The circuit
The 42\times176 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 |
|---|---|
| out 0 (75) | ·1···1·1·1··1·1·11···11··11··1·····1·1·1··1··1··1····1··1·····1···1·11··11111·1111111·1···1·1·1·1··1·1··1···11··111··1···1··1·1··1··1··1·······1·····1·····1··1·11·1111·11111111 |
| out 1 (67) | 1·····11·1·····1···1·1·11·111····1··1·1···1·11······1·1·111·1·····1111·········1111111·····11·1·····1·····1·1·11·1··1··1··1··1···1·11·········1·1····1111·····1111········111111 |
| out 2 (47) | 1····11··1··1··11·1·····1·1·······1····1·····11········11·····1··1·1·1··1··1····1···11····11··1·1···11···1·····1·1······1·····1·····11··············11·····1···1·1·1··1····1···1 |
| out 3 (67) | ·····11111··1······1···1··1·11···1··1·······1···1···1··1111·1··1···111·····1··1111111··1··11111·1·········1···1··1··11·1··1········1···1······1·····11111···1··111····1··1111111 |
| out 4 (51) | ····111111·········1······1·1·······1·····1·············111·1······111·1······111111·····111111·······1···1······1··1·····1······1···················1111······1111······111111· |
| out 5 (71) | 1····11·1···11·1··1··1····1··1··11·1·11·111·11··11·····1··1····11···11····1·1·11··11·1····11·1··1·1·1····1··1····1·1·111·1··11·111·11··1·1··········1··1····11··11···1··111··11· |
| out 6 (63) | 1····1·····11·111·1···1·111·······1··1·1·1···11··1·····111··1······1·1··1·111···1···11····1····11··111···1···1·111······1···1·1·1···11···1··1·······111·1······1·1·1·11·1··1···1 |
| out 7 (75) | 11··1111···1·11·1····111·11·1···1··11·1······1·111······111·1·····11··11·1111·····1·111··1111··1··11·1······111·11··1·1··11··1······1·11·1···········1111·····11··1·111·1····1·1 |
| out 8 (63) | 1···11···11····11·1····1111·······1····11·1··11·111·1··11·····11···1·1··1··1····1·1··1···11···1·····11···1····1111······1·····11·1··11·1·1···11·····11·····11··1·1·1··1····1·1·· |
| out 9 (67) | 1········1··11·1···1·1······1···11·11111111·11···1······111·1··11··11·····1·1···1···11········1·1·1·1·····1·1·······1·11·11·111111·11····11··········1111···11·11····1··1··1···1 |
| out 10 (79) | ·1····1··1··1·1··1·1···11···11····111111····1·1·11···11111··1·1·1·11·····1111·1·11··1·1·1··1··1·1··1····1·1···11····11··111·111····1·1·1·1·····11···111·1··1·111····111·11·11··1 |
| out 11 (71) | ····11······1·1111·1··1··11··1····1111·······111·1··11111··1··111··11··1···1··1·11·······11·····1··111··1·1··1··11···1··111·1·······111··1····111···11·····111·11·1···1··1·11··· |
| out 12 (71) | ·1···1·········1···1·1····1·11····1···111·····111···1·1·111·1··11·111··1111·1·1··11·1·1···1·········1·····1·1····1··11··1····111·····111···1··1·1····1111···11111·1111··11··11·1 |
| out 13 (67) | 11····11······11·11··11···1··1···1·1·111··1···1·1····1·1········1···1····1111·111·11·11····11······11···11··11···1···1·1·1··111··1···1·1·······1····1········1··1···11111111·11· |
| out 14 (71) | ·1·11···1···111·11·1·1···11·1···11··1···111··1·111···111·11·1···········11·····1·111··1··1···1··1·11·1··1·1·1···11··1·11··1····111··1·11·1·····11···1·111··········11·····1·111· |
| out 15 (71) | ·1··11··1··1··11···1···1··1·11····1··11111····11····1·1·1·1·····1·111··111····1··11·1·1··11··1·1···11·····1···1··1··11··1···11111····11·······1·1····1·1··1··1111·111····1··11·1 |
| out 16 (31) | ···········11·11·····················11··11·········1····1··1·············1·1···1···1··········11··11·······················11··11············1···1···1·1············1··1··1···1 |
