11···········11·1111····11···········1··1···
out 35 (87)1··1····1·1···11·1·1·11·111111····11····111····1111··11·1···1·1······11·1·1··111·1··1·1···11·1···1···1····1·····1·1···11··1··1·11·111111·····1·1····111·11·1··111···1·1···11·1···1···111··1·····11···11··1···1··
out 36 (79)·1·1····1·1···11·1·11·1···1111····11····111······1·1·11·1···1·1······11·1··1···1·1··1·1···11·1···11···1···1·····1·1···11··1··11·1···1111·····1·1····111··1··1·111···1·1···11·1····1····1··1·····11···11··1···111
out 37 (61)11··11·1··1···1···1··1·····1··1··1·1··1··11··1··1····1······1···11····1···1··11··1··1·····1·······1··11····11··1··1···1··1··1··1·····1···1··1··111···1111·····1·····1···1··1·····1···11···1·····1····1········1·
out 38 (123)····111·111···11·111111111·1111·11·1··1·11······11111111···1··11111··111·1·····1·1111·11·111·111111·1······1111·111···11··1·11111111·111·1·11··111··11··11·11111··1···1111111·1········1··1·1··1111·111··111111·
out 39 (77)11·1·1··1111···1··11··11·11·111··11··············1····1····111··11···1·1····11·1······1····111··11···11···1·1···1111···1····11··11·11·11·1··11···1·······1·····1··1·11··1·1·1·······11·1·········1·1··11·1··11··
out 40 (85)1···1·11·11···1··1·······111····1·11·····1···1··111·11·····1···1··1···1··1111·1··1111··1·11···111·1111·····1·111·11···1···1········111·····1·1·1·····1·111·1·11···1····1·1·1··1··11·1·1···1·1··11·1·11····111·1·
out 41 (31)11·······11············1··1·······1·1·············11········11··············1··1···1··1··············11··········11··············1··1········1·············11·······11··············1··1·······1·1··············
out 42 (87)·1·1···11·····11····1·1···1111·1···1····111··1··11·1·11·1··1111······11····1···1·1111·1···11·1··111···1···1····11·····11······1·1···1111··1····1····111111··1·111·1·111···11····1·1····1··1·1··111···11··1··111·
out 43 (123)···1·11·11·1··11·111111··1·111·111·1··1·11······1111111·1··1·1·1111··111·1··11·1·1·11·11·111·111111·1·····1·111·11·1··11··1·11111··1·111··111··111··11··11·111111·1··1·111111·1·····11·1··11···1111·111··111111·
check 44 (124)··11111·111···11·111··1·1111111·11·1··1·111··1··111111·····1··11111··111·1111····1111·11·111··11·11·1····111111·111···11··1·11··1·111111·1·11··111··111111·1111···1···1111111·1··11·1·····1·1··1111·111···11·11·
check 45 (88)·1·111··11····1······1·111··1111·1·······11··1··1·1···1·1··11·11··1···1·111·1·····11··11·11··1111·1···1···111···11····1········1·111··11·11·1········1111··1···11·1·1·11·1·1·11··1··1·······1··1·11·11···1111·1·
check 46 (88)1··1··11·1·1······1···1··1111·1···1···1·1·1··1····111···1···11···11··11··1·11111··1·1··1··111·1·111··1····1···11·1·1········1···1··1111··1···1··1···1·11···111··1···11··1111··1···1·1111····1···1·1··111··1·111·
check 47 (126)·1···1·11·1111·111·111···1···11·1·1·11·1··11111111···1·1·111··1·11·11111111·1···1····11·1·111·1·1·11·1·11111··1··1··111·11·11···111·111·1·1·1·11111111····1111·111·111·1·1·····11·11·111111111111·111···11·1·1·1
check 48 (102)1··1··11·····1··1·······1·1·11··1·111·111·11·1·111·1··1··1·111···1··111·1111··11··111·1······1·11··1···1·111·11··1·1·1·1··1·1·1·1······11·111111·11·111·1··111··1····11·111···11··11·11·1·1···11···11·1·1111··1·
check 49 (102)···1··11111····111111··1····1····1··1···11··11··11·1··111·1·1··1·1········11··1··111·1111··11·11·1·11·11···1·1··11·111·11·11·11111·11·111··1···1·11··········1·1··1·1111·111··11·1·1···1·1··1111··1·1·1·1·1·11·1
check 50 (102)·1··11··1·····1·1111111·1··11111·1111·111·····11··1···111··1··1·11·111111·11··1·1·1··1·······1··1·11··1·111·1····111···11·1·1·····1·····111·1·1·11·1·1··11·1··1···1111··1···11··1··1··1·111·1··111···11111··11·1
check 51 (92)1·····111·11·····1·1·11·1·1····1····1·1····111111··11····1111··1111·····1·1·1·1···1111·1·1····111··111······1····1·······1·11·1···1·1····1·1111111····1··1··11···1·1·11111··1·111·111·1····1·11··11·1····1···111
check 52 (112)·111··11··1·11111·1·1·11·1···1·11111·1·1·1·11·111···1·1·1·····11111111···1111··1·111··1111·11·1·1111··1··11···11··1·11··1111·1·1·1·1···1··1111·111111·11·1···1·11·····1····111·11··11··1··1·1··1·1·1·1··11·1···1
check 53 (100)·1···1··1··1··11···11··1····1·1···111111···11111·111·11·····1·111······11····1··1·1·111·1·1··1·11··11·1·····11··1··1··11·····11···1111·11·111·1···11111·1··11·11····1·11····11······1·111111·111··111·111·1··111

