Magic State Factory Catalog

[\![208,\,40,\,3]\!] factory

\mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3}\cdot \mathrm{T}_{4}\cdot \mathrm{T}_{5}\cdot \mathrm{T}_{6}\cdot \mathrm{T}_{7}\cdot \mathrm{T}_{8}\cdot \mathrm{T}_{9}\cdot \mathrm{T}_{10}\cdot \mathrm{T}_{11}\cdot \mathrm{T}_{12}\cdot \mathrm{T}_{13}\cdot \mathrm{T}_{14}\cdot \mathrm{T}_{15}\cdot \mathrm{T}_{16}\cdot \mathrm{T}_{17}\cdot \mathrm{T}_{18}\cdot \mathrm{T}_{19}\cdot \mathrm{T}_{20}\cdot \mathrm{T}_{21}\cdot \mathrm{T}_{22}\cdot \mathrm{T}_{23}\cdot \mathrm{T}_{}\cdots

distance, proved here
d=3
error coefficient
A_{3}=398
\gamma_\rho
1.5007
V_{\mathrm{ex}}
40
wires
N=52
Clifford correction
none with T powers
found by
AI search pure T

The circuit

The 52\times208 matrix. Rows are wires and columns are rotations. A 1 means the wire takes part in the rotation, and the number beside each wire is its row weight. Scroll sideways to see every column.

  • output wires
  • check wires, measured and postselected
wire
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
out 0 (91)··1·1···1···1·111·1111···1··11·····1·1···11111··1·11··1·11···1···1···1·111···1·····11··111·1···1111··1·1·······1··1···1·111···1111·1···11·······1·11111·1·11·1··111··1·1·1····111·1···1···11·111·1··111··1·1····
out 1 (91)········11·11·11111·11······11·····1·1···11111··1·11··11·1·····1·11·····11··11·····11··111·1···1111··1·1··········11·111111··11·11·····11·······1·11111·1·11·11··11···1111······1·1··11···11·111·1··111··1·1····
out 2 (89)·1·1···11··1····11111····1·1·1·1·1·1·1···11·1···1······111··11···1··11·1·1·111······1···111···1·11····1···111·1·111··1····1··1111··1··1·1·1·····1·11·1··1·····11111·11·1·1···111··1·111····1··111··1·1····1··1·1
out 3 (61)····1···1··1·1·1···11····1···1······1·····111··1···1····1···1··1·1·····11···1···1··········1······111··1·1·1···1··1··1···1·····11··1····1······1···111·1·1·1····1···1·11··1····11····1···1·······1····111··11··1
out 4 (75)·1·11··11···1·11·1···1·1·1·11··1·1·11··1·1·1······1···1··1·····1····1···1·11·······1···1·11····1·11···11··1·1·11·11···1·11···1···111··11··1····1··1·1·····1··1·1·1····1······1··1··11·····1··1·11···1·1···11·1··
out 5 (47)····1···1········1··1······1········1·1···1·11·1···1····1···1········1··1·····1············1·1····1·11·1···1···1··1··········1··1·····1········1···1·111···1····1···1·········1·1·······1········11···1·11·1···1
out 6 (77)····1···1··11·11111·11···1··1·······1······1····1·11··1111···1·1·1······11···1··1··11··11···1··11···1··1·······1··1··11·111··11·11·1···1·······1····1···1·11·11·111··111·1······1·1···1··111·11·····11··1··1····
out 7 (91)····1···1··11·11111·11···1··1······1·1···11·1···1·11··1111···1·1·1······11···1··1··11··11111···11111···1·······1··1··11·111··11·11·1···1······1·1·11·1··1·11·11·111··111·1······1·1···1··111·11111··1111···1····
out 8 (103)·1·1···111·1···11·111··1·1·1···1·1·111···1·1···11··1···11···1····1·····1·11111··1···1···1111··1·11111·11·11·1·1··111·1···11·1·111·11··1···1····11·1·1··111·1··111·1·1··1·11····1··11111··1·1··1111·1·1111·1111··
out 9 (83)·1·11··1·1·1··1····1·1·1···11··1·1··········1··1··1···1··1··11··1···11·11·111······1···1··11··11··111·1··11·1·11·1·1·1··1······1·11···11··1··········1·1·11··1·1·1··11····1··1111··111····1··1··11·11·111·1·11··
out 10 (93)·1·1···111··1·11···1·1·1·11·11·1·1··1·····1·1·····11··1·11···1·1····11·11·11··1·1··1···1··1···11···11·11··111·1··111··1·11·····1·111·1·11·1····1···1·1····11·1·111···11······1111··11··1111··1··1··11··11·11·1·1
