The circuit
The 54\times211 matrix. Rows are wires and columns are rotations. A 1 means the wire takes part in the rotation, and the number beside each wire is its row weight. Scroll sideways to see every column.
| wire | 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 |
|---|---|
| out 0 (55) | ·1······1···1·11·····1··1······1···1········1·1111········1····11·1·······1··1······1··1·······1·1·111··1········1····1·1·1···1···1········1··1·······11·111··1·······1····1·1·1·····1···1········1··1······1·1·111 |
| out 1 (55) | 1·······1··1···1·····1····1····1···1···1·1····1·111······11··1····1·······1···1·····1··1··1·1····1··111·········11·1······1···1·····1······1··1···11···1··111········11·1······1·····1·····1······1··1··1·1···1··11 |
| out 2 (123) | 11·1··111·11·11··11···1111··111·1·11·11··111·11····1111111·1··11·111··111111··1111·111··1·11···1··11111·1··1··11·1111··11··1···1·111···111··111·11·111·1·····1·1·11··1·1··11··11111111·1···1111··11·1·1·11··1··1111 |
| out 3 (123) | 11·1··111·11·11··11···1111··111·1·1·11111111····11·1111111·1··11·111··111111··1111·111·1·······11111··1·1··1··11·1111··11··1···1·111···111··11·1111111····11·1·1·11··1·1··11··11111111·1···1111··11·11······11111·· |
| out 4 (31) | ······1····11·1·········1·1·················1····1···········1·11············11················1·····1········1····1··1·1·········1·1·················1····1·······1····1··1·1···········1·1················1·····1 |
| out 5 (63) | 1·······11·11·11·····1··1······1···1···1·1·····1111·······11·1·11·1·······1··1······1··1··1·1······1111··········1·1··1·1·1···1···1········1··1···11·····1111·········111··1·1·1·····1···1········1··1··1·1·····111 |
| out 6 (123) | ·1·1··111··111·1·11··111·1··11111·111111·1·1···1··11111111·11·1·11·1··1111·1·11111·1·1·····1·1·1111·11··1··1··11·1·11·11··11··11·1·1···111·111111111·1···1··11·1·11··1·1··1··11·11111··1·1·1111··1··1····1·11111·11 |
| out 7 (123) | ·1·1··11··11111··11··11·····11111·111111·1·11·11··1111111111··1··111··1111·1111111·1·1·····1·1··1·1·11··1··1··11··111·111··1··11·······111·111111111·111·1··11·1·11··111··1···1111111··11111111··1··1····1·1·1·1·11 |
| out 8 (55) | 1··1····1··11·1·········1·1········1···1·11····11·1·······1··1·11············11········1··1·11·····11·1····1·····1·1··1·1·········1·1·········1···111····11·1···1·····1·1··1·1···········1·1·········1··1·11····11· |
| out 9 (123) | ·1·1··11··11111··11··111·1··111·1111111····1···1··1111111111··1··111··1111·1·11111·11·····1111·1111·11··1··1··11··111·111··1··11·1·1···111··111111···1···1··11·1·11··111··1···1111111··1·1·1111··11·····11111111·11 |
| out 10 (123) | ·1·1··111·11·111·11··111111·11111·1··111··111··1·1111111·1·1··11·1·1··1111·1···111·1·1·11··11···111·1···1··1·111·1111··11·11··11·1111··111·111··111·111··1·111·1·1···1·1··11··1·11111··1····111··1··111··11··111·1· |
| out 11 (55) | ········1··1···1·······11··········11····11·1·1·11········1··1···11·········11·········11···11·1·1··11···········1·1·····11······11···········11···11·11··11··········1·1·····11········11···········11···111·1··11 |
| out 12 (123) | 11·1··11··1·11···111·11··1··111·1·1··111···1··1111·111111111·11·1111··1·11·1111111·111·11··111·11·1···1·1··1··11··1·1·11···1··11···1···111··11··111··1·1·111·1·1·11··1111·1··11111·11··111·1111··11·111··11111·1··· |
| out 13 (31) | ···································1·1·1·1··1·11·1·····································1·11·1··1·1·1·1········································1·1·11··11·1·1·········································1··1·1·1·1·1·1 |