| out 17 (83) | 11··1··111····11·11····1·11··1···1·1·111··1·1···1····1111·1·····1··11··111111··11111·11··1··111····11···11····1·11···1·1·1··111··1·1···1·······11··111·1·····1·11·11111·1·11111· |
| out 18 (67) | ·1·····1···11··1·1···1·······1···111···1·111··1·11··11·1·11·1·1·1·1·1····1·1··11··11··1·····1··11···1···1···1········1·111····1·11···1·1·1····11····1·111··1·11·1···1·1··11··11· |
| out 19 (83) | 1···11111··1·····111··111·1·11·1·1·1···1·11····11···11111·1·····1··1····11111··11·1111···11111·1········111··111·1··11·1·1····1·11····11······111···11·1·····1·1···1111·1·11·111 |
| out 20 (67) | ·1··1··11··1·····1·····1·····1···111··11·1··111·11··111··11·1·1·····11·1·1·····1··11··1··1··11·1········1·····1······1·111···11·1··111·111····111·····111··1····111·1·····1··11· |
| out 21 (75) | ·1···1·11···111··1······1····1···111·111··1·111··1···1····1···111···11·1·11·1·11··111·1···1·11··1·11····1······1·····1·111·1111··1·111···1·····1·······1···111··111·11··111··111 |
| out 22 (71) | ·1······1····11·1111·1···11·11··1···1··1111··1···1··1111·1··11·········111·1···1··11··1······1····11·1··11111···11··111···1···1111··1····1····111···1·1·11········111·1···1··11· |
| out 23 (15) | ···········1··1··1··1···········1··1·1···1·····················································1···1····1·············1··1··1···1··············································· |
| out 24 (75) | ·····1·1···11·1·1··1·111·11·1····11·11·1·1··1·1·····1·111···········11··1·111··111111·····1·1··11··1·1·1··1·111·11··1··11·1·1·1·1··1·1········1·1···11··········11·1·11·1·111111 |
| out 25 (79) | ·1···11·1···1··1···1·111··1·11····1····1······1111···11·111·1·1·1·111··1111·1·1·11··1·1···11·1··11··1·····1·111··1··11··1·····1······111·1·····11····1111··1·1111·1111··11·11··1 |
| out 26 (71) | 11···11·····1·1111·1···1·1··111····1111···1·1···1···111··11·1···1····1···11·1·1··1··111···11····1··111··1·1···1·1···11···11·11···1·1···1······111·····111····1···1··11··11··1··1 |
| out 27 (67) | ····111111·11····1·1······1·1····1·11····1··········1··1111········111·1··11··1·11111····11111111·······1·1······1··1··1·11·····1·············1··1··1111·······1111··11··1·11111 |
| out 28 (83) | ·1·····1·····111·111·111··1·11··1········1··11·111···11·111·1··11·11·1·11·111·1··11·1·1·····1·····111···111·111··1··111·········1·111·11·1·····11····1111···1111·111·11·11··11·1 |
| out 29 (79) | 11··1·1·11·11·1·1·11·11·11··11······11···11··1·1····1·11··1····11·1·11·1·1·1·11··1··111··1·1·1111··1·1···11·11·11···11····1·1···11··1·1·······1·1···1··1····111·111·1·1··1··1··1 |
| out 30 (67) | ····111111·11····1·1······1·1····1·11····1·········11··11·1·1······111·1···11·11111·1····11111111·······1·1······1··1··1·11·····1·············1·····11·11······1111···1·111111·1 |
| out 31 (79) | 111·111·11·11··1·111·11·1·1·11···1111····11·····11··11··1·1·······11·1·1·1·1····1···111··111·1111···1···111·11·1·1··11·1111·····11·····1·1····11·····1·1······11·11·1·1····1···1 |
| check 32 (72) | 11··11······1·1··11···11111·1·····11·11·····1····11····1111·1·1···1··1·1··111···1·1·111··11·····1··1····11···11111··1···11··11·····1·····1···1······11111··1··1··11··11·1··1·1·1 |
| check 33 (76) | ····11·····1·1·1111··11·····11··11···1111····111·1···1·1111·11·······1··11111···11··1····11····1··1·11··11··11······1111····1111····111··1·····1····111111·······1·1111·1··11··1 |