out 11 (77)·1·1···1····1·11···1·1··11·111·1·1··1·····11······11··1·11··1··1····11·11·······1··1···1··1···11······11··111·1··1····1·11·····1·1·1··111·1····1···11·····11·1·111··1·1······1111········11··1··1··11·····11·1·1
out 12 (99)·1·1···111··1·11···1·1·1·1···1·1·1·1·····11111·1··11··1·11···1·1····11·11·11··111··1···1·111··11·1··111···111·1··111··1·11·····1·111····1·1·······111111··11·1·111···11······1111··11···111··1·111·11···111··1·1
out 13 (31)···········1·····1··1······1········1·····1····1······1········1·····1··1·····1············1······1····1·····1·······1·······1··1·····1········1···1···1·····1········1·······1·1·······1········1····1····1····
out 14 (29)·····1··1·····1········1·····1······1·····1··1··············1·············1·····1··········1······1··1············1·····1·········1·····1······1···1··1···········1·1····1·········1·····1·······1····1··1······
out 15 (53)········1··1···1·1·11··1·1·1··············11·1··············1··1·····1·11·1·1·1····1··············11·1···1········1··1···1···1·11·11··1····1·······11·1··1··········1·1···1···111··1·1··1·1···········11·1··1···
out 16 (91)·1··1··11······1111·1····1···1·1····11····1·11·11······1····11·1·1··11···1·1111·····1·1·1·11··1·1····111·1111··1·11······11··11·1··1····1·1····11··1·11111····1···1·1111·11··11···1·111·1··1··1·11·1·1···11111·1
out 17 (91)·1·1···1····1·1··1·111···1·111·1·1··1·····1111····11··1·11··1··1····1··1···1····1··1···1··1···111···1111·1111·1··1····1·1····1·111·1··111·1····1·1·1111··111·1·111··1·1···1··1·1····1····11··1··1··111··111111·1
out 18 (83)·1··········1·1·1·1··1·····111·······1····1111·11·1··111·1··11···1···1···1··111····11··11··1···11·1··1···1·11·········1·1·1···1··1····111·······1··11111111··11··11·11·1·11···1···1··11·1·11·11··1··111··1··1··1
out 19 (61)····1···1······1·1·11····1·1········1·····1111·····1····1···1······1·1·11···1·1············1······1111···1·1···1··1······1·1·1·11··1··1········1···1111··1·1····1···1·····1·1·111····1··1········1····1111··1··1
out 20 (95)·1·1···11······11111·····1·1···111·1·1···11·····1······11···11·1·1··11·111·111··1···11··111···1·11··1·1··1111·1··11······11··111···1··1···1·····1·11····11····111·1·1111·11··1111·1·111··1·11·111··1·1··1·1·11·1
out 21 (61)1···1···1··1···1·1·1·····1·1·1············11·1·1···11···1···1··1·····1·1····1·1·1·················11·1·1·1·1···1··1··1···1···1·1···1··1·1··········11·11·1·11···1···1·1···1···11·····1··11············11·1·11··1
out 22 (83)·1·11··1·1·1····11111··1···1·1·1·1··11····1··11·1······1·········1··11·1·111·1······1···1·11··1·1···11····111·11·1·1·1····1··1111·1···1·1·1····11··1··1·1·····11··1····1·1···111··111·1····1··1·11·1·1··11···1·1
out 23 (91)····1····1·11·111111·1···1·111·····111···1···1·11·····1111··1··1·1·····1·1···11····11··111·····111·111·····1···1···1·11·111··111·1·1··111······11·1···111····11·111·1·11·1·····1··1···1·1·11·111····11·111····11
out 24 (91)·1·1···1····1·11···1·1·1····1··1·1·1·····11··1·1··11··1·11··1··1····11·11·11111····1···1·111··11·1·1·11···111·1··1····1·11·····1·11·1··1··1·······11··11··11·1·111··1·1······1111··1111·1·1··1·111·11··1·11··1·1
out 25 (31)······1·1··············1·1·1·1············1·1·············1·1·············1·1·1·1·················1·1·············1···············11··1·1··········1·1·············11··············1·1··11············1·1·······
out 26 (91)·1·1···111111·11····11·1·1··1··1·1·1·····11··1·1··11··1·11···11·····11····11··1····1···1·111··11·1·1·11···111·1··111111·11······1111···1··1·······11··11··11·1·111···1·······11····11···1·1··1·111·11··1·11··1·1