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 n208Noisy T states per run.
outputs k44
distance d3Exact, proved and witnessed here.
error coefficient A_{3}1,398Faults of weight 3 in the circuit shown that pass every check and damage the output. The output error is A_{3}\,p^{3} to leading order.
wires N54
checks r10
V_{\mathrm{ex}}44T states per run. CCZ counts as two.
\gamma_\rho = \log(n/V_{\mathrm{ex}})/\log d1.4139Overhead per extractable T state. Lower is better.
\gamma = \log(n/k)/\log d1.4139Overhead 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 width44Rank of the output rows modulo the check rows.

Output gate

44 T0 CS0 CCZon 44 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}\cdot \mathrm{T}_{14}\cdot \mathrm{T}_{15}\cdot \mathrm{T}_{16}\cdot \mathrm{T}_{17}\cdot \mathrm{T}_{18}\cdot \mathrm{T}_{19}\cdot \mathrm{T}_{20}\cdot \mathrm{T}_{21}\cdot \mathrm{T}_{22}\cdot \mathrm{T}_{23}\cdot \mathrm{T}_{24}\cdot \mathrm{T}_{25}\cdot \mathrm{T}_{26}\cdot \mathrm{T}_{27}\cdot \mathrm{T}_{28}\cdot \mathrm{T}_{29}\cdot \mathrm{T}_{30}\cdot \mathrm{T}_{31}\cdot \mathrm{T}_{32}\cdot \mathrm{T}_{33}\cdot \mathrm{T}_{34}\cdot \mathrm{T}_{35}\cdot \mathrm{T}_{36}\cdot \mathrm{T}_{37}\cdot \mathrm{T}_{38}\cdot \mathrm{T}_{39}\cdot \mathrm{T}_{40}\cdot \mathrm{T}_{41}\cdot \mathrm{T}_{42}\cdot \mathrm{T}_{43}

As monomials: 0+1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37+38+39+40+41+42+43 (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 51 single-wire gates and 522 \mathrm{CZ} gates: \mathrm{Z} on output 0, \mathrm{S}^\dagger on output 1, \mathrm{Z} on output 2, \mathrm{Z} on output 3, \mathrm{Z} on output 4, \mathrm{S}^\dagger on output 5, \mathrm{S} on output 6, \mathrm{S}^\dagger on output 7, \mathrm{Z} on output 8, \mathrm{S} on output 9, \mathrm{Z} on output 10, \mathrm{Z} on output 11, \mathrm{Z} on output 12, \mathrm{S}^\dagger on output 13, \mathrm{Z} on output 14, \mathrm{Z} on output 15, \mathrm{Z} on output 16, \mathrm{Z} on output 17, \mathrm{Z} on output 18, \mathrm{Z} on output 19, \mathrm{Z} on output 20, \mathrm{S} on output 21, \mathrm{Z} on output 22, \mathrm{S} on output 23 and 549 more (listed below). The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.