| out 14 (123) | ·1·1··11··11111··11··11·111·11111·1··111··111·11··1111111111··1··111··1111·11···11·1·1·11··11···1·1·11··1··1··11··111·111··1··11··1111·111·111··111·1111·1··11·1·11··111··1···1111111··11····11··1··111··11··1·1·11 |
| out 15 (123) | 11·1··111·1·11·1·11··111·1··11111·1··111·1·1···111··111111·1·11·11·1··1111·1·11111·1·1·11··1·1·1111···111··1··11·11·1·11··11··11·1·1···111·111··1111·1···111···1·11··1·11·1··11·11111··1·1·1111··1··111··1·11111··· |
| out 16 (55) | 1·······1···1········1·········1···1···1·1··1·11111·······1····1··········11········1··1··1·1··1·1·1111··········1····1·······1·1··········1··1···11··11·1111·········1····1·········1·1··········1··1··1·1·1·1·111 |
| out 17 (47) | 1·······1··1·······················11··1·1··1·1·111·1·····1··1·························11·1·1··1·1··111··1·······1·1··························11··11··11··111··1······1·1····························11·1·1·1·1··11 |
| out 18 (47) | ·······················11·11·······11····11·1·11·1··························111··1·····11···11·1·1·1·1···························11·1·········11···11·11·1·1····························11·1··1······11···111·1·1·1 |
| out 19 (31) | ························1·1···11····1·····1·1····1···························11·····1···1····1·1·····1····························1·1····1·1···1····1·1····1·····························1·1····1·1···1····11·····1 |
| out 20 (123) | ·1·1··11··11111··11··111111··11·1·11·11···11··1···1111111111··1··111··1111·1···111·111··1·111··11·1111··1··1··11··111·111··1··11·1111···11··111·11··11·1····11·1·11··111··1···1111111··1····1111·11·1·1·111·11·1111 |
| out 21 (47) | ········1··11111·······11···················1·1·11········1··1·11·1·········11·················1·1··11···········1·1··111·1······11···················11··11··········1·1··111·1········11··················1·1··11 |
| out 22 (125) | 11·1·1111·1·11·1··1···11111·11111·1··111·1·1··1·11·1111·11·1·11·11·11·111111···111·1·1·11··1·1·11·11··1·1··11·11·11·1·11··11···1·1111··111·111··1111·1·1··11·1·1·11··1·11·1··11·111111·1····111··1··111··1·111·11·· |
| out 23 (31) | 1···1·······1·1·········1·1········1·····1········1···1········11············11········1····1·········1···············1·1·········1·1·········1····1········1····1·········1·1···········1·1·········1····1········ |
| out 24 (123) | 11·1··11··1··1··111···1··1··111·1·11·11··1111·1····111111111·111111···111111111111·111··1·11····1·11111·1··1··11··1·1··1···1···1···1···111··111·11·11111·····1·1·11··1111·11·111·11111·111·1111··11·1·1·11···1·1111 |
| out 25 (47) | ·······11···1·1·········1·1········11····11···11··········1····11············11········11···11···1·1···········1·1····1·1·········1·1·········11···11··1·1··········1·1····1·1···········1·1·········11···11··1·1·· |
| out 26 (123) | 11·1··111·11·1·1·11·111111··111·1·1·1111·1·1···11··1111111·1··1111·1··11·1·1··1111·111·1···1·1·1111··11·1··1··11·1111··1··11··11·111···111··11·11111·1···11··1·1·11··1·1··11·11·111·1··1···1111··11·11···1·11111··1 |
| out 27 (123) | 11·1··11··1·111··11···1··1··111·1·1·111····1···1·1·111111111·1···111··111111111111·111·1··1111·1111·1·1·1··1··11··1·11111··1···1···1···111··11·111···1···1·1·1·1·11··1111·····11111111·111·1111··11·11··11111111·1· |
| out 28 (123) | 11·1··111·111111·11··11111··111·1·1··111·1111··111·1111111·1··1··1·1···111·1··1111·111·11··1····111···1·1··1··11·1111·111·111·11·111···111··11··1111111··111·1·1·11··1·1··1···1·1·111··1···1111··11·111··1···111··· |