out 27 (29)···············1·1·1···1················1··1·1·······················1·1··1····················1···1·1···1···············1···1·1··1·················1·1··1················1···11···1················1··1·1··1···
out 28 (59)·········1·1···1·1·····1·1···1·····11····1111··1·············1·1·····1····1·1···1···1····1·1·····1111··1·1·········1·1···1···1····11····1···1··1··1111·1·1···········11···1···1····1·1···1·1···1·1····111··11···
out 29 (47)········1··1········1····1·1·1······1·····1·1··1···1········1··1········1···1·1·1·······1··1······1·1··1···1······1··1··········1··1··1·1····1·1···1·1·1···1········1·1·········1····1··11····1··1····1·1··1···1
out 30 (75)·1·11··1·1·1····1······1···1·1·1·1··11····111···1······1·········1··1·1·1111·1······1···1·11··1·1··1··1···111·11·1·1·1····1·······1···1·1·1····11··111··1·····11··1····1·1···1··1·111·1····1··1·11·1·1·1··1··1·1
out 31 (43)····1···1······1····11···1···1·····1·····1···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·····11···1···1·····1·········1··1··1
out 32 (45)····1···1······1···1··1··1·1········1······1···1···1····1···1··········1·1··1·1············1·······1···1·1·1···1··1······1·····1···1··1········1····1··1·1·1····1···1·····1····1··1··1··1········1·····1···11··1
out 33 (31)····1···1·······1······1···1········1·······11··········1···1·············1···1············1········11·········1··1·······1·······1···1········1·····11·········1···1······1·······1····1········1······11······
out 34 (45)·······1·1·············1·1·········11····11111·1···········1·1············1·1············1·1·····11111·1·········1·1··············11···········1··111111·············1·············1·1·········1·1····1111·1····
out 35 (83)·1·11··1········111·1··1·1·····1·11··1······1··11··1···1·····1·1·1··11···111111·1···1··11·1···1·1·1···11··1·1·11·1········1··11·1·11······1·····1····1·11··1··11··1··111·1···11···11111·11·1·11·1··1·11···11·1··
out 36 (47)····1······1·····1·······1·1·11·····1·····1··1·1···1····1······1·····1······1·1·11·········1······1··1·1···1···1·····1·······1·····1··1·11·····1···1··11···1····1·····1·······1······1··11·······1····1··1·1···1
out 37 (45)····1···1······1···11··1···1········1······1·1···1······1···1··········11·1···1············1·······1·1···11····1··1······1·····11·1···1········1····1·1··1······1···1·····1····11··1····1········1·····1·1··11··
out 38 (71)····1···11·1·····1·11····1·1·1·1····1·····1111·1···1····1···11·1·····1·11···1·1·1·1········1······1111·1···1···1··11·1·······1·11··1··1·1·1····1···11111···1····1···111·······111····1··11·······1····1111·1···1
out 39 (89)·1·11··1·1·1····11111··1·······1·1·1·1···111·1·········1·········1··11·1·111·11·1···1···111···1·11·1111·1·111·11·1·1·1····1··1111·1·······1·····1·111·1·······11··1····1·1···111··111·1·11·1··111··1·1·1111··1·1
check 40 (122)·1·····1··1·1··11·11·1···11·1111·1·1·1······11··1·1·1·11111·11·1·11111··1111·11····111·11·11·1111111··11·1111····1··1·1··11···11·1·1·1·1111·····1····11·1·1·11111111111111··111·1·111·1·1·11111·11111111··1111·1
check 41 (82)···11···111·1·11·11···········11···1·1····1·11·1··1······1··11·1·1·11·····1111·····11·····11·····1·1·1·1··11··11··111·1·11···11··········11·····1··1·111·11······11·1111·11·11·····1111···11····11·····1·1·1·1·1
check 42 (82)···1···1·1111··1···111·······1···1·1·1···1·11··11·11······1·1····1·······11111··1····1·11·11·····1·1111···1···1··1·1111··1·····111······1·······1·1·11·11111···1···11··1··1·······11111··1··111·11·····1111··1··