All 573 correction gates

\mathrm{Z} on output 0, \mathrm{S}^\dagger on output 1, \mathrm{Z} on output 2, \mathrm{Z} on output 3, \mathrm{Z} on output 4, \mathrm{S}^\dagger on output 5, \mathrm{S} on output 6, \mathrm{S}^\dagger on output 7, \mathrm{Z} on output 8, \mathrm{S} on output 9, \mathrm{Z} on output 10, \mathrm{Z} on output 11, \mathrm{Z} on output 12, \mathrm{S}^\dagger on output 13, \mathrm{Z} on output 14, \mathrm{Z} on output 15, \mathrm{Z} on output 16, \mathrm{Z} on output 17, \mathrm{Z} on output 18, \mathrm{Z} on output 19, \mathrm{Z} on output 20, \mathrm{S} on output 21, \mathrm{Z} on output 22, \mathrm{S} on output 23, \mathrm{Z} on output 24, \mathrm{Z} on output 25, \mathrm{S} on output 26, \mathrm{S}^\dagger on output 27, \mathrm{Z} on output 28, \mathrm{Z} on output 29, \mathrm{S}^\dagger on output 30, \mathrm{S} on output 31, \mathrm{S}^\dagger on output 32, \mathrm{Z} on output 33, \mathrm{S} on output 34, \mathrm{S} on output 35, \mathrm{S} on output 36, \mathrm{Z} on output 37, \mathrm{S}^\dagger on output 38, \mathrm{Z} on output 39, \mathrm{Z} on output 40, \mathrm{S} on output 41, \mathrm{S} on output 42, \mathrm{S}^\dagger on output 43, \mathrm{Z} on check 44, \mathrm{S} on check 47, \mathrm{S} on check 48, \mathrm{S} on check 49, \mathrm{S} on check 50, \mathrm{Z} on check 51, \mathrm{Z} on check 53, \mathrm{CZ} on output 0, output 1, \mathrm{CZ} on output 0, output 2, \mathrm{CZ} on output 0, output 5, \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 13, \mathrm{CZ} on output 0, output 14, \mathrm{CZ} on output 0, output 15, \mathrm{CZ} on output 0, output 18, \mathrm{CZ} on output 0, output 19, \mathrm{CZ} on output 0, output 27, \mathrm{CZ} on output 0, output 29, \mathrm{CZ} on output 0, output 30, \mathrm{CZ} on output 0, output 32, \mathrm{CZ} on output 0, output 38, \mathrm{CZ} on output 0, output 40, \mathrm{CZ} on output 0, output 43, \mathrm{CZ} on output 0, check 44, \mathrm{CZ} on output 0, check 47, \mathrm{CZ} on output 0, check 49, \mathrm{CZ} on output 0, check 51, \mathrm{CZ} on output 0, check 53, \mathrm{CZ} on output 1, output 2, \mathrm{CZ} on output 1, output 3, \mathrm{CZ} on output 1, output 4, \mathrm{CZ} on output 1, output 8, \mathrm{CZ} on output 1, output 10, \mathrm{CZ} on output 1, output 11, \mathrm{CZ} on output 1, output 12, \mathrm{CZ} on output 1, output 14, \mathrm{CZ} on output 1, output 15, \mathrm{CZ} on output 1, output 16, \mathrm{CZ} on output 1, output 17, \mathrm{CZ} on output 1, output 18, \mathrm{CZ} on output 1, output 19, \mathrm{CZ} on output 1, output 20, \mathrm{CZ} on output 1, output 22, \mathrm{CZ} on output 1, output 24, \mathrm{CZ} on output 1, output 25, \mathrm{CZ} on output 1, output 28, \mathrm{CZ} on output 1, output 29, \mathrm{CZ} on output 1, output 33, \mathrm{CZ} on output 1, output 37, \mathrm{CZ} on output 1, output 39, \mathrm{CZ} on output 1, output 40, \mathrm{CZ} on output 1, check 47, \mathrm{CZ} on output 1, check 48, \mathrm{CZ} on output 1, check 49, \mathrm{CZ} on output 1, check 50, \mathrm{CZ} on output 2, output 5, \mathrm{CZ} on output 2, output 7, \mathrm{CZ} on output 2, output 10, \mathrm{CZ} on output 2, output 11, \mathrm{CZ} on output 2, output 13, \mathrm{CZ} on output 2, output 14, \mathrm{CZ} on output 2, output 15, \mathrm{CZ} on output 2, output 18, \mathrm{CZ} on output 2, output 19, \mathrm{CZ} on output 2, output 27, \mathrm{CZ} on output 2, output 29, \mathrm{CZ} on output 2, output 30, \mathrm{CZ} on output 2, output 32, \mathrm{CZ} on output 2, output 38, \mathrm{CZ} on output 2, output 40, \mathrm{CZ} on output 2, output 43, \mathrm{CZ} on output 2, check 44, \mathrm{CZ} on output 2, check 47, \mathrm{CZ} on output 2, check 49, \mathrm{CZ} on output 2, check 51, \mathrm{CZ} on output 2, check 53, \mathrm{CZ} on output 3, output 4, \mathrm{CZ} on output 3, output 5, \mathrm{CZ} on output 3, output 7, \mathrm{CZ} on output 3, output 8, \mathrm{CZ} on output 3, output 12, \mathrm{CZ} on output 3, output 13, \mathrm{CZ} on