| out 29 (123) | 11·1··111·11·11··11···1·111·1·111·1··111···11··1···1111111·1··11·111··1111111··11111·1·11··111··111·111·1··1··11·1111··11··1···1··111··1·1·111··111··11··1···1·1·11··1·1··11··11111111·11···111··1··111··111·111·11 |
| out 30 (123) | 11·1··11··1··11··11··11··1··111·1·11·111·1·11·111··111111111·111·111··111··1111111·111··1··1·1··1·1··11·1··1··11··1·1··11··1·111···1···111··111·1111·111·11··1·1·11··1111·11··111111···111·1111··11·1·1··1·1·1·1··1 |
| out 31 (53) | ··1·····1···1·1······1····1····1·········1··1·1111···1····1····11·········1···1·····1·······1··1·1·111····1······1··1·1·1·····1·····1······1·······1··11·111··········1··1·1·1·······1·····1······1·······1·1·1·111 |
| out 32 (123) | 11·1··111·11·11··11···1··1··11111··1·111·1·11·1····1111111·1··11·111··111111111111·1·11·1··1·1··1·11111·1··1··11·1111··11··1···1···1···111·11·1·1111·111·····1·1·11··1·1··11··11111111·111·1111··1··1·1··1·1·1·1111 |
| out 33 (123) | 11·1··111·11·11··11···1111··111·1·1··1·1·1·11·111··1111111·1··11·111··111111··1111·111·11··1·1··1·1··11·1··1··11·1111··11··1···1·111···111··11··1·11·111·11··1·1·11··1·1··11··11111111·1···1111··11·1111·1·1·1·1··1 |
| out 34 (123) | 11·1··111·11·11··11···1111··111·1·1·111····1···1···1111111·1··11·111··111111··1111·111·1··1111·111··111·1··1··11·1111··11··1···1·111···111··11·111···1··11···1·1·11··1·1··11··11111111·1···1111··11·11··1111111··11 |
| out 35 (123) | ·1·1··11··11111··11··11·11··111···1··111·1·11·11111111111111··1··111··1111·11·1111·111·11··1·1··1·1·····1··1··11··111·111··1··11··11···111···1··1111·111·11111·1·11··111··1···1111111··11··1111··111111··1·1·1·1··· |
| out 36 (123) | 11·1··11··1··1·1·11··1···1··111·1·11·11····11·1·11·111111111·11111·1··1111·1111111·111··1·1111··1·11··1·1··1··11··1·1··1··11··1····1···111··111·11···111··11·1·1·11··1111·11·11·11111·1111·1111··11·1·1·1111·1·11·· |
| out 37 (31) | ···································11····1111···11·····································11···1111····11········································11···1111···11·········································11···111····11 |
| out 38 (123) | 11·1··11··1·1111·1···11··1··111·1·1··11··1·11··111·111111111·11··1·1·11111·1111111·111·11·11·1··111···1·1··1··11··1·1·111·1···11···1···111··11··11·1·11··111·1·1·11··1111·1···1·11111··111·1111··11·111·11·1·111··· |
| out 39 (123) | ·1·1··11··11111··11··11··11·111·1·11·111···1··111·1111111111··1··111··1111·111·111··11··1··111·11·1··1··1··1··11··111·111··1··11···11··1111·111·111··1·1·11·11·1·11··111··1···1111111··111··111···1·1·1··11111·1··1 |
| out 40 (47) | 1·······1···1········1··1·1····1···1···1·1······1·1·······1····1··········1··11·1···1··1··1·1·······1·1··········1····1·······1···1·1·1····1··1···11······1·1·········1····1·········1···1·1·1····1··1··1·1······1· |
| check 41 (88) | 1··1·11······1····1··11111···1··1·1···11·1··1··11·111····1·1·11··1·1···1·11··1·1·1·1····1·1111···1··111····11·1········1···1··11·111····1···11···111··1··11·11·1·····1·11·1···1·1·1·11···1··1·1··1····1·1111··1··11 |
| check 42 (88) | ·1···1·11·111111·····11··11···1···11111··11····11···1·1··11·····1··1··11·····11·11·1·1·11·······11·1·1··1···1··1·111··111·1···11···11····1···11111·11····11····1·1···11······1··111······1·1·11··1··111······11·1·1 |
| check 43 (92) | ·11··1111···11···11··1··111·111······111···11·1·111111·1111···1·······11··111·1····1···11·1·····1···11··1·1·1·11·1··1·11···1··1···111··111······111··111··1111·1··1··11···1······11··1·11··1·····1···11·1····1···11 |
