The circuit
The 62\times255 matrix. Rows are wires and columns are rotations. A 1 means the wire takes part in the rotation, and the number beside each wire is its row weight. Scroll sideways to see every column.
| wire | 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 |
|---|---|
| out 0 (131) | 111111111111111111111111111································111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111····111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111··1···1·111·111··111··11··1·1··11·11··1111·1···1··1·111·11····111111·········1·11·1··1 |
| out 1 (73) | 11111111111111·············1111111111111····································································································································1··11··1·11·11·111·111··1·1·1··1·11·11··11·1·1··1·11·1·····11111111··111··11··1··11 |
| out 2 (135) | 111111········111111·······1111111······1111111············111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111····111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111···111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1······11111111··111··11··1··11 |
| out 3 (7) | ······11············11············11···········1··············································································································································································································· |
| out 4 (135) | 11111·11111···111·····111··1111···11····111·····11·········111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111····111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111···1··11··1·11·11·111·111··1·1·1··1·11·11··11·1·1··1·11·1·····11111111··111··11··1··11 |
| out 5 (137) | 1111··1·1111··1··11·1······111·11·1·1···11111··1111········111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111····111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111··11·1··1111·1···1111···11·1···11·1111···11··1··1···11·11·111·1·11···1111·1·111··1··11 |
| out 6 (131) | 11····111·····1111·1111··········1··111·11·1·11············111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111····111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111··111·1·1····1111·1··11··111·1·1·1·11·11·1··11·111···11···111·1·11···1111·1·111··1··11 |
| out 7 (63) | ····1111·1··1·111111111111·11·1111··111···1111··1····························································································································111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111····························· |
| out 8 (133) | 111·1·111·1···1···1·111···1···111···11··111·1···1··········111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111····111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111···111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1······11111111··111··11··1··11 |
| out 9 (133) | 1····1··11·111····1···11··1111·1·1······1111111·1·1········111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111····111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111··11·1··1111·1···1111···11·1···11·1111···11··1··1···11·11·111·1·11···1111·1·111··1··11 |
| out 10 (75) | 1·1·1·1··1·11·····111·11·1·11··1··1··1··111··1·1·11111111·····················································································································1·1··11·1111·11·····11·11·1··1···1···1·1111··11·1·111111·1·11··1·11·1··11·11·1··1 |
| out 11 (103) | ·························································111111111111111111111111111·······························111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1···11·11··1·1·1·1···111·1111··1111···1··1·1·1·111·1·1····11·1·1·11··1·1111·1·1·····1111 |
| out 12 (71) | ·························································11···11···11··11·1111·1·1···11···11···111·1··1·1··1·111···························································1111111·11·······1·11····11····1111··1·11···1·1111··1111·1·1·11··1·····1111·····1111 |
| out 13 (103) | ·························································11···1·111··11··1···1·1·1···11···1·111···1·11·11··1·111···111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1···11·11··1·1·1·1···111·1111··1111···1··1·1·1·111·1·1····11·1·1·11··1·1111·1·1·····1111 |
| out 14 (7) | ·······························································11············11············11···········1······················································································································································ |
| out 15 (101) | ·························································11····1···1·11··11111··1····11·11·1·11···1···1·1111·111···111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1···1·1·11····1·11·1111···11·1·11··1····11111··1··1···11·111·1·1·11··1·····1111·····1111 |
| out 16 (101) | ·························································1111··1·1111··1··11·1······111·11·1·1···11111··1111·······111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1···11··11·11···1·1·1·11·11···1·1·1·1·1·11·1111····11··1·1··1·1111·1·1·11111·1·········· |
| out 17 (89) | ·························································11····111·····1111·1111··········1··111·11·1·11···········111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1···1·111···1111··11··1···1·111·11·11····111··1·111·111·····1·1111·1·1·················· |
| out 18 (69) | ·····························································1111·1··1·111111111111·11·1111··111···1111··1··································································111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1·············11111·1·········· |
| out 19 (101) | ·························································11·1··1··11·11·1·1·1·1··1·11·······1·····1··11·11·1·111···111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1···11·11··1·1·1·1···111·1111··1111···1··1·1·1·111·1·1····11·1·1·11··1·1111·1·1·····1111 |
| out 20 (97) | ·························································1····1··11·111····1···11··1111·1·1······1111111·1·1·······111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1···11··11·11···1·1·1·11·11···1·1·1·1·1·11·1111····11··1·1··1·1111·1·1·11111·1·········· |
| out 21 (61) | ·························································1··1··111··1·111·1·····111··1·1·1·111····1·······1·1···11·························································111·1·1····1111·1··11··111·1·1·1·11·11·1··11·111···11···1·1111·1·1·················· |
| out 22 (73) | ··················································································································1···1·1·1····11··11·1·11··1··111··1·1·1·11··1111·111·1···1··11111·11··111····11·1···1··1··1···11111111····11·11··11·1·1·11·11·1···1111·1·11·· |
| out 23 (69) | ··················································································································1·11··11·1··1·1·111·111··1··1·11·1··1··1·11·1·1··1·11·1······1·1111111·1····1··11·1·11··1··11···1·1··11··1·11111·1·········11·1···11·11111·1· |
| out 24 (73) | ··················································································································1·11····1·11·1·1···1111··1··1·11·111·11·1··1·1···1·11·1··1··11111·11··111····11·1···1··1··1···11111111····11·11··11·1·1·11·11·1···1111·1·11·· |
| out 25 (7) | ························································································································11············11············11···········1····························································································· |
| out 26 (73) | ··················································································································1·11·111·1·1·1·1·11·11·111··1·1·1····11·1···1·1·1··11·1···11·1·1··1·111·1·1···11···1·1·1·1··1··11··111111···111·1·1·1·1·11·11·1···1111·1·11·· |
| out 27 (73) | ··················································································································1111··1·1111··1··11·1······111·11·1·1···11111··1111·······11·1·1··1·111·1·1···11···1·1·1·1··1··11··111111···111·1·1·1·1·11·11·1···1111·1·11·· |
| out 28 (67) | ··················································································································11····111·····1111·1111··········1··111·11·1·11··········1··11111·11··111····11·1···1··1··1···11111111····11·11··11·1·1·11·11·1···1111·1·11·· |
| out 29 (63) | ······················································································································1111·1··1·111111111111·11·1111··111···1111··1········1111·1·1··111·1··1··1·11··111···11·1·1··11···111·111···11··························· |
| out 30 (71) | ··················································································································1·1··111···1·1··11·······111·111··111·1·1·····1·11·11·1··1··11111·11··111····11·1···1··1··1···11111111····11·11··11·1·1·11·11·1···1111·1·11·· |