output 3, output 16, \mathrm{CZ} on output 3, output 17, \mathrm{CZ} on output 3, output 20, \mathrm{CZ} on output 3, output 22, \mathrm{CZ} on output 3, output 24, \mathrm{CZ} on output 3, output 25, \mathrm{CZ} on output 3, output 27, \mathrm{CZ} on output 3, output 28, \mathrm{CZ} on output 3, output 30, \mathrm{CZ} on output 3, output 32, \mathrm{CZ} on output 3, output 33, \mathrm{CZ} on output 3, output 37, \mathrm{CZ} on output 3, output 38, \mathrm{CZ} on output 3, output 39, \mathrm{CZ} on output 3, output 43, \mathrm{CZ} on output 3, check 44, \mathrm{CZ} on output 3, check 48, \mathrm{CZ} on output 3, check 50, \mathrm{CZ} on output 3, check 51, \mathrm{CZ} on output 3, check 53, \mathrm{CZ} on output 4, output 5, \mathrm{CZ} on output 4, output 7, \mathrm{CZ} on output 4, output 8, \mathrm{CZ} on output 4, output 12, \mathrm{CZ} on output 4, output 13, \mathrm{CZ} on output 4, output 16, \mathrm{CZ} on output 4, output 17, \mathrm{CZ} on output 4, output 20, \mathrm{CZ} on output 4, output 22, \mathrm{CZ} on output 4, output 24, \mathrm{CZ} on output 4, output 25, \mathrm{CZ} on output 4, output 27, \mathrm{CZ} on output 4, output 28, \mathrm{CZ} on output 4, output 30, \mathrm{CZ} on output 4, output 32, \mathrm{CZ} on output 4, output 33, \mathrm{CZ} on output 4, output 37, \mathrm{CZ} on output 4, output 38, \mathrm{CZ} on output 4, output 39, \mathrm{CZ} on output 4, output 43, \mathrm{CZ} on output 4, check 44, \mathrm{CZ} on output 4, check 48, \mathrm{CZ} on output 4, check 50, \mathrm{CZ} on output 4, check 51, \mathrm{CZ} on output 4, check 53, \mathrm{CZ} on output 5, output 8, \mathrm{CZ} on output 5, output 10, \mathrm{CZ} on output 5, output 11, \mathrm{CZ} on output 5, output 12, \mathrm{CZ} on output 5, output 14, \mathrm{CZ} on output 5, output 15, \mathrm{CZ} on output 5, output 16, \mathrm{CZ} on output 5, output 17, \mathrm{CZ} on output 5, output 18, \mathrm{CZ} on output 5, output 19, \mathrm{CZ} on output 5, output 20, \mathrm{CZ} on output 5, output 22, \mathrm{CZ} on output 5, output 24, \mathrm{CZ} on output 5, output 25, \mathrm{CZ} on output 5, output 28, \mathrm{CZ} on output 5, output 29, \mathrm{CZ} on output 5, output 33, \mathrm{CZ} on output 5, output 37, \mathrm{CZ} on output 5, output 39, \mathrm{CZ} on output 5, output 40, \mathrm{CZ} on output 5, check 47, \mathrm{CZ} on output 5, check 48, \mathrm{CZ} on output 5, check 49, \mathrm{CZ} on output 5, check 50, \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, output 12, \mathrm{CZ} on output 7, output 14, \mathrm{CZ} on output 7, output 15, \mathrm{CZ} on output 7, output 16, \mathrm{CZ} on output 7, output 17, \mathrm{CZ} on output 7, output 18, \mathrm{CZ} on output 7, output 19, \mathrm{CZ} on output 7, output 20, \mathrm{CZ} on output 7, output 22, \mathrm{CZ} on output 7, output 24, \mathrm{CZ} on output 7, output 25, \mathrm{CZ} on output 7, output 28, \mathrm{CZ} on output 7, output 29, \mathrm{CZ} on output 7, output 33, \mathrm{CZ} on output 7, output 37, \mathrm{CZ} on output 7, output 39, \mathrm{CZ} on output 7, output 40, \mathrm{CZ} on output 7, check 47, \mathrm{CZ} on output 7, check 48, \mathrm{CZ} on output 7, check 49, \mathrm{CZ} on output 7, check 50, \mathrm{CZ} on output 8, output 12, \mathrm{CZ} on output 8, output 13, \mathrm{CZ} on output 8, output 16, \mathrm{CZ} on output 8, output 17, \mathrm{CZ} on output 8, output 20, \mathrm{CZ} on output 8, output 22, \mathrm{CZ} on output 8, output 24, \mathrm{CZ} on output 8, output 25, \mathrm{CZ} on output 8, output 27, \mathrm{CZ} on output 8, output 28, \mathrm{CZ} on output 8, output 30, \mathrm{CZ} on output 8, output 32, \mathrm{CZ} on output 8, output 33, \mathrm{CZ} on output 8, output 37, \mathrm{CZ} on output 8, output 38, \mathrm{CZ} on output 8, output 39, \mathrm{CZ} on output 8, output 43, \mathrm{CZ} on output 8, check 44, \mathrm{CZ} on output 8, check 48, \mathrm{CZ} on output 8, check 50, \mathrm{CZ} on output 8, check 51, \mathrm{CZ} on output 8, check 