| check 44 (94) | 1····1111·1·1·1··11··1111···11111·1····1·1·1···1111····111·1·1·11·····1111··11·····1·1····1·1··1111···1·····1·11·11···1·1··1··11·11····111·111····11·1···1111·····1··1·11··1·1···1111···11·······1··1···1·1·1111··· |
| check 45 (76) | ··1··1·······1···11··1111··········11111··111···11···1·1······111111··1111··11·············11···1111······1·1·······1··1···1··11·11···········11111·111···11··············11·11111111···11···············11··1111·· |
| check 46 (80) | 1·······1···1········1··111·111·1·11111···11···11·1·······1····1··········111··111·111····111··1111··11··········1····1·······1···111··111··111111··11···11·1·········1····1·········1·11···111··11·1···111·1111··1 |
| check 47 (124) | 1111111111··11··1··11111·1111·1·11····111·1·1111···11···11··111·111·111··1·1·1·11·111·1111··1·1···1·11·1·1···1··1·11·1··111111··1·1··11·1·11·1111··1·1··1·111·1·11·11·11·1·11···1·11·11·1·11··111··11··111·1111·1·· |
| check 48 (124) | ·1·1·111111·11·111·1··1111111·1·1···111·11·11···1·11111·11···11·11···11·1111···11·111111··1··1··1111·1·111·····11111111··111·1····1·11·1111··1··1··111·1·1111···1111···1····11··1111·1111·1··11···11111111··11·1··· |
| check 49 (124) | ·1·11111·····11·1111··11·111111111····11···1·1····1111··11111111·11···1·1111·1·11··1··1111·111·1·11111···1·1··1·1·111·1·1·111·1··11·1·111·1111··1·1111·1··11111··11·1·11··1·1·1·1·111·11··1·1·1·1···1111·1··11·11·· |
| check 50 (124) | 1111111111···1·1111·1·1··1·1111111·1·11·1··11·1·1··11···11··111111····11·11111111··1··1·1·1·11··1·11·11·1·····11··111··111·1·11····1111111···11·1111111111·1·1·1111··111··11···11111··1111····1··1111·1··1···1···1· |
| check 51 (88) | 1········111··1··11111····1··1···11·1····111·1··1·1········111··1···11·11·1···1···1··11·1···111·1···1·1······11111······1···11·1····1··1·1111··1···11·1111·11·····11111······1···1··1·1····1···11111··1···111·111·1 |
| check 52 (80) | ········1··11···11111··11············1··1111·11111········1··1·1···111·11···11···········1·1111·11·111···········1·111·1111·1111·11···········11·11···1·1·············1·111·1111·1··111·11···········1111···1··1··· |
| check 53 (124) | ·1·1··11··11111··11··11··1·11·1·11····1·1···11····1111111111··1··111··1111·111111·111·11111·111··11111··1··1··11··111·111··1··11111·111·1·11·1111·1111·1111111·1·11··111··1···1111111·····1···111··11··1·1··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
Output gate
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}
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 (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 49 single-wire gates and 91 \mathrm{CZ} gates: \mathrm{S} on output 0, \mathrm{S} on output 1, \mathrm{S}^\dagger on output 2, \mathrm{S}^\dagger on output 3, \mathrm{S} on output 4, \mathrm{S} on output 5, \mathrm{S}^\dagger on output 6, \mathrm{S}^\dagger on output 7, \mathrm{S} on output 8, \mathrm{S}^\dagger on output 9, \mathrm{S}^\dagger on output 10, \mathrm{S} on output 11, \mathrm{S}^\dagger on output 12, \mathrm{S} on output 13, \mathrm{S}^\dagger on output 14, \mathrm{S}^\dagger on output 15, \mathrm{S} on output 16, \mathrm{S} on output 17, \mathrm{S} on output 18, \mathrm{S} on output 19, \mathrm{S}^\dagger on output 20, \mathrm{S} on output 21, \mathrm{Z} on output 22, \mathrm{S} on output 23 and 116 more (listed below). The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.