check 43 (62)··············1·111·11···················1··1··11·1···············111··1·11111···············1111·11·11···11············1·1··11·11················1··1·1111·············1·1·11·1··11111···········111111·11··1·1
check 44 (92)···1····11·1···1···1···1·11·1111·1··11···11·11··1·11········11·1·······1··1·1·1····111·1111··1111··1··11··1···1···11·1···1·····1··11·1·1111····11·11·11·1111···1····111···1····1···1·1··1·1111111·1111·1··11·1··
check 45 (124)111··111··111111·11···1·1··11·1111111111111·111111··11·1··11··1111···11··1·1··11·11··111·1·11111·11·1111111··1111·11···1··111··1111111··1··1111··1··1·····11·1·111··11··1··111·111·1·111·111··1·1····1·1······11
check 46 (106)··1111···1·111···111111·11·11·······11·1···1·11·111··111·1·111···1··1·11·1········11··11··1·1··1··1··1·111··1·1··1····1·1·1·11·1·1·11··11·1·1111·1·1111··111·1·111··1·1···11···11···1·1·1111·11····111··111111·1
check 47 (116)·111·1·1···1111·1··11·11···111·1·1111···11···1·1········1····1··1···1·11··11·1·1····1··111111·11111·11111·11111·1111·1··11···11111111·1···1111··111·111·1·11·111111·1··1··11·111·11111···11····11··1··11·11··11·
check 48 (84)····11····111111·1111··11·····11···1·1···11··1·1·1······1·····1111···1111·11·····11······1··1····11··1·1·11···1·1·······11·11··11··11······11·11····1111··11···1·1·······11·11·11····11···11····11·····111·1··11
check 49 (100)·1·11·1·111111··11111·111··1·1·11·11······1··11··1·····1··111·1·1···11·1··1··1······1111111·1·1·1···11··1··11·11·1········1··11··1·11·1·1·1··1111··1···11111··11··1··111·1···11·111·11·····1·····1·1·1··1·11111·
check 50 (96)·1·11··1·1·1···111····1·11·····1·1·1·1111··11·11·1·1···1·········1··111·1···111·1···1···111··1·11·11···111··1·11·1·1·1·11··11·11111·1·····1·····1111··1·1·1···11··1····11··11··1·1111···11·1··111··1····111··111
check 51 (100)········1···1·111·1·11········11111·1·111·1··11··11···11·1···1···1···1··11··11·11111111·1···111·1·1111111·11······1···1·111···1·11···1111··1111·111·11·111·1·11··11··1·1·1····1·1·1··1111····1·111··1·······1·1·

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 n208Noisy T states per run.
outputs k40
distance d3Exact, proved and witnessed here.
error coefficient A_{3}398Faults of weight 3 in the circuit shown that pass every check and damage the output. The output error is A_{3}\,p^{3} to leading order.
wires N52
checks r12
V_{\mathrm{ex}}40T states per run. CCZ counts as two.
\gamma_\rho = \log(n/V_{\mathrm{ex}})/\log d1.5007Overhead per extractable T state. Lower is better.
\gamma = \log(n/k)/\log d1.5007Overhead 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 width40Rank of the output rows modulo the check rows.

Output gate

40 T0 CS0 CCZon 40 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}

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 (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 511 \mathrm{CZ} gates: \mathrm{S}^\dagger on output 0, \mathrm{S}^\dagger on output 1, \mathrm{Z} on output 3, \mathrm{S}^\dagger on output 4, \mathrm{S} on output 5, \mathrm{Z} on output 6, \mathrm{S}^\dagger on output 7, \mathrm{S} on output 8, \mathrm{S}^\dagger on output 9, \mathrm{Z} on output 10, \mathrm{Z} on output 11, \mathrm{S}^\dagger on output 12, \mathrm{S} on output 13, \mathrm{Z} on output 14, \mathrm{Z} on output 15, \mathrm{S}^\dagger on output 16, \mathrm{S}^\dagger on output 17, \mathrm{S}^\dagger on output 18, \mathrm{Z} on output 19, \mathrm{S} on output 20, \mathrm{Z} on output 21, \mathrm{S}^\dagger on output 22, \mathrm{S}^\dagger on output 23, \mathrm{S}^\dagger on output 24 and 536 more (listed below). The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.