53, \mathrm{CZ} on output 10, output 11, \mathrm{CZ} on output 10, output 13, \mathrm{CZ} on output 10, output 14, \mathrm{CZ} on output 10, output 15, \mathrm{CZ} on output 10, output 18, \mathrm{CZ} on output 10, output 19, \mathrm{CZ} on output 10, output 27, \mathrm{CZ} on output 10, output 29, \mathrm{CZ} on output 10, output 30, \mathrm{CZ} on output 10, output 32, \mathrm{CZ} on output 10, output 38, \mathrm{CZ} on output 10, output 40, \mathrm{CZ} on output 10, output 43, \mathrm{CZ} on output 10, check 44, \mathrm{CZ} on output 10, check 47, \mathrm{CZ} on output 10, check 49, \mathrm{CZ} on output 10, check 51, \mathrm{CZ} on output 10, check 53, \mathrm{CZ} on output 11, output 13, \mathrm{CZ} on output 11, output 14, \mathrm{CZ} on output 11, output 15, \mathrm{CZ} on output 11, output 18, \mathrm{CZ} on output 11, output 19, \mathrm{CZ} on output 11, output 27, \mathrm{CZ} on output 11, output 29, \mathrm{CZ} on output 11, output 30, \mathrm{CZ} on output 11, output 32, \mathrm{CZ} on output 11, output 38, \mathrm{CZ} on output 11, output 40, \mathrm{CZ} on output 11, output 43, \mathrm{CZ} on output 11, check 44, \mathrm{CZ} on output 11, check 47, \mathrm{CZ} on output 11, check 49, \mathrm{CZ} on output 11, check 51, \mathrm{CZ} on output 11, check 53, \mathrm{CZ} on output 12, output 13, \mathrm{CZ} on output 12, output 16, \mathrm{CZ} on output 12, output 17, \mathrm{CZ} on output 12, output 20, \mathrm{CZ} on output 12, output 22, \mathrm{CZ} on output 12, output 24, \mathrm{CZ} on output 12, output 25, \mathrm{CZ} on output 12, output 27, \mathrm{CZ} on output 12, output 28, \mathrm{CZ} on output 12, output 30, \mathrm{CZ} on output 12, output 32, \mathrm{CZ} on output 12, output 33, \mathrm{CZ} on output 12, output 37, \mathrm{CZ} on output 12, output 38, \mathrm{CZ} on output 12, output 39, \mathrm{CZ} on output 12, output 43, \mathrm{CZ} on output 12, check 44, \mathrm{CZ} on output 12, check 48, \mathrm{CZ} on output 12, check 50, \mathrm{CZ} on output 12, check 51, \mathrm{CZ} on output 12, check 53, \mathrm{CZ} on output 13, output 14, \mathrm{CZ} on output 13, output 15, \mathrm{CZ} on output 13, output 16, \mathrm{CZ} on output 13, output 17, \mathrm{CZ} on output 13, output 18, \mathrm{CZ} on output 13, output 19, \mathrm{CZ} on output 13, output 20, \mathrm{CZ} on output 13, output 22, \mathrm{CZ} on output 13, output 24, \mathrm{CZ} on output 13, output 25, \mathrm{CZ} on output 13, output 28, \mathrm{CZ} on output 13, output 29, \mathrm{CZ} on output 13, output 33, \mathrm{CZ} on output 13, output 37, \mathrm{CZ} on output 13, output 39, \mathrm{CZ} on output 13, output 40, \mathrm{CZ} on output 13, check 47, \mathrm{CZ} on output 13, check 48, \mathrm{CZ} on output 13, check 49, \mathrm{CZ} on output 13, check 50, \mathrm{CZ} on output 14, output 15, \mathrm{CZ} on output 14, output 18, \mathrm{CZ} on output 14, output 19, \mathrm{CZ} on output 14, output 27, \mathrm{CZ} on output 14, output 29, \mathrm{CZ} on output 14, output 30, \mathrm{CZ} on output 14, output 32, \mathrm{CZ} on output 14, output 38, \mathrm{CZ} on output 14, output 40, \mathrm{CZ} on output 14, output 43, \mathrm{CZ} on output 14, check 44, \mathrm{CZ} on output 14, check 47, \mathrm{CZ} on output 14, check 49, \mathrm{CZ} on output 14, check 51, \mathrm{CZ} on output 14, check 53, \mathrm{CZ} on output 15, output 18, \mathrm{CZ} on output 15, output 19, \mathrm{CZ} on output 15, output 27, \mathrm{CZ} on output 15, output 29, \mathrm{CZ} on output 15, output 30, \mathrm{CZ} on output 15, output 32, \mathrm{CZ} on output 15, output 38, \mathrm{CZ} on output 15, output 40, \mathrm{CZ} on output 15, output 43, \mathrm{CZ} on output 15, check 44, \mathrm{CZ} on output 15, check 47, \mathrm{CZ} on output 15, check 49, \mathrm{CZ} on output 15, check 51, \mathrm{CZ} on output 15, check 53, \mathrm{CZ} on output 16, output 17, \mathrm{CZ} on output 16, output 20, \mathrm{CZ} on output 16, output 22, \mathrm{CZ} on output 16, output 24, \mathrm{CZ} on