| out 31 (69) | ··················································································································1····1··11·111····1···11··1111·1·1······1111111·1·1·······11·1·1··1·111·1·1···11···1·1·1·1··1··11··111111···111·1·1·1·1·11·11·1···1111·1·11·· |
| out 32 (65) | ··················································································································1··1··111··1·111·1·····111··1·1·1·111····1·······1·1···111111111·11·······1·11····11····1111··1·11···1·1111··1111··········11·1···11·11111·1· |
| out 33 (31) | ··············································································································································································1111····11·11·111·1·1·11·11···1·111···1111·111·1··11····························· |
| out 34 (31) | ··············································································································································································1·1··1··111·1···1111··1··1·11·111·1111·11·1··11·1··111··························· |
| out 35 (7) | ·············································································································································································1··1····1·1·····························1·····1···1······························· |
| out 36 (7) | ·················································································································································································11············11············11···········1···································· |
| out 37 (25) | ············································································································································································111···1··1··1·111··1···1·111··1···11·1···1··1111·11································ |
| out 38 (25) | ·············································································································································································11·1··1·····1·11··11·1·11·111···11·1···1·1·1··11··1·1····························· |
| out 39 (25) | ············································································································································································1·1·111···1111·11·······1··1·11··111····11···1·11·1··1····························· |
| out 40 (25) | ············································································································································································1·111·11·1··11··1···1··111···11··1··1····1··1·1·1111······························· |
| out 41 (25) | ·············································································································································································11··111···1··111·1111···1111···1··11····11·11·····1······························· |
| out 42 (25) | ············································································································································································1····1·1·1·11··1····11··1··111·11·111···1·1··111···1·1····························· |
| out 43 (25) | ············································································································································································11··11·1·1·11·1····1·1·1111··1··11·11··1····111··1································· |
| out 44 (7) | ····································································································································································································································111··1111·················· |
| out 45 (7) | ·············································································································································································································································111··1111········· |
| out 46 (7) | ······················································································································································································································································111··1111 |
| check 47 (62) | ·1·1··1·111·11·11·111·11··111···111·1·1····11··111111························································································································111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111····························· |
| check 48 (78) | ·111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1······················································································································11····11·1··111·1·1·1·1·111··111·1···1··111··111·1···11111··1··11·1111·1·1·11111·1· |
| check 49 (72) | ··111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111····················································································································1111111·11·······1·11····11····1111··1·11···1·1111··1111··········11··111···11111·1· |
| check 50 (62) | 1···1·111·111··111··11··1·1··11·11··1111·1···1··1·111·11···111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111··············································································································································· |
| check 51 (78) | 1111·1·1··111·1··1··1·11··111···11·1·1··11···111·111···11··················································································································111·1·1····1111·1··11··111·1·1·1·11·11·1··11·111···11···111·1·11···1111·1·111··1··11 |
| check 52 (62) | 111·1·1····1111·1··11··111·1·1·1·11·11·1··11·111···11···1···························································111··111··1111·1111·1·1··1··111··111··1·1··1·1··1·111······················································································ |
| check 53 (68) | ··························································1·1··1·111·11·11·111·11··111···111·1·1····11··111111······························································111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1·············11111·1·········· |
| check 54 (76) | ··························································111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1·····························································11····11·1··111·1·1·1·1·111··111·1···1··111··111·1···11111·1·11···1111·1·1·····1111 |
| check 55 (62) | ·························································1···1·111·111··111··11··1·1··11·11··1111·1···1··1·111·11··111·1·1·1111··11··1·1··11···111··1·1·1·11··1111·111·1······················································································· |
| check 56 (68) | ·························································1111·1·1··111·1··1··1·11··111···11·1·1··11···111·111···11·························································111·1·1····1111·1··11··111·1·1·1·11·11·1··11·111···11···1·1111·1·1·················· |
| check 57 (62) | ·························································111·1·1····1111·1··11··111·1·1·1·11·11·1··11·111···11···1··························································1·1··1·111·11·11·111·11··111···111·1·1····11··111111······························· |
| check 58 (62) | ···················································································································1·1··1·111·11·11·111·11··111···111·1·1····11··111111····1111·1·1··111·1··1··1·11··111···11·1·1··11···111·111···11··························· |
| check 59 (78) | ··················································································································1···1·111·111··111··11··1·1··11·11··1111·1···1··1·111·11·111·1·1····1111·1··11··111·1·1·1·11·11·1··11·111···11···111·1·11···1111·1·111·1·11·· |
| check 60 (72) | ··················································································································1111·1·1··111·1··1··1·11··111···11·1·1··11···111·111···111111111·11·······1·11····11····1111··1·11···1·1111··1111··········11·1···11·11111·1· |
| check 61 (80) | ··················································································································111·1·1····1111·1··11··111·1·1·1·11·11·1··11·111···11···11···1·111·111··111··11··1·1··11·11··1111·1···1··1·111·11··11111·1··1111·1·1·11111·1· |
The gate is read off the output rows. An output row of odd weight gives a T, a pair of output rows with odd overlap gives a CS, and an odd triple overlap gives a CCZ. The circuit is a valid factory because every such parity that touches a check row is even.