All 560 correction gates

\mathrm{S}^\dagger on output 0, \mathrm{S}^\dagger on output 1, \mathrm{Z} on output 3, \mathrm{S}^\dagger on output 4, \mathrm{S} on output 5, \mathrm{Z} on output 6, \mathrm{S}^\dagger on output 7, \mathrm{S} on output 8, \mathrm{S}^\dagger on output 9, \mathrm{Z} on output 10, \mathrm{Z} on output 11, \mathrm{S}^\dagger on output 12, \mathrm{S} on output 13, \mathrm{Z} on output 14, \mathrm{Z} on output 15, \mathrm{S}^\dagger on output 16, \mathrm{S}^\dagger on output 17, \mathrm{S}^\dagger on output 18, \mathrm{Z} on output 19, \mathrm{S} on output 20, \mathrm{Z} on output 21, \mathrm{S}^\dagger on output 22, \mathrm{S}^\dagger on output 23, \mathrm{S}^\dagger on output 24, \mathrm{S} on output 25, \mathrm{S}^\dagger on output 26, \mathrm{Z} on output 27, \mathrm{S}^\dagger on output 28, \mathrm{S} on output 29, \mathrm{S}^\dagger on output 30, \mathrm{S}^\dagger on output 31, \mathrm{Z} on output 32, \mathrm{S} on output 33, \mathrm{Z} on output 34, \mathrm{S}^\dagger on output 35, \mathrm{S} on output 36, \mathrm{Z} on output 37, \mathrm{S} on output 38, \mathrm{S}^\dagger on check 40, \mathrm{S}^\dagger on check 41, \mathrm{S}^\dagger on check 42, \mathrm{S} on check 43, \mathrm{Z} on check 44, \mathrm{Z} on check 45, \mathrm{S}^\dagger on check 46, \mathrm{Z} on check 47, \mathrm{Z} on check 48, \mathrm{Z} on check 49, \mathrm{Z} on check 51, \mathrm{CZ} on output 0, output 4, \mathrm{CZ} on output 0, output 6, \mathrm{CZ} on output 0, output 12, \mathrm{CZ} on output 0, output 14, \mathrm{CZ} on output 0, output 16, \mathrm{CZ} on output 0, output 18, \mathrm{CZ} on output 0, output 22, \mathrm{CZ} on output 0, output 24, \mathrm{CZ} on output 0, output 26, \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 34, \mathrm{CZ} on output 0, output 35, \mathrm{CZ} on output 0, check 42, \mathrm{CZ} on output 0, check 45, \mathrm{CZ} on output 0, check 46, \mathrm{CZ} on output 0, check 47, \mathrm{CZ} on output 0, check 48, \mathrm{CZ} on output 0, check 49, \mathrm{CZ} on output 1, output 4, \mathrm{CZ} on output 1, output 6, \mathrm{CZ} on output 1, output 12, \mathrm{CZ} on output 1, output 14, \mathrm{CZ} on output 1, output 16, \mathrm{CZ} on output 1, output 18, \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 28, \mathrm{CZ} on output 1, output 30, \mathrm{CZ} on output 1, output 31, \mathrm{CZ} on output 1, output 34, \mathrm{CZ} on output 1, output 35, \mathrm{CZ} on output 1, check 42, \mathrm{CZ} on output 1, check 45, \mathrm{CZ} on output 1, check 46, \mathrm{CZ} on output 1, check 47, \mathrm{CZ} on output 1, check 48, \mathrm{CZ} on output 1, check 49, \mathrm{CZ} on output 2, output 6, \mathrm{CZ} on output 2, output 8, \mathrm{CZ} on output 2, output 9, \mathrm{CZ} on output 2, output 10, \mathrm{CZ} on output 2, output 11, \mathrm{CZ} on output 2, output 14, \mathrm{CZ} on output 2, output 16, \mathrm{CZ} on output 2, output 17, \mathrm{CZ} on output 2, output 18, \mathrm{CZ} on output 2, output 20, \mathrm{CZ} on output 2, output 28, \mathrm{CZ} on output 2, output 31, \mathrm{CZ} on output 2, output 34, \mathrm{CZ} on output 2, output 39, \mathrm{CZ} on output 2, check 40, \mathrm{CZ} on output 2, check 44, \mathrm{CZ} on output 2, check 45, \mathrm{CZ} on output 2, check 48, \mathrm{CZ} on output 2, check 50, \mathrm{CZ} on output 3, output 8, \mathrm{CZ} on output 3, output 9, \mathrm{CZ} on output 3, output 15, \mathrm{CZ} on output 3, output 16, \mathrm{CZ} on output 3, output 17, \mathrm{CZ} on output 3, output 18, \mathrm{CZ} on output 3, output 19, \mathrm{CZ} on output 3, output 20, \mathrm{CZ} on output 3, output 21, \mathrm{CZ} on output 3, output 27, \mathrm{CZ} on output 3, output 28, \mathrm{CZ} on output 3, output 31, \mathrm{CZ} on output 3, output 32, \mathrm{CZ} on output 