| check 34 (80) | ···111··1··1··11·1·111····1·111·1··1·1·1111111·1····1·11···1···11····11·1··1··1·111··1···1·11··1111·1·111···111···1···1·····11···1·11···1111···1·1···1··1··1·111·11···1·11····1· |
| check 35 (84) | 1·11·11··1·1·1111·1···11·1·1··111·111··11····1·1····1·1111··1··11·11·····1·1·1·11·11111···1·1··11··1·11··1··11·111···1······1···1·1·11···11·1·····1··11·1·1····1·111·1111··11··1 |
| check 36 (88) | ··1··1······11111·111··1··111·1···11··11·1···1·111··11·111111···11·1111·1··1·111···1···1····1···1··11·1·11··11··1··1·····11·1·11·1·11111111···111··1·1·1··111111······11·1·1·1·1 |
| check 37 (88) | ·111·····1·1········1·1·11·1····111·1··1111·11·11···111111·11··11111·111·11·1·1··11···1·1···11·1·11111···11··11·····111··11···1····1·1·1··111·1··1111111······11····1111·1··1·11 |
| check 38 (92) | 1···1·1·1··1·1··1·11111··1111····111·1······1·111···111·1·11··1···111·11··1111·1·1·111····1·1·11··1··1111···11··1·11·1·11111·11111·1·11·11111·111···1·1·1··1·····1···11·11·1·1·· |
| check 39 (88) | ······11··1·1·11·11··1111·1·1·11·1·1111·1···11·11·1·11··11111··1··11·1···1···11111·11······11····111·1111·1·1111·1··1···11·11·1··1·11·11········1·111·1·1···1·11·1·1··11··1··11· |
| check 40 (68) | ····111111··1·1····11·····1·1···11··11·············11··11·1·1······111·1···11·11111·1····11111111·······1·1······1··1··1·11·····1·············1··1··1111·······1111··11··1·11111 |
| check 41 (88) | ·1··1··1·1·11111··1··11··1·111111·11··1·······1111·····1·11·1·11111···1···1·11··111···1··1··1·111·111····1·1··11·111···1··111·1111···111·1··········1·11111····111111·11·11···11 |
The gate is read off the output rows. An output row of odd weight gives a T, a pair of output rows with odd overlap gives a CS, and an odd triple overlap gives a CCZ. The circuit is a valid factory because every such parity that touches a check row is even.
Figures of merit
inputs n176Noisy T states per run.
outputs k32
distance d3Exact, proved and witnessed here.
error coefficient A_{3}864Faults 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 N42
checks r10
V_{\mathrm{ex}}32T states per run. CCZ counts as two.
\gamma_\rho = \log(n/V_{\mathrm{ex}})/\log d1.5517Overhead per extractable T state. Lower is better.
\gamma = \log(n/k)/\log d1.5517Overhead 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 width32Rank of the output rows modulo the check rows.
Output gate
32 T0 CS0 CCZon 32 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}
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 (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 36 single-wire gates and 128 \mathrm{CZ} gates: \mathrm{S}^\dagger on output 0, \mathrm{S}^\dagger on output 1, \mathrm{S} on output 2, \mathrm{S}^\dagger on output 3, \mathrm{S}^\dagger on output 4, \mathrm{S} on output 5, \mathrm{S} on output 6, \mathrm{S}^\dagger on output 7, \mathrm{S} on output 8, \mathrm{S}^\dagger on output 9, \mathrm{S} on output 10, \mathrm{S} on output 11, \mathrm{S} on output 12, \mathrm{S}^\dagger on output 13, \mathrm{S} on output 14, \mathrm{S} on output 15, \mathrm{S} on output 16, \mathrm{S}^\dagger on output 17, \mathrm{S}^\dagger on output 18, \mathrm{S}^\dagger on output 19, \mathrm{S}^\dagger on output 20, \mathrm{S}^\dagger on output 21, \mathrm{S} on output 22, \mathrm{S} on output 23 and 140 more (listed below). The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.