Figures of merit
inputs n255Noisy T states per run.
outputs k47
distance d3Exact, proved and witnessed here.
error coefficient A_{3}1,995Faults 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 N62
checks r15
V_{\mathrm{ex}}47T states per run. CCZ counts as two.
\gamma_\rho = \log(n/V_{\mathrm{ex}})/\log d1.5393Overhead per extractable T state. Lower is better.
\gamma = \log(n/k)/\log d1.5393Overhead 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 width47Rank of the output rows modulo the check rows.
Output gate
47 T0 CS0 CCZon 47 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}\cdot \mathrm{T}_{44}\cdot \mathrm{T}_{45}\cdot \mathrm{T}_{46}
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+44+45+46 (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 43 single-wire gates and 584 \mathrm{CZ} gates: \mathrm{S}^\dagger on output 0, \mathrm{S} on output 2, \mathrm{S} on output 3, \mathrm{S} on output 4, \mathrm{S}^\dagger on output 6, \mathrm{S} on output 7, \mathrm{Z} on output 8, \mathrm{Z} on output 9, \mathrm{S}^\dagger on output 10, \mathrm{S} on output 11, \mathrm{S} on output 12, \mathrm{S} on output 13, \mathrm{S} on output 14, \mathrm{Z} on output 15, \mathrm{Z} on output 16, \mathrm{Z} on output 18, \mathrm{Z} on output 19, \mathrm{Z} on output 21, \mathrm{Z} on output 23, \mathrm{S} on output 25, \mathrm{S}^\dagger on output 28, \mathrm{S} on output 29, \mathrm{S} on output 30, \mathrm{Z} on output 31 and 603 more (listed below). The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.
All 627 correction gates
\mathrm{S}^\dagger on output 0, \mathrm{S} on output 2, \mathrm{S} on output 3, \mathrm{S} on output 4, \mathrm{S}^\dagger on output 6, \mathrm{S} on output 7, \mathrm{Z} on output 8, \mathrm{Z} on output 9, \mathrm{S}^\dagger on output 10, \mathrm{S} on output 11, \mathrm{S} on output 12, \mathrm{S} on output 13, \mathrm{S} on output 14, \mathrm{Z} on output 15, \mathrm{Z} on output 16, \mathrm{Z} on output 18, \mathrm{Z} on output 19, \mathrm{Z} on output 21, \mathrm{Z} on output 23, \mathrm{S} on output 25, \mathrm{S}^\dagger on output 28, \mathrm{S} on output 29, \mathrm{S} on output 30, \mathrm{Z} on output 31, \mathrm{S} on output 33, \mathrm{S} on output 34, \mathrm{S} on output 35, \mathrm{S} on output 36, \mathrm{S} on output 44, \mathrm{S} on output 45, \mathrm{S} on output 46, \mathrm{S} on check 47, \mathrm{S} on check 48, \mathrm{S} on check 50, \mathrm{S} on check 51, \mathrm{S} on check 52, \mathrm{Z} on check 53, \mathrm{Z} on check 54, \mathrm{S} on check 55, \mathrm{Z} on check 56, \mathrm{S} on check 57, \mathrm{S} on check 58, \mathrm{S} on check 59, \mathrm{CZ} on output 0, output 1, \mathrm{CZ} on output 0, output 2, \mathrm{CZ} on output 0, output 4, \mathrm{CZ} on output 0, output 7, \mathrm{CZ} on output 0, output 9, \mathrm{CZ} on output 0, output 22, \mathrm{CZ} on output 0, output 23, \mathrm{CZ} on output 0, output 24, \mathrm{CZ} on output 0, output 26, \mathrm{CZ} on output 0, output 27, \mathrm{CZ} on output 0, output 28, \mathrm{CZ} on output 0, output 30, \mathrm{CZ} on output 0, output 31, \mathrm{CZ} on output 0, output 32, \mathrm{CZ} on output 0, output 37, \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 49, \mathrm{CZ} on output 0, check 50, \mathrm{CZ} on output 0, check 52, \mathrm{CZ} on output 0, check 59, \mathrm{CZ} on output 0, check 60, \mathrm{CZ} on output 0, check 61, \mathrm{CZ} on output 1, output 4, \mathrm{CZ} on output 1, output 5, \mathrm{CZ} on output 1, output 7, \mathrm{CZ} on output 1, output 9, \mathrm{CZ} on output 1, output 10, \mathrm{CZ} on output 1, output 12, \mathrm{CZ} on output 1, output 15, \mathrm{CZ} on output 1, output 16, \mathrm{CZ} on output 1, output 18, \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 26, \mathrm{CZ} on output 1, output 27, \mathrm{CZ} on output 1, output 28, \mathrm{CZ} on output 1, output 30, \mathrm{CZ} on output 1, output 31, \mathrm{CZ} on output 1, check 47, \mathrm{CZ} on output 1, check 49, \mathrm{CZ} on output 1, check 51, \mathrm{CZ} on output 1, check 53, \mathrm{CZ} on output 1, check 