output 16, output 25, \mathrm{CZ} on output 16, output 27, \mathrm{CZ} on output 16, output 28, \mathrm{CZ} on output 16, output 30, \mathrm{CZ} on output 16, output 32, \mathrm{CZ} on output 16, output 33, \mathrm{CZ} on output 16, output 37, \mathrm{CZ} on output 16, output 38, \mathrm{CZ} on output 16, output 39, \mathrm{CZ} on output 16, output 43, \mathrm{CZ} on output 16, check 44, \mathrm{CZ} on output 16, check 48, \mathrm{CZ} on output 16, check 50, \mathrm{CZ} on output 16, check 51, \mathrm{CZ} on output 16, check 53, \mathrm{CZ} on output 17, output 20, \mathrm{CZ} on output 17, output 22, \mathrm{CZ} on output 17, output 24, \mathrm{CZ} on output 17, output 25, \mathrm{CZ} on output 17, output 27, \mathrm{CZ} on output 17, output 28, \mathrm{CZ} on output 17, output 30, \mathrm{CZ} on output 17, output 32, \mathrm{CZ} on output 17, output 33, \mathrm{CZ} on output 17, output 37, \mathrm{CZ} on output 17, output 38, \mathrm{CZ} on output 17, output 39, \mathrm{CZ} on output 17, output 43, \mathrm{CZ} on output 17, check 44, \mathrm{CZ} on output 17, check 48, \mathrm{CZ} on output 17, check 50, \mathrm{CZ} on output 17, check 51, \mathrm{CZ} on output 17, check 53, \mathrm{CZ} on output 18, output 19, \mathrm{CZ} on output 18, output 27, \mathrm{CZ} on output 18, output 29, \mathrm{CZ} on output 18, output 30, \mathrm{CZ} on output 18, output 32, \mathrm{CZ} on output 18, output 38, \mathrm{CZ} on output 18, output 40, \mathrm{CZ} on output 18, output 43, \mathrm{CZ} on output 18, check 44, \mathrm{CZ} on output 18, check 47, \mathrm{CZ} on output 18, check 49, \mathrm{CZ} on output 18, check 51, \mathrm{CZ} on output 18, check 53, \mathrm{CZ} on output 19, output 27, \mathrm{CZ} on output 19, output 29, \mathrm{CZ} on output 19, output 30, \mathrm{CZ} on output 19, output 32, \mathrm{CZ} on output 19, output 38, \mathrm{CZ} on output 19, output 40, \mathrm{CZ} on output 19, output 43, \mathrm{CZ} on output 19, check 44, \mathrm{CZ} on output 19, check 47, \mathrm{CZ} on output 19, check 49, \mathrm{CZ} on output 19, check 51, \mathrm{CZ} on output 19, check 53, \mathrm{CZ} on output 20, output 22, \mathrm{CZ} on output 20, output 24, \mathrm{CZ} on output 20, output 25, \mathrm{CZ} on output 20, output 27, \mathrm{CZ} on output 20, output 28, \mathrm{CZ} on output 20, output 30, \mathrm{CZ} on output 20, output 32, \mathrm{CZ} on output 20, output 33, \mathrm{CZ} on output 20, output 37, \mathrm{CZ} on output 20, output 38, \mathrm{CZ} on output 20, output 39, \mathrm{CZ} on output 20, output 43, \mathrm{CZ} on output 20, check 44, \mathrm{CZ} on output 20, check 48, \mathrm{CZ} on output 20, check 50, \mathrm{CZ} on output 20, check 51, \mathrm{CZ} on output 20, check 53, \mathrm{CZ} on output 22, output 24, \mathrm{CZ} on output 22, output 25, \mathrm{CZ} on output 22, output 27, \mathrm{CZ} on output 22, output 28, \mathrm{CZ} on output 22, output 30, \mathrm{CZ} on output 22, output 32, \mathrm{CZ} on output 22, output 33, \mathrm{CZ} on output 22, output 37, \mathrm{CZ} on output 22, output 38, \mathrm{CZ} on output 22, output 39, \mathrm{CZ} on output 22, output 43, \mathrm{CZ} on output 22, check 44, \mathrm{CZ} on output 22, check 48, \mathrm{CZ} on output 22, check 50, \mathrm{CZ} on output 22, check 51, \mathrm{CZ} on output 22, check 53, \mathrm{CZ} on output 24, output 25, \mathrm{CZ} on output 24, output 27, \mathrm{CZ} on output 24, output 28, \mathrm{CZ} on output 24, output 30, \mathrm{CZ} on output 24, output 32, \mathrm{CZ} on output 24, output 33, \mathrm{CZ} on output 24, output 37, \mathrm{CZ} on output 24, output 38, \mathrm{CZ} on output 24, output 39, \mathrm{CZ} on output 24, output 43, \mathrm{CZ} on output 24, check 44, \mathrm{CZ} on output 24, check 48, \mathrm{CZ} on output 24, check 50, \mathrm{CZ} on output 24, check 51, \mathrm{CZ} on output 24, check 53, \mathrm{CZ} on output 25, output 27, \mathrm{CZ} on output 25, output 28, \mathrm{CZ} on output 25, output 30, \mathrm{CZ} on output 25, output 