All 140 correction gates
\mathrm{S} on output 0, \mathrm{S} on output 1, \mathrm{S}^\dagger on output 2, \mathrm{S}^\dagger on output 3, \mathrm{S} on output 4, \mathrm{S} on output 5, \mathrm{S}^\dagger on output 6, \mathrm{S}^\dagger on output 7, \mathrm{S} on output 8, \mathrm{S}^\dagger on output 9, \mathrm{S}^\dagger on output 10, \mathrm{S} on output 11, \mathrm{S}^\dagger on output 12, \mathrm{S} on output 13, \mathrm{S}^\dagger on output 14, \mathrm{S}^\dagger on output 15, \mathrm{S} on output 16, \mathrm{S} on output 17, \mathrm{S} on output 18, \mathrm{S} on output 19, \mathrm{S}^\dagger on output 20, \mathrm{S} on output 21, \mathrm{Z} on output 22, \mathrm{S} on output 23, \mathrm{S}^\dagger on output 24, \mathrm{S} on output 25, \mathrm{S}^\dagger on output 26, \mathrm{S}^\dagger on output 27, \mathrm{S}^\dagger on output 28, \mathrm{S}^\dagger on output 29, \mathrm{S}^\dagger on output 30, \mathrm{Z} on output 31, \mathrm{S}^\dagger on output 32, \mathrm{S}^\dagger on output 33, \mathrm{S}^\dagger on output 34, \mathrm{S}^\dagger on output 35, \mathrm{S}^\dagger on output 36, \mathrm{S} on output 37, \mathrm{S}^\dagger on output 38, \mathrm{S}^\dagger on output 39, \mathrm{S} on output 40, \mathrm{Z} on check 43, \mathrm{S} on check 44, \mathrm{Z} on check 45, \mathrm{Z} on check 47, \mathrm{Z} on check 48, \mathrm{Z} on check 49, \mathrm{Z} on check 50, \mathrm{Z} on check 53, \mathrm{CZ} on output 2, output 22, \mathrm{CZ} on output 2, output 31, \mathrm{CZ} on output 2, check 44, \mathrm{CZ} on output 3, output 22, \mathrm{CZ} on output 3, output 31, \mathrm{CZ} on output 3, check 44, \mathrm{CZ} on output 6, output 22, \mathrm{CZ} on output 6, output 31, \mathrm{CZ} on output 6, check 44, \mathrm{CZ} on output 7, output 22, \mathrm{CZ} on output 7, output 31, \mathrm{CZ} on output 7, check 44, \mathrm{CZ} on output 9, output 22, \mathrm{CZ} on output 9, output 31, \mathrm{CZ} on output 9, check 44, \mathrm{CZ} on output 10, output 22, \mathrm{CZ} on output 10, output 31, \mathrm{CZ} on output 10, check 44, \mathrm{CZ} on output 12, output 22, \mathrm{CZ} on output 12, output 31, \mathrm{CZ} on output 12, check 44, \mathrm{CZ} on output 14, output 22, \mathrm{CZ} on output 14, output 31, \mathrm{CZ} on output 14, check 44, \mathrm{CZ} on output 15, output 22, \mathrm{CZ} on output 15, output 31, \mathrm{CZ} on output 15, check 44, \mathrm{CZ} on output 20, output 22, \mathrm{CZ} on output 20, output 31, \mathrm{CZ} on output 20, check 44, \mathrm{CZ} on output 22, output 24, \mathrm{CZ} on output 22, output 26, \mathrm{CZ} on output 22, output 27, \mathrm{CZ} on output 22, output 28, \mathrm{CZ} on output 22, output 29, \mathrm{CZ} on output 22, output 30, \mathrm{CZ} on output 22, output 31, \mathrm{CZ} on output 22, output 32, \mathrm{CZ} on output 22, output 33, \mathrm{CZ} on output 22, output 34, \mathrm{CZ} on output 22, output 35, \mathrm{CZ} on output 22, output 36, \mathrm{CZ} on output 22, output 38, \mathrm{CZ} on output 22, output 39, \mathrm{CZ} on output 22, check 43, \mathrm{CZ} on output 22, check 45, \mathrm{CZ} on output 22, check 47, \mathrm{CZ} on output 22, check 48, \mathrm{CZ} on output 22, check 49, \mathrm{CZ} on output 22, check 50, \mathrm{CZ} on output 22, check 53, \mathrm{CZ} on output 24, output 31, \mathrm{CZ} on output 24, check 44, \mathrm{CZ} on output 26, output 31, \mathrm{CZ} on output 26, check 44, \mathrm{CZ} on output 