61, \mathrm{CZ} on output 2, output 5, \mathrm{CZ} on output 2, output 6, \mathrm{CZ} on output 2, output 7, \mathrm{CZ} on output 2, output 8, \mathrm{CZ} on output 2, output 10, \mathrm{CZ} on output 2, output 12, \mathrm{CZ} on output 2, output 15, \mathrm{CZ} on output 2, output 16, \mathrm{CZ} on output 2, output 18, \mathrm{CZ} on output 2, output 20, \mathrm{CZ} on output 2, output 22, \mathrm{CZ} on output 2, output 24, \mathrm{CZ} on output 2, output 26, \mathrm{CZ} on output 2, output 27, \mathrm{CZ} on output 2, output 28, \mathrm{CZ} on output 2, output 30, \mathrm{CZ} on output 2, output 31, \mathrm{CZ} on output 2, check 49, \mathrm{CZ} on output 2, check 50, \mathrm{CZ} on output 2, check 51, \mathrm{CZ} on output 2, check 52, \mathrm{CZ} on output 2, check 53, \mathrm{CZ} on output 2, check 61, \mathrm{CZ} on output 4, output 5, \mathrm{CZ} on output 4, output 7, \mathrm{CZ} on output 4, output 12, \mathrm{CZ} on output 4, output 15, \mathrm{CZ} on output 4, output 16, \mathrm{CZ} on output 4, output 18, \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 26, \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 31, \mathrm{CZ} on output 4, check 47, \mathrm{CZ} on output 4, check 49, \mathrm{CZ} on output 4, check 51, \mathrm{CZ} on output 4, check 53, \mathrm{CZ} on output 4, check 61, \mathrm{CZ} on output 5, output 7, \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 13, \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 19, \mathrm{CZ} on output 5, output 20, \mathrm{CZ} on output 5, output 21, \mathrm{CZ} on output 5, output 22, \mathrm{CZ} on output 5, output 23, \mathrm{CZ} on output 5, output 24, \mathrm{CZ} on output 5, output 26, \mathrm{CZ} on output 5, output 27, \mathrm{CZ} on output 5, output 28, \mathrm{CZ} on output 5, output 30, \mathrm{CZ} on output 5, output 31, \mathrm{CZ} on output 5, output 32, \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, output 42, \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 52, \mathrm{CZ} on output 5, check 56, \mathrm{CZ} on output 5, check 60, \mathrm{CZ} on output 6, output 8, \mathrm{CZ} on output 6, output 9, \mathrm{CZ} on output 6, output 10, \mathrm{CZ} on output 6, output 11, \mathrm{CZ} on output 6, output 12, \mathrm{CZ} on output 6, output 13, \mathrm{CZ} on output 6, output 15, \mathrm{CZ} on output 6, output 16, \mathrm{CZ} on output 6, output 17, \mathrm{CZ} on output 6, output 19, \mathrm{CZ} on output 6, output 20, \mathrm{CZ} on output 6, output 21, \mathrm{CZ} on output 6, output 22, \mathrm{CZ} on output 6, output 23, \mathrm{CZ} on output 6, output 24, \mathrm{CZ} on output 6, output 26, \mathrm{CZ} on output 6, output 27, \mathrm{CZ} on output 6, output 28, \mathrm{CZ} on output 6, output 30, \mathrm{CZ} on output 6, output 31, \mathrm{CZ} on output 6, output 32, \mathrm{CZ} on output 6, output 37, \mathrm{CZ} on output 6, output 39, \mathrm{CZ} on output 6, output 40, \mathrm{CZ} on output 6, output 42, \mathrm{CZ} on output 6, check 48, \mathrm{CZ} on output 6, check 49, \mathrm{CZ} on output 6, check 50, \mathrm{CZ} on output 6, check 56, \mathrm{CZ} on output 6, check 60, \mathrm{CZ} on output 7, output 9, \mathrm{CZ} on output 7, check 47, \mathrm{CZ} on output 7, check 50, \mathrm{CZ} on output 7, check 51, \mathrm{CZ} on output 7, check 52, \mathrm{CZ} on output 8, output 10, \mathrm{CZ} on output 8, output 12, \mathrm{CZ} on output 8, output 15, \mathrm{CZ} on output 8, output 16, \mathrm{CZ} on output 8, output 18, \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 26, \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 31, \mathrm{CZ} on output 8, check 49, \mathrm{CZ} on output 8, check 50, \mathrm{CZ} on output 8, check 52, \mathrm{CZ} on output 8, check 53, \mathrm{CZ} on output 8, check 61, \mathrm{CZ} on output 9, output 11, \mathrm{CZ} on output 9, output 12, \mathrm{CZ} on output 9, output 13, \mathrm{CZ} on output 9, output 15, \mathrm{CZ} on output 9, output 16, \mathrm{CZ} on output 9, output 17, \mathrm{CZ} on output 9, output 19, \mathrm{CZ} on output 9, output 20, \mathrm{CZ} on output 9, output 