32, \mathrm{CZ} on output 25, output 33, \mathrm{CZ} on output 25, output 37, \mathrm{CZ} on output 25, output 38, \mathrm{CZ} on output 25, output 39, \mathrm{CZ} on output 25, output 43, \mathrm{CZ} on output 25, check 44, \mathrm{CZ} on output 25, check 48, \mathrm{CZ} on output 25, check 50, \mathrm{CZ} on output 25, check 51, \mathrm{CZ} on output 25, check 53, \mathrm{CZ} on output 27, output 28, \mathrm{CZ} on output 27, output 29, \mathrm{CZ} on output 27, output 33, \mathrm{CZ} on output 27, output 37, \mathrm{CZ} on output 27, output 39, \mathrm{CZ} on output 27, output 40, \mathrm{CZ} on output 27, check 47, \mathrm{CZ} on output 27, check 48, \mathrm{CZ} on output 27, check 49, \mathrm{CZ} on output 27, check 50, \mathrm{CZ} on output 28, output 30, \mathrm{CZ} on output 28, output 32, \mathrm{CZ} on output 28, output 33, \mathrm{CZ} on output 28, output 37, \mathrm{CZ} on output 28, output 38, \mathrm{CZ} on output 28, output 39, \mathrm{CZ} on output 28, output 43, \mathrm{CZ} on output 28, check 44, \mathrm{CZ} on output 28, check 48, \mathrm{CZ} on output 28, check 50, \mathrm{CZ} on output 28, check 51, \mathrm{CZ} on output 28, check 53, \mathrm{CZ} on output 29, output 30, \mathrm{CZ} on output 29, output 32, \mathrm{CZ} on output 29, output 38, \mathrm{CZ} on output 29, output 40, \mathrm{CZ} on output 29, output 43, \mathrm{CZ} on output 29, check 44, \mathrm{CZ} on output 29, check 47, \mathrm{CZ} on output 29, check 49, \mathrm{CZ} on output 29, check 51, \mathrm{CZ} on output 29, check 53, \mathrm{CZ} on output 30, output 33, \mathrm{CZ} on output 30, output 37, \mathrm{CZ} on output 30, output 39, \mathrm{CZ} on output 30, output 40, \mathrm{CZ} on output 30, check 47, \mathrm{CZ} on output 30, check 48, \mathrm{CZ} on output 30, check 49, \mathrm{CZ} on output 30, check 50, \mathrm{CZ} on output 32, output 33, \mathrm{CZ} on output 32, output 37, \mathrm{CZ} on output 32, output 39, \mathrm{CZ} on output 32, output 40, \mathrm{CZ} on output 32, check 47, \mathrm{CZ} on output 32, check 48, \mathrm{CZ} on output 32, check 49, \mathrm{CZ} on output 32, check 50, \mathrm{CZ} on output 33, output 37, \mathrm{CZ} on output 33, output 38, \mathrm{CZ} on output 33, output 39, \mathrm{CZ} on output 33, output 43, \mathrm{CZ} on output 33, check 44, \mathrm{CZ} on output 33, check 48, \mathrm{CZ} on output 33, check 50, \mathrm{CZ} on output 33, check 51, \mathrm{CZ} on output 33, check 53, \mathrm{CZ} on output 37, output 38, \mathrm{CZ} on output 37, output 39, \mathrm{CZ} on output 37, output 43, \mathrm{CZ} on output 37, check 44, \mathrm{CZ} on output 37, check 48, \mathrm{CZ} on output 37, check 50, \mathrm{CZ} on output 37, check 51, \mathrm{CZ} on output 37, check 53, \mathrm{CZ} on output 38, output 39, \mathrm{CZ} on output 38, output 40, \mathrm{CZ} on output 38, check 47, \mathrm{CZ} on output 38, check 48, \mathrm{CZ} on output 38, check 49, \mathrm{CZ} on output 38, check 50, \mathrm{CZ} on output 39, output 43, \mathrm{CZ} on output 39, check 44, \mathrm{CZ} on output 39, check 48, \mathrm{CZ} on output 39, check 50, \mathrm{CZ} on output 39, check 51, \mathrm{CZ} on output 39, check 53, \mathrm{CZ} on output 40, output 43, \mathrm{CZ} on output 40, check 44, \mathrm{CZ} on output 40, check 47, \mathrm{CZ} on output 40, check 49, \mathrm{CZ} on output 40, check 51, \mathrm{CZ} on output 40, check 53, \mathrm{CZ} on output 43, check 47, \mathrm{CZ} on output 43, check 48, \mathrm{CZ} on output 43, check 49, \mathrm{CZ} on output 43, check 50, \mathrm{CZ} on check 44, check 47, \mathrm{CZ} on check 44, check 48, \mathrm{CZ} on check 44, check 49, \mathrm{CZ} on check 44, check 50, \mathrm{CZ} on check 47, check 49, \mathrm{CZ} on check 47, check 51, \mathrm{CZ} on check 47, check 53, \mathrm{CZ} on check 48, check 50, \mathrm{CZ} on check 48, check 51, \mathrm{CZ} on check 48, check 53, \mathrm{CZ} on check 49, check 51, \mathrm{CZ} on check 49, check 53, \mathrm{CZ} on check 50, check 51, \mathrm{CZ} on check 50, check 53