27, output 31, \mathrm{CZ} on output 27, check 44, \mathrm{CZ} on output 28, output 31, \mathrm{CZ} on output 28, check 44, \mathrm{CZ} on output 29, output 31, \mathrm{CZ} on output 29, check 44, \mathrm{CZ} on output 30, output 31, \mathrm{CZ} on output 30, check 44, \mathrm{CZ} on output 31, output 32, \mathrm{CZ} on output 31, output 33, \mathrm{CZ} on output 31, output 34, \mathrm{CZ} on output 31, output 35, \mathrm{CZ} on output 31, output 36, \mathrm{CZ} on output 31, output 38, \mathrm{CZ} on output 31, output 39, \mathrm{CZ} on output 31, check 43, \mathrm{CZ} on output 31, check 45, \mathrm{CZ} on output 31, check 47, \mathrm{CZ} on output 31, check 48, \mathrm{CZ} on output 31, check 49, \mathrm{CZ} on output 31, check 50, \mathrm{CZ} on output 31, check 53, \mathrm{CZ} on output 32, check 44, \mathrm{CZ} on output 33, check 44, \mathrm{CZ} on output 34, check 44, \mathrm{CZ} on output 35, check 44, \mathrm{CZ} on output 36, check 44, \mathrm{CZ} on output 38, check 44, \mathrm{CZ} on output 39, check 44, \mathrm{CZ} on check 43, check 44, \mathrm{CZ} on check 44, check 45, \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 44, check 53
Or no Clifford gate at all: run \mathrm{T}^3 on rotations 2, 3, 7, 13, 21, 24, 29, 30, 34, 37, 39, 52, 61, 63, 64, 65, 66, 67, 70, 72, 76, 79, 80, 82 and 58 more (listed below); \mathrm{T}^5 on rotations 1, 4, 23, 28, 31, 43, 50, 104; \mathrm{T}^\dagger on rotations 5, 6, 9, 10, 11, 15, 17, 19, 20, 22, 25, 26, 27, 33, 35, 36, 38, 75, 102, 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 82 rotations at T³
2, 3, 7, 13, 21, 24, 29, 30, 34, 37, 39, 52, 61, 63, 64, 65, 66, 67, 70, 72, 76, 79, 80, 82, 86, 87, 93, 94, 97, 98, 99, 100, 105, 106, 107, 111, 112, 119, 121, 123, 124, 125, 128, 134, 136, 139, 140, 141, 142, 143, 151, 153, 156, 157, 158, 159, 160, 161, 162, 164, 170, 171, 172, 173, 174, 177, 179, 181, 183, 184, 185, 187, 189, 191, 193, 194, 197, 200, 201, 203, 205, 209
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/211.41.3.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-211.41.3.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 204 faults of weight 3 are undetectable and damaging, so A_{3}=204. 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 [0,4,140].
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
a paired-column contraction of an existing triorthogonal code, 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): contraction of an existing code
Discovery: AI search
1 recorded source
[
{
"regime": "AI search (gamma frontier): contraction of an existing code",
"file": "gamma_frontier_release/protocols/paired_column_contraction_211_41_3_f30d08466e84.json",
"label": "paired_column_contraction_211_41_3_f30d08466e84",
"N": 54,
"d": 3,
"d_is_exact": true,
"origin": "Existing triorthogonal code with physical paired-column contraction",
"provenance": "gamma frontier release record paired_column_contraction_211_41_3_f30d08466e84 (source catalogue id paired_column_contraction_211_41_3_f30d08466e84); construction: Existing triorthogonal code with physical paired-column contraction; matrix f30d08466e8406d4...; the release states d=3 (exact), gamma=1.4912322424118487 (exact)",
"notes": "matrix gamma_frontier_release/matrices/f30d08466e8406d49178fc9f14e44ea331ac83397ed4c4d9af3bca6f077a7864.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/211.41.3.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[854].