21, \mathrm{CZ} on output 9, output 22, \mathrm{CZ} on output 9, output 23, \mathrm{CZ} on output 9, output 24, \mathrm{CZ} on output 9, output 26, \mathrm{CZ} on output 9, output 27, \mathrm{CZ} on output 9, output 28, \mathrm{CZ} on output 9, output 30, \mathrm{CZ} on output 9, output 31, \mathrm{CZ} on output 9, output 32, \mathrm{CZ} on output 9, output 37, \mathrm{CZ} on output 9, output 39, \mathrm{CZ} on output 9, output 40, \mathrm{CZ} on output 9, output 42, \mathrm{CZ} on output 9, check 47, \mathrm{CZ} on output 9, check 48, \mathrm{CZ} on output 9, check 49, \mathrm{CZ} on output 9, check 50, \mathrm{CZ} on output 9, check 51, \mathrm{CZ} on output 9, check 56, \mathrm{CZ} on output 9, check 60, \mathrm{CZ} on output 10, output 11, \mathrm{CZ} on output 10, output 13, \mathrm{CZ} on output 10, output 17, \mathrm{CZ} on output 10, output 18, \mathrm{CZ} on output 10, output 19, \mathrm{CZ} on output 10, output 21, \mathrm{CZ} on output 10, output 22, \mathrm{CZ} on output 10, output 24, \mathrm{CZ} on output 10, output 26, \mathrm{CZ} on output 10, output 27, \mathrm{CZ} on output 10, output 28, \mathrm{CZ} on output 10, output 30, \mathrm{CZ} on output 10, output 31, \mathrm{CZ} on output 10, output 38, \mathrm{CZ} on output 10, output 39, \mathrm{CZ} on output 10, output 42, \mathrm{CZ} on output 10, output 43, \mathrm{CZ} on output 10, check 48, \mathrm{CZ} on output 10, check 49, \mathrm{CZ} on output 10, check 50, \mathrm{CZ} on output 10, check 53, \mathrm{CZ} on output 10, check 56, \mathrm{CZ} on output 10, check 59, \mathrm{CZ} on output 11, output 16, \mathrm{CZ} on output 11, output 17, \mathrm{CZ} on output 11, output 18, \mathrm{CZ} on output 11, output 19, \mathrm{CZ} on output 11, output 37, \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 48, \mathrm{CZ} on output 11, check 51, \mathrm{CZ} on output 11, check 54, \mathrm{CZ} on output 11, check 55, \mathrm{CZ} on output 11, check 56, \mathrm{CZ} on output 11, check 57, \mathrm{CZ} on output 11, check 59, \mathrm{CZ} on output 11, check 61, \mathrm{CZ} on output 12, output 22, \mathrm{CZ} on output 12, output 23, \mathrm{CZ} on output 12, output 24, \mathrm{CZ} on output 12, output 26, \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 31, \mathrm{CZ} on output 12, output 32, \mathrm{CZ} on output 12, output 37, \mathrm{CZ} on output 12, output 38, \mathrm{CZ} on output 12, output 40, \mathrm{CZ} on output 12, output 43, \mathrm{CZ} on output 12, check 48, \mathrm{CZ} on output 12, check 49, \mathrm{CZ} on output 12, check 51, \mathrm{CZ} on output 12, check 54, \mathrm{CZ} on output 12, check 56, \mathrm{CZ} on output 12, check 59, \mathrm{CZ} on output 12, check 60, \mathrm{CZ} on output 12, check 61, \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 37, \mathrm{CZ} on output 13, output 38, \mathrm{CZ} on output 13, output 40, \mathrm{CZ} on output 13, output 43, \mathrm{CZ} on output 13, check 48, \mathrm{CZ} on output 13, check 51, \mathrm{CZ} on output 13, check 54, \mathrm{CZ} on output 13, check 55, \mathrm{CZ} on output 13, check 56, \mathrm{CZ} on output 13, check 57, \mathrm{CZ} on output 13, check 59, \mathrm{CZ} on output 13, check 61, \mathrm{CZ} on output 15, output 20, \mathrm{CZ} on output 15, output 21, \mathrm{CZ} on output 15, output 22, \mathrm{CZ} on output 15, output 23, \mathrm{CZ} on output 15, output 24, \mathrm{CZ} on output 15, output 26, \mathrm{CZ} on output 15, output 27, \mathrm{CZ} on output 15, output 28, \mathrm{CZ} on output 15, output 30, \mathrm{CZ} on output 15, output 31, \mathrm{CZ} on output 15, output 32, \mathrm{CZ} on output 15, output 37, \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 48, \mathrm{CZ} on output 15, check 49, \mathrm{CZ} on output 15, check 51, \mathrm{CZ} on output 15, check 54, \mathrm{CZ} on output 15, check 56, \mathrm{CZ} on output 15, check 59, \mathrm{CZ} on output 15, check 60, \mathrm{CZ} on output 15, check 61, \mathrm{CZ} on output 16, output 17, \mathrm{CZ} on output 16, output 19, \mathrm{CZ} on output 16, output 21, \mathrm{CZ} on output 16, output 23, \mathrm{CZ} on output 