3, output 37, \mathrm{CZ} on output 3, check 41, \mathrm{CZ} on output 3, check 42, \mathrm{CZ} on output 3, check 43, \mathrm{CZ} on output 3, check 44, \mathrm{CZ} on output 3, check 45, \mathrm{CZ} on output 3, check 46, \mathrm{CZ} on output 3, check 48, \mathrm{CZ} on output 3, check 50, \mathrm{CZ} on output 4, output 7, \mathrm{CZ} on output 4, output 9, \mathrm{CZ} on output 4, output 10, \mathrm{CZ} on output 4, output 11, \mathrm{CZ} on output 4, output 17, \mathrm{CZ} on output 4, output 22, \mathrm{CZ} on output 4, output 23, \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, output 34, \mathrm{CZ} on output 4, output 35, \mathrm{CZ} on output 4, check 41, \mathrm{CZ} on output 4, check 44, \mathrm{CZ} on output 4, check 46, \mathrm{CZ} on output 4, check 49, \mathrm{CZ} on output 4, check 51, \mathrm{CZ} on output 6, output 7, \mathrm{CZ} on output 6, output 8, \mathrm{CZ} on output 6, output 14, \mathrm{CZ} on output 6, output 16, \mathrm{CZ} on output 6, output 18, \mathrm{CZ} on output 6, output 20, \mathrm{CZ} on output 6, output 22, \mathrm{CZ} on output 6, output 23, \mathrm{CZ} on output 6, output 30, \mathrm{CZ} on output 6, output 35, \mathrm{CZ} on output 6, output 39, \mathrm{CZ} on output 6, check 40, \mathrm{CZ} on output 6, check 41, \mathrm{CZ} on output 6, check 45, \mathrm{CZ} on output 6, check 46, \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 51, \mathrm{CZ} on output 7, output 12, \mathrm{CZ} on output 7, output 14, \mathrm{CZ} on output 7, output 16, \mathrm{CZ} on output 7, output 18, \mathrm{CZ} on output 7, output 22, \mathrm{CZ} on output 7, output 24, \mathrm{CZ} on output 7, output 26, \mathrm{CZ} on output 7, output 28, \mathrm{CZ} on output 7, output 30, \mathrm{CZ} on output 7, output 31, \mathrm{CZ} on output 7, output 34, \mathrm{CZ} on output 7, output 35, \mathrm{CZ} on output 7, check 42, \mathrm{CZ} on output 7, check 45, \mathrm{CZ} on output 7, check 46, \mathrm{CZ} on output 7, check 47, \mathrm{CZ} on output 7, check 48, \mathrm{CZ} on output 7, check 49, \mathrm{CZ} on output 8, output 10, \mathrm{CZ} on output 8, output 11, \mathrm{CZ} on output 8, output 14, \mathrm{CZ} on output 8, output 15, \mathrm{CZ} on output 8, output 19, \mathrm{CZ} on output 8, output 21, \mathrm{CZ} on output 8, output 27, \mathrm{CZ} on output 8, output 32, \mathrm{CZ} on output 8, output 34, \mathrm{CZ} on output 8, output 37, \mathrm{CZ} on output 8, output 39, \mathrm{CZ} on output 8, check 40, \mathrm{CZ} on output 8, check 41, \mathrm{CZ} on output 8, check 42, \mathrm{CZ} on output 8, check 43, \mathrm{CZ} on output 8, check 46, \mathrm{CZ} on output 9, output 10, \mathrm{CZ} on output 9, output 11, \mathrm{CZ} on output 9, output 12, \mathrm{CZ} on output 9, output 15, \mathrm{CZ} on output 9, output 16, \mathrm{CZ} on output 9, output 18, \mathrm{CZ} on output 9, output 19, \mathrm{CZ} on output 9, output 21, \mathrm{CZ} on output 9, output 22, \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 35, \mathrm{CZ} on output 9, output 37, \mathrm{CZ} on output 9, output 39, \mathrm{CZ} on output 9, check 40, \mathrm{CZ} on output 9, check 41, \mathrm{CZ} on output 9, check 43, \mathrm{CZ} on output 9, check 45, \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 10, output 11, \mathrm{CZ} on output 10, output 12, \mathrm{CZ} on output 10, output 17, \mathrm{CZ} on output 10, output 20, \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 30, \mathrm{CZ} on output 10, output 35, \mathrm{CZ} on output 10, output 39, \mathrm{CZ} on output 10, check 40, \mathrm{CZ} on output 10, check 42, \mathrm{CZ} on output 10, check 44, \mathrm{CZ} on output 10, check 46, \mathrm{CZ} on output 10, check 47, \mathrm{CZ} on output 10, check 49, \mathrm{CZ} on output 10, check 50, \mathrm{CZ} on output 11, output 