All 164 correction gates
\mathrm{S}^\dagger on output 0, \mathrm{S}^\dagger on output 1, \mathrm{S} on output 2, \mathrm{S}^\dagger on output 3, \mathrm{S}^\dagger on output 4, \mathrm{S} on output 5, \mathrm{S} on output 6, \mathrm{S}^\dagger on output 7, \mathrm{S} on output 8, \mathrm{S}^\dagger on output 9, \mathrm{S} on output 10, \mathrm{S} on output 11, \mathrm{S} on output 12, \mathrm{S}^\dagger on output 13, \mathrm{S} on output 14, \mathrm{S} on output 15, \mathrm{S} on output 16, \mathrm{S}^\dagger on output 17, \mathrm{S}^\dagger on output 18, \mathrm{S}^\dagger on output 19, \mathrm{S}^\dagger on output 20, \mathrm{S}^\dagger on output 21, \mathrm{S} on output 22, \mathrm{S} on output 23, \mathrm{S}^\dagger on output 24, \mathrm{S} on output 25, \mathrm{S} on output 26, \mathrm{S}^\dagger on output 27, \mathrm{S}^\dagger on output 28, \mathrm{S} on output 29, \mathrm{S}^\dagger on output 30, \mathrm{S} on output 31, \mathrm{Z} on check 33, \mathrm{Z} on check 35, \mathrm{Z} on check 38, \mathrm{Z} on check 40, \mathrm{CZ} on output 0, check 38, \mathrm{CZ} on output 0, check 39, \mathrm{CZ} on output 0, check 41, \mathrm{CZ} on output 1, check 34, \mathrm{CZ} on output 1, check 35, \mathrm{CZ} on output 1, check 36, \mathrm{CZ} on output 1, check 37, \mathrm{CZ} on output 1, check 39, \mathrm{CZ} on output 3, check 38, \mathrm{CZ} on output 3, check 39, \mathrm{CZ} on output 3, check 41, \mathrm{CZ} on output 4, check 38, \mathrm{CZ} on output 4, check 39, \mathrm{CZ} on output 4, check 41, \mathrm{CZ} on output 5, check 34, \mathrm{CZ} on output 5, check 35, \mathrm{CZ} on output 5, check 36, \mathrm{CZ} on output 5, check 37, \mathrm{CZ} on output 5, check 38, \mathrm{CZ} on output 5, check 41, \mathrm{CZ} on output 7, check 34, \mathrm{CZ} on output 7, check 35, \mathrm{CZ} on output 7, check 36, \mathrm{CZ} on output 7, check 37, \mathrm{CZ} on output 7, check 39, \mathrm{CZ} on output 9, check 34, \mathrm{CZ} on output 9, check 35, \mathrm{CZ} on output 9, check 36, \mathrm{CZ} on output 9, check 37, \mathrm{CZ} on output 9, check 39, \mathrm{CZ} on output 10, check 34, \mathrm{CZ} on output 10, check 35, \mathrm{CZ} on output 10, check 36, \mathrm{CZ} on output 10, check 37, \mathrm{CZ} on output 10, check 38, \mathrm{CZ} on output 10, check 41, \mathrm{CZ} on output 11, check 34, \mathrm{CZ} on output 11, check 35, \mathrm{CZ} on output 11, check 36, \mathrm{CZ} on output 11, check 37, \mathrm{CZ} on output 11, check 38, \mathrm{CZ} on output 11, check 41, \mathrm{CZ} on output 12, check 34, \mathrm{CZ} on output 12, check 35, \mathrm{CZ} on output 12, check 36, \mathrm{CZ} on output 12, check 37, \mathrm{CZ} on output 12, check 38, \mathrm{CZ} on output 12, check 41, \mathrm{CZ} on output 13, check 38, \mathrm{CZ} on output 13, check 39, \mathrm{CZ} on output 13, check 41, \mathrm{CZ} on output 15, check 34, \mathrm{CZ} on output 15, check 35, \mathrm{CZ} on output 15, check 36, \mathrm{CZ} on output 15, check 37, \mathrm{CZ} on output 15, check 38, \mathrm{CZ} on output 15, check 41, \mathrm{CZ} on output 17, check 34, \mathrm{CZ} on output 17, check 35, \mathrm{CZ} on output 17, check 36, \mathrm{CZ} on output 17, check 37, \mathrm{CZ} on output 17, check 39, \mathrm{CZ} on output 18, check 38, \mathrm{CZ} on output 18, check 39, \mathrm{CZ} on output 18, check 41, \mathrm{CZ} on output 19, check 34, \mathrm{CZ} on output 19, check 35, \mathrm{CZ} on output 19, check 36, \mathrm{CZ} on output 19, check 37, \mathrm{CZ} on output 19, check 39, \mathrm{CZ} on output 20, check 34, \mathrm{CZ} on output 20, check 35, \mathrm{CZ} on output 20, check 36, \mathrm{CZ} on output 