16, output 32, \mathrm{CZ} on output 16, output 37, \mathrm{CZ} on output 16, output 39, \mathrm{CZ} on output 16, output 40, \mathrm{CZ} on output 16, output 42, \mathrm{CZ} on output 16, check 48, \mathrm{CZ} on output 16, check 49, \mathrm{CZ} on output 16, check 51, \mathrm{CZ} on output 16, check 54, \mathrm{CZ} on output 16, check 56, \mathrm{CZ} on output 16, check 57, \mathrm{CZ} on output 16, check 59, \mathrm{CZ} on output 16, check 60, \mathrm{CZ} on output 17, output 19, \mathrm{CZ} on output 17, output 21, \mathrm{CZ} on output 17, output 22, \mathrm{CZ} on output 17, output 24, \mathrm{CZ} on output 17, output 26, \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 31, \mathrm{CZ} on output 17, output 37, \mathrm{CZ} on output 17, output 39, \mathrm{CZ} on output 17, output 40, \mathrm{CZ} on output 17, output 42, \mathrm{CZ} on output 17, check 48, \mathrm{CZ} on output 17, check 51, \mathrm{CZ} on output 17, check 54, \mathrm{CZ} on output 17, check 55, \mathrm{CZ} on output 17, check 56, \mathrm{CZ} on output 17, check 59, \mathrm{CZ} on output 18, output 22, \mathrm{CZ} on output 18, output 23, \mathrm{CZ} on output 18, output 24, \mathrm{CZ} on output 18, output 26, \mathrm{CZ} on output 18, output 27, \mathrm{CZ} on output 18, output 28, \mathrm{CZ} on output 18, output 30, \mathrm{CZ} on output 18, output 31, \mathrm{CZ} on output 18, output 32, \mathrm{CZ} on output 18, check 49, \mathrm{CZ} on output 18, check 55, \mathrm{CZ} on output 18, check 56, \mathrm{CZ} on output 18, check 57, \mathrm{CZ} on output 18, check 60, \mathrm{CZ} on output 19, output 37, \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 48, \mathrm{CZ} on output 19, check 51, \mathrm{CZ} on output 19, check 54, \mathrm{CZ} on output 19, check 55, \mathrm{CZ} on output 19, check 57, \mathrm{CZ} on output 19, check 59, \mathrm{CZ} on output 19, check 61, \mathrm{CZ} on output 20, output 23, \mathrm{CZ} on output 20, output 32, \mathrm{CZ} on output 20, output 37, \mathrm{CZ} on output 20, output 39, \mathrm{CZ} on output 20, output 40, \mathrm{CZ} on output 20, output 42, \mathrm{CZ} on output 20, check 48, \mathrm{CZ} on output 20, check 49, \mathrm{CZ} on output 20, check 51, \mathrm{CZ} on output 20, check 54, \mathrm{CZ} on output 20, check 55, \mathrm{CZ} on output 20, check 59, \mathrm{CZ} on output 20, check 60, \mathrm{CZ} on output 21, output 22, \mathrm{CZ} on output 21, output 24, \mathrm{CZ} on output 21, output 26, \mathrm{CZ} on output 21, output 27, \mathrm{CZ} on output 21, output 28, \mathrm{CZ} on output 21, output 30, \mathrm{CZ} on output 21, output 31, \mathrm{CZ} on output 21, output 37, \mathrm{CZ} on output 21, output 39, \mathrm{CZ} on output 21, output 40, \mathrm{CZ} on output 21, output 42, \mathrm{CZ} on output 21, check 48, \mathrm{CZ} on output 21, check 51, \mathrm{CZ} on output 21, check 54, \mathrm{CZ} on output 21, check 55, \mathrm{CZ} on output 21, check 59, \mathrm{CZ} on output 22, output 23, \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 37, \mathrm{CZ} on output 22, output 39, \mathrm{CZ} on output 22, output 40, \mathrm{CZ} on output 22, output 42, \mathrm{CZ} on output 22, check 51, \mathrm{CZ} on output 22, check 53, \mathrm{CZ} on output 22, check 54, \mathrm{CZ} on output 22, check 56, \mathrm{CZ} on output 22, check 60, \mathrm{CZ} on output 23, output 24, \mathrm{CZ} on output 23, output 27, \mathrm{CZ} on output 23, output 29, \mathrm{CZ} on output 23, output 30, \mathrm{CZ} on output 23, output 31, \mathrm{CZ} on output 23, output 32, \mathrm{CZ} on output 23, output 37, \mathrm{CZ} on output 23, output 39, \mathrm{CZ} on output 23, output 41, \mathrm{CZ} on output 23, output 43, \mathrm{CZ} on output 23, check 49, \mathrm{CZ} on output 23, check 51, \mathrm{CZ} on output 23, check 53, \mathrm{CZ} on output 23, check 58, \mathrm{CZ} on output 23, check 59, \mathrm{CZ} on output 24, output 26, \mathrm{CZ} on output 24, output 27, \mathrm{CZ} on output 24, output 28, \mathrm{CZ} on output 24, output 29, \mathrm{CZ} on output 24, output 37, \mathrm{CZ} on output 24, output 39, \mathrm{CZ} on output 