12, \mathrm{CZ} on output 11, output 17, \mathrm{CZ} on output 11, output 20, \mathrm{CZ} on output 11, output 22, \mathrm{CZ} on output 11, output 24, \mathrm{CZ} on output 11, output 26, \mathrm{CZ} on output 11, output 30, \mathrm{CZ} on output 11, output 35, \mathrm{CZ} on output 11, output 39, \mathrm{CZ} on output 11, check 40, \mathrm{CZ} on output 11, check 42, \mathrm{CZ} on output 11, check 44, \mathrm{CZ} on output 11, check 46, \mathrm{CZ} on output 11, check 47, \mathrm{CZ} on output 11, check 49, \mathrm{CZ} on output 11, check 50, \mathrm{CZ} on output 12, output 17, \mathrm{CZ} on output 12, output 22, \mathrm{CZ} on output 12, output 23, \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 34, \mathrm{CZ} on output 12, output 35, \mathrm{CZ} on output 12, check 41, \mathrm{CZ} on output 12, check 44, \mathrm{CZ} on output 12, check 46, \mathrm{CZ} on output 12, check 49, \mathrm{CZ} on output 12, check 51, \mathrm{CZ} on output 14, output 16, \mathrm{CZ} on output 14, output 18, \mathrm{CZ} on output 14, output 20, \mathrm{CZ} on output 14, output 22, \mathrm{CZ} on output 14, output 23, \mathrm{CZ} on output 14, output 30, \mathrm{CZ} on output 14, output 35, \mathrm{CZ} on output 14, output 39, \mathrm{CZ} on output 14, check 40, \mathrm{CZ} on output 14, check 41, \mathrm{CZ} on output 14, check 45, \mathrm{CZ} on output 14, check 46, \mathrm{CZ} on output 14, check 48, \mathrm{CZ} on output 14, check 49, \mathrm{CZ} on output 14, check 50, \mathrm{CZ} on output 14, check 51, \mathrm{CZ} on output 15, output 16, \mathrm{CZ} on output 15, output 17, \mathrm{CZ} on output 15, output 18, \mathrm{CZ} on output 15, output 19, \mathrm{CZ} on output 15, output 20, \mathrm{CZ} on output 15, output 21, \mathrm{CZ} on output 15, output 27, \mathrm{CZ} on output 15, output 28, \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, check 41, \mathrm{CZ} on output 15, check 42, \mathrm{CZ} on output 15, check 43, \mathrm{CZ} on output 15, check 44, \mathrm{CZ} on output 15, check 45, \mathrm{CZ} on output 15, check 46, \mathrm{CZ} on output 15, check 48, \mathrm{CZ} on output 15, check 50, \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 22, \mathrm{CZ} on output 16, output 23, \mathrm{CZ} on output 16, output 27, \mathrm{CZ} on output 16, output 28, \mathrm{CZ} on output 16, output 30, \mathrm{CZ} on output 16, output 31, \mathrm{CZ} on output 16, output 32, \mathrm{CZ} on output 16, output 35, \mathrm{CZ} on output 16, output 37, \mathrm{CZ} on output 16, output 39, \mathrm{CZ} on output 16, check 40, \mathrm{CZ} on output 16, check 42, \mathrm{CZ} on output 16, check 43, \mathrm{CZ} on output 16, check 44, \mathrm{CZ} on output 16, check 49, \mathrm{CZ} on output 16, check 51, \mathrm{CZ} on output 17, output 18, \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 32, \mathrm{CZ} on output 17, output 35, \mathrm{CZ} on output 17, output 37, \mathrm{CZ} on output 17, output 39, \mathrm{CZ} on output 17, check 40, \mathrm{CZ} on output 17, check 41, \mathrm{CZ} on output 17, check 43, \mathrm{CZ} on output 17, check 45, \mathrm{CZ} on output 17, check 47, \mathrm{CZ} on output 17, check 48, \mathrm{CZ} on output 17, check 49, \mathrm{CZ} on output 18, output 19, \mathrm{CZ} on output 18, output 21, \mathrm{CZ} on output 18, output 22, \mathrm{CZ} on output 18, output 23, \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, output 35, \mathrm{CZ} on output 18, output 37, \mathrm{CZ} on output 18, output 39, \mathrm{CZ} on output 18, check 40, \mathrm{CZ} on output 18, check 42, \mathrm{CZ} on output 18, check 43, \mathrm{CZ} on output 18, check 44, \mathrm{CZ} on output 18, check 49, \mathrm{CZ} on output 18, check 51, \mathrm{CZ} on output 19, output 20, \mathrm{CZ} on output 19, output 21, \mathrm{CZ} on output 19, output 