20, check 37, \mathrm{CZ} on output 20, check 39, \mathrm{CZ} on output 21, check 34, \mathrm{CZ} on output 21, check 35, \mathrm{CZ} on output 21, check 36, \mathrm{CZ} on output 21, check 37, \mathrm{CZ} on output 21, check 39, \mathrm{CZ} on output 24, check 34, \mathrm{CZ} on output 24, check 35, \mathrm{CZ} on output 24, check 36, \mathrm{CZ} on output 24, check 37, \mathrm{CZ} on output 24, check 39, \mathrm{CZ} on output 25, check 34, \mathrm{CZ} on output 25, check 35, \mathrm{CZ} on output 25, check 36, \mathrm{CZ} on output 25, check 37, \mathrm{CZ} on output 25, check 38, \mathrm{CZ} on output 25, check 41, \mathrm{CZ} on output 26, check 34, \mathrm{CZ} on output 26, check 35, \mathrm{CZ} on output 26, check 36, \mathrm{CZ} on output 26, check 37, \mathrm{CZ} on output 26, check 38, \mathrm{CZ} on output 26, check 41, \mathrm{CZ} on output 27, check 38, \mathrm{CZ} on output 27, check 39, \mathrm{CZ} on output 27, check 41, \mathrm{CZ} on output 28, check 38, \mathrm{CZ} on output 28, check 39, \mathrm{CZ} on output 28, check 41, \mathrm{CZ} on output 29, check 34, \mathrm{CZ} on output 29, check 35, \mathrm{CZ} on output 29, check 36, \mathrm{CZ} on output 29, check 37, \mathrm{CZ} on output 29, check 38, \mathrm{CZ} on output 29, check 41, \mathrm{CZ} on output 30, check 38, \mathrm{CZ} on output 30, check 39, \mathrm{CZ} on output 30, check 41, \mathrm{CZ} on check 33, check 34, \mathrm{CZ} on check 33, check 35, \mathrm{CZ} on check 33, check 36, \mathrm{CZ} on check 33, check 37, \mathrm{CZ} on check 33, check 39, \mathrm{CZ} on check 34, check 35, \mathrm{CZ} on check 34, check 38, \mathrm{CZ} on check 35, check 36, \mathrm{CZ} on check 35, check 37, \mathrm{CZ} on check 35, check 41, \mathrm{CZ} on check 36, check 38, \mathrm{CZ} on check 37, check 38, \mathrm{CZ} on check 38, check 39, \mathrm{CZ} on check 38, check 40, \mathrm{CZ} on check 39, check 40, \mathrm{CZ} on check 40, check 41
Or no Clifford gate at all: run \mathrm{T}^3 on rotations 0, 9, 12, 13, 14, 16, 23, 26, 27, 37, 38, 40, 41, 44, 57, 59, 63, 64, 65, 66, 67, 73, 75, 76 and 38 more (listed below); \mathrm{T}^5 on rotations 4, 5, 8, 10, 21, 22, 25, 43, 48, 50, 61, 85, 88, 89; \mathrm{T}^\dagger on rotations 1, 2, 3, 6, 11, 15, 45, 46, 86, 96, 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 62 rotations at T³
0, 9, 12, 13, 14, 16, 23, 26, 27, 37, 38, 40, 41, 44, 57, 59, 63, 64, 65, 66, 67, 73, 75, 76, 77, 79, 80, 82, 84, 92, 95, 98, 99, 100, 103, 104, 107, 108, 111, 113, 116, 120, 127, 129, 130, 133, 136, 137, 139, 141, 143, 146, 148, 150, 158, 160, 163, 165, 166, 169, 170, 173
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/176.32.3.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-176.32.3.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 864 faults of weight 3 are undetectable and damaging, so A_{3}=864. 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 [0,13,49].
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/restricted_light_176_32_3.json",
"label": "restricted_light_176_32_3",
"N": 42,
"d": 3,
"d_is_exact": true,
"origin": "Wills-derived restricted-light construction",
"provenance": "gamma frontier release record restricted_light_176_32_3 (source catalogue id restricted_light_176_32_3); construction: Wills-derived restricted-light construction; matrix 6c43caa529aade15...; the release states d=3 (exact), gamma=1.5517285850726805 (exact)",
"notes": "matrix gamma_frontier_release/matrices/6c43caa529aade15983143ffb037a54e3ee556b1bda0b23937ae7c2ae6908556.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/176.32.3.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[808].