24, output 40, \mathrm{CZ} on output 24, output 42, \mathrm{CZ} on output 24, check 51, \mathrm{CZ} on output 24, check 53, \mathrm{CZ} on output 24, check 54, \mathrm{CZ} on output 24, check 56, \mathrm{CZ} on output 24, check 60, \mathrm{CZ} on output 26, output 27, \mathrm{CZ} on output 26, output 29, \mathrm{CZ} on output 26, output 30, \mathrm{CZ} on output 26, output 32, \mathrm{CZ} on output 26, output 37, \mathrm{CZ} on output 26, output 38, \mathrm{CZ} on output 26, output 41, \mathrm{CZ} on output 26, output 42, \mathrm{CZ} on output 26, check 51, \mathrm{CZ} on output 26, check 53, \mathrm{CZ} on output 26, check 54, \mathrm{CZ} on output 26, check 56, \mathrm{CZ} on output 26, check 58, \mathrm{CZ} on output 26, check 59, \mathrm{CZ} on output 26, check 60, \mathrm{CZ} on output 26, check 61, \mathrm{CZ} on output 27, output 29, \mathrm{CZ} on output 27, output 30, \mathrm{CZ} on output 27, output 37, \mathrm{CZ} on output 27, output 38, \mathrm{CZ} on output 27, output 41, \mathrm{CZ} on output 27, output 42, \mathrm{CZ} on output 27, check 51, \mathrm{CZ} on output 27, check 53, \mathrm{CZ} on output 27, check 54, \mathrm{CZ} on output 27, check 56, \mathrm{CZ} on output 27, check 58, \mathrm{CZ} on output 27, check 59, \mathrm{CZ} on output 28, output 30, \mathrm{CZ} on output 28, output 31, \mathrm{CZ} on output 28, output 37, \mathrm{CZ} on output 28, output 39, \mathrm{CZ} on output 28, output 40, \mathrm{CZ} on output 28, output 42, \mathrm{CZ} on output 28, check 51, \mathrm{CZ} on output 28, check 53, \mathrm{CZ} on output 28, check 54, \mathrm{CZ} on output 28, check 56, \mathrm{CZ} on output 28, check 61, \mathrm{CZ} on output 29, output 31, \mathrm{CZ} on output 29, output 38, \mathrm{CZ} on output 29, output 39, \mathrm{CZ} on output 29, output 40, \mathrm{CZ} on output 29, output 41, \mathrm{CZ} on output 29, check 58, \mathrm{CZ} on output 29, check 59, \mathrm{CZ} on output 29, check 60, \mathrm{CZ} on output 29, check 61, \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, output 42, \mathrm{CZ} on output 30, check 51, \mathrm{CZ} on output 30, check 53, \mathrm{CZ} on output 30, check 54, \mathrm{CZ} on output 30, check 56, \mathrm{CZ} on output 31, output 32, \mathrm{CZ} on output 31, output 37, \mathrm{CZ} on output 31, output 38, \mathrm{CZ} on output 31, output 41, \mathrm{CZ} on output 31, output 42, \mathrm{CZ} on output 31, check 51, \mathrm{CZ} on output 31, check 53, \mathrm{CZ} on output 31, check 54, \mathrm{CZ} on output 31, check 56, \mathrm{CZ} on output 31, check 58, \mathrm{CZ} on output 31, check 60, \mathrm{CZ} on output 31, check 61, \mathrm{CZ} on output 32, output 37, \mathrm{CZ} on output 32, output 38, \mathrm{CZ} on output 32, output 40, \mathrm{CZ} on output 32, output 43, \mathrm{CZ} on output 32, check 49, \mathrm{CZ} on output 32, check 51, \mathrm{CZ} on output 32, check 53, \mathrm{CZ} on output 34, output 40, \mathrm{CZ} on output 34, output 41, \mathrm{CZ} on output 34, output 42, \mathrm{CZ} on output 34, output 43, \mathrm{CZ} on output 37, output 38, \mathrm{CZ} on output 37, output 39, \mathrm{CZ} on output 37, output 40, \mathrm{CZ} on output 37, output 41, \mathrm{CZ} on output 37, output 42, \mathrm{CZ} on output 37, output 43, \mathrm{CZ} on output 37, check 49, \mathrm{CZ} on output 37, check 51, \mathrm{CZ} on output 37, check 56, \mathrm{CZ} on output 37, check 59, \mathrm{CZ} on output 37, check 60, \mathrm{CZ} on output 37, check 61, \mathrm{CZ} on output 38, output 39, \mathrm{CZ} on output 38, output 40, \mathrm{CZ} on output 38, output 41, \mathrm{CZ} on output 38, output 42, \mathrm{CZ} on output 38, output 43, \mathrm{CZ} on output 38, check 48, \mathrm{CZ} on output 38, check 49, \mathrm{CZ} on output 38, check 54, \mathrm{CZ} on output 38, check 58, \mathrm{CZ} on output 38, check 60, \mathrm{CZ} on output 38, check 61, \mathrm{CZ} on output 39, output 40, \mathrm{CZ} on output 39, output 41, \mathrm{CZ} on output 39, output 42, \mathrm{CZ} on output 39, output 43, \mathrm{CZ} on output 39, check 48, \mathrm{CZ} on output 39, check 51, \mathrm{CZ} on output 