Or no Clifford gate at all: run \mathrm{T}^3 on rotations 5, 9, 10, 12, 13, 14, 26, 29, 32, 37, 40, 42, 55, 56, 60, 61, 62, 64, 65, 67, 68, 70, 71, 72 and 69 more (listed below); \mathrm{T}^5 on rotations 11, 15, 20, 22, 27, 30, 57, 63, 100, 108, 129; \mathrm{T}^\dagger on rotations 0, 2, 4, 6, 16, 18, 25, 28, 35, 39, 50, 53, 104, 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 93 rotations at T³

5, 9, 10, 12, 13, 14, 26, 29, 32, 37, 40, 42, 55, 56, 60, 61, 62, 64, 65, 67, 68, 70, 71, 72, 77, 79, 80, 87, 88, 89, 91, 92, 96, 98, 101, 103, 105, 107, 111, 112, 115, 116, 117, 118, 120, 122, 125, 126, 127, 128, 131, 132, 134, 135, 136, 137, 138, 140, 141, 146, 147, 148, 151, 152, 153, 154, 155, 158, 160, 162, 168, 171, 174, 177, 178, 181, 183, 184, 185, 188, 191, 192, 193, 194, 195, 196, 198, 199, 200, 201, 202, 203, 205

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/208.44.3.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-208.44.3.a.txt", "w") as out:
    out.write("\n".join(lines) + "\n")

Distance

d=3, exact. No fault of weight below 3 is both undetectable and damaging. One of weight 3 is.

Exactly 1,398 faults of weight 3 are undetectable and damaging, so A_{3}=1{,}398. If each rotation fails independently with probability p, an accepted run has a damaged output with probability A_{3}\,p^{3}+O(p^{4}).

A damaging fault of weight 3 sits on columns [0,1,96].

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

built from one of Wills's parent codes (weighted Reed-Muller, restricted-light or one-heavy) by searched punctures, paired-column contractions or logical-pair checks, in the pure-T distillation-exponent release (gamma = log(n/k)/log d, n < 1000); a verified witness, not a maximum -- only the release's own frontier points are the lowest gamma it knows

Found by: AI search (gamma frontier): from Wills parent codes

Discovery: AI search

1 recorded source
[
 {
  "regime": "AI search (gamma frontier): from Wills parent codes",
  "file": "gamma_frontier_release/protocols/paired_column_contraction_208_44_3_3700258a70b8.json",
  "label": "paired_column_contraction_208_44_3_3700258a70b8",
  "N": 54,
  "d": 3,
  "d_is_exact": true,
  "origin": "Wills weighted Reed\u2013Muller parent (weights 1^6,2^2, cutoff 3), with searched graph punctures; 2 paired-column contractions",
  "provenance": "gamma frontier release record paired_column_contraction_208_44_3_3700258a70b8 (source catalogue id paired_column_contraction_208_44_3_3700258a70b8); construction: Wills weighted Reed\u2013Muller parent (weights 1^6,2^2, cutoff 3), with searched graph punctures; 2 paired-column contractions; matrix 3700258a70b884f8...; the release states d=3 (exact), gamma=1.4139186879715695 (exact)",
  "notes": "matrix gamma_frontier_release/matrices/3700258a70b884f8cf15432bd9b092663dc7daa72cb451670610b037e40aee1b.matrix"
 }
]

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/208.44.3.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[848].