39, check 54, \mathrm{CZ} on output 39, check 56, \mathrm{CZ} on output 39, check 58, \mathrm{CZ} on output 39, check 59, \mathrm{CZ} on output 40, output 41, \mathrm{CZ} on output 40, output 42, \mathrm{CZ} on output 40, output 43, \mathrm{CZ} on output 40, check 49, \mathrm{CZ} on output 40, check 51, \mathrm{CZ} on output 40, check 56, \mathrm{CZ} on output 40, check 58, \mathrm{CZ} on output 40, check 59, \mathrm{CZ} on output 40, check 60, \mathrm{CZ} on output 40, check 61, \mathrm{CZ} on output 41, output 42, \mathrm{CZ} on output 41, output 43, \mathrm{CZ} on output 41, check 58, \mathrm{CZ} on output 42, output 43, \mathrm{CZ} on output 42, check 48, \mathrm{CZ} on output 42, check 51, \mathrm{CZ} on output 42, check 54, \mathrm{CZ} on output 42, check 56, \mathrm{CZ} on output 42, check 59, \mathrm{CZ} on output 43, check 48, \mathrm{CZ} on output 43, check 49, \mathrm{CZ} on output 43, check 54, \mathrm{CZ} on output 43, check 60, \mathrm{CZ} on output 43, check 61, \mathrm{CZ} on check 48, check 51, \mathrm{CZ} on check 48, check 56, \mathrm{CZ} on check 48, check 59, \mathrm{CZ} on check 48, check 61, \mathrm{CZ} on check 49, check 51, \mathrm{CZ} on check 49, check 53, \mathrm{CZ} on check 49, check 59, \mathrm{CZ} on check 49, check 60, \mathrm{CZ} on check 51, check 56, \mathrm{CZ} on check 51, check 60, \mathrm{CZ} on check 53, check 60, \mathrm{CZ} on check 54, check 56, \mathrm{CZ} on check 54, check 61, \mathrm{CZ} on check 56, check 59, \mathrm{CZ} on check 59, check 60
Or no Clifford gate at all: run \mathrm{T}^3 on rotations 2, 5, 6, 13, 15, 16, 23, 25, 30, 36, 38, 39, 40, 42, 44, 45, 47, 51, 60, 63, 73, 76, 79, 86 and 79 more (listed below); \mathrm{T}^5 on rotations 3, 10, 62, 66, 68, 71, 115, 119, 122, 125, 128, 174, 178, 181, 246; \mathrm{T}^\dagger on rotations 0, 1, 4, 7, 14, 57, 61, 64, 72, 117, 121, 129, 172, 175, 185, 228, 237, 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 103 rotations at T³
2, 5, 6, 13, 15, 16, 23, 25, 30, 36, 38, 39, 40, 42, 44, 45, 47, 51, 60, 63, 73, 76, 79, 86, 89, 90, 94, 97, 98, 99, 104, 105, 107, 109, 110, 111, 112, 114, 118, 120, 130, 133, 136, 143, 146, 147, 151, 154, 155, 156, 161, 162, 164, 166, 167, 168, 169, 171, 179, 182, 186, 187, 189, 191, 192, 193, 195, 197, 198, 199, 200, 202, 203, 204, 207, 209, 210, 212, 213, 214, 215, 217, 218, 219, 222, 223, 224, 231, 232, 233, 235, 238, 239, 240, 241, 242, 243, 244, 245, 247, 250, 251, 254
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/255.47.3.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-255.47.3.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 1,995 faults of weight 3 are undetectable and damaging, so A_{3}=1{,}995. 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 [0,1,38].
A fault is a set of faulty columns. It is undetectable when their check parts cancel, and damaging when their output parts do not.
Original source
S. P. Jain, A. Wills, and S. Singh, "Symmetry and AI-assisted discovery of magic-state factories," arXiv:2610.06535 (2026). arxiv.org/abs/2610.06535
Provenance
found by an AI search that glues catalogued small factories along a graph of paired-column contractions; exact distance verified from columns; a verified witness, not a maximum
Found by: AI search: graph gluing
Discovery: AI search
1 recorded source
[
{
"regime": "AI search: graph gluing",
"file": "master_catalog/imports/2026-09-18_graph_gluing_d3/factories.json",
"label": "graph gluing [[255,47,3]]",
"N": 62,
"d": 3,
"d_is_exact": true,
"origin": null,
"provenance": "AI search: graph gluing -- four [[63,11,3]] and three [[15,1,3]] glued along K7 by 21 paired-column contraction(s) (dependent ports); construction and witness in master_catalog/imports/2026-09-18_graph_gluing_d3/witnesses/graph_heterogeneous_n255_k47_d3.json",
"notes": "Exact A3=1995 independently counted; details in master_catalog/imports/2026-09-18_graph_gluing_d3/README.md. General gluing has prior art; publication novelty is not asserted."
}
]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/255.47.3.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[921].