27, \mathrm{CZ} on output 19, output 28, \mathrm{CZ} on output 19, output 31, \mathrm{CZ} on output 19, output 32, \mathrm{CZ} on output 19, output 37, \mathrm{CZ} on output 19, check 41, \mathrm{CZ} on output 19, check 42, \mathrm{CZ} on output 19, check 43, \mathrm{CZ} on output 19, check 44, \mathrm{CZ} on output 19, check 45, \mathrm{CZ} on output 19, check 46, \mathrm{CZ} on output 19, check 48, \mathrm{CZ} on output 19, check 50, \mathrm{CZ} on output 20, output 21, \mathrm{CZ} on output 20, output 27, \mathrm{CZ} on output 20, output 32, \mathrm{CZ} on output 20, output 34, \mathrm{CZ} on output 20, output 37, \mathrm{CZ} on output 20, output 39, \mathrm{CZ} on output 20, check 40, \mathrm{CZ} on output 20, check 41, \mathrm{CZ} on output 20, check 42, \mathrm{CZ} on output 20, check 43, \mathrm{CZ} on output 20, check 46, \mathrm{CZ} on output 21, output 27, \mathrm{CZ} on output 21, output 28, \mathrm{CZ} on output 21, output 31, \mathrm{CZ} on output 21, output 32, \mathrm{CZ} on output 21, output 37, \mathrm{CZ} on output 21, check 41, \mathrm{CZ} on output 21, check 42, \mathrm{CZ} on output 21, check 43, \mathrm{CZ} on output 21, check 44, \mathrm{CZ} on output 21, check 45, \mathrm{CZ} on output 21, check 46, \mathrm{CZ} on output 21, check 48, \mathrm{CZ} on output 21, check 50, \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, check 41, \mathrm{CZ} on output 22, check 42, \mathrm{CZ} on output 22, check 44, \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 51, \mathrm{CZ} on output 23, output 24, \mathrm{CZ} on output 23, output 26, \mathrm{CZ} on output 23, output 28, \mathrm{CZ} on output 23, output 30, \mathrm{CZ} on output 23, output 31, \mathrm{CZ} on output 23, output 34, \mathrm{CZ} on output 23, output 35, \mathrm{CZ} on output 23, check 42, \mathrm{CZ} on output 23, check 45, \mathrm{CZ} on output 23, check 46, \mathrm{CZ} on output 23, check 47, \mathrm{CZ} on output 23, check 48, \mathrm{CZ} on output 23, check 49, \mathrm{CZ} on output 24, output 28, \mathrm{CZ} on output 24, output 30, \mathrm{CZ} on output 24, output 31, \mathrm{CZ} on output 24, output 34, \mathrm{CZ} on output 24, output 35, \mathrm{CZ} on output 24, check 41, \mathrm{CZ} on output 24, check 44, \mathrm{CZ} on output 24, check 46, \mathrm{CZ} on output 24, check 49, \mathrm{CZ} on output 24, check 51, \mathrm{CZ} on output 26, output 28, \mathrm{CZ} on output 26, output 30, \mathrm{CZ} on output 26, output 31, \mathrm{CZ} on output 26, output 34, \mathrm{CZ} on output 26, output 35, \mathrm{CZ} on output 26, check 41, \mathrm{CZ} on output 26, check 44, \mathrm{CZ} on output 26, check 46, \mathrm{CZ} on output 26, check 49, \mathrm{CZ} on output 26, check 51, \mathrm{CZ} on output 27, output 28, \mathrm{CZ} on output 27, output 31, \mathrm{CZ} on output 27, output 32, \mathrm{CZ} on output 27, output 37, \mathrm{CZ} on output 27, check 41, \mathrm{CZ} on output 27, check 42, \mathrm{CZ} on output 27, check 43, \mathrm{CZ} on output 27, check 44, \mathrm{CZ} on output 27, check 45, \mathrm{CZ} on output 27, check 46, \mathrm{CZ} on output 27, check 48, \mathrm{CZ} on output 27, check 50, \mathrm{CZ} on output 28, output 32, \mathrm{CZ} on output 28, output 34, \mathrm{CZ} on output 28, output 37, \mathrm{CZ} on output 28, output 39, \mathrm{CZ} on output 28, check 40, \mathrm{CZ} on output 28, check 43, \mathrm{CZ} on output 28, check 44, \mathrm{CZ} on output 28, check 45, \mathrm{CZ} on output 28, check 46, \mathrm{CZ} on output 28, check 47, \mathrm{CZ} on output 28, check 48, \mathrm{CZ} on output 28, check 51, \mathrm{CZ} on output 30, check 41, \mathrm{CZ} on output 30, check 42, \mathrm{CZ} on output 30, check 44, \mathrm{CZ} on output 30, check 45, \mathrm{CZ} on output 30, check 47, \mathrm{CZ} on output 30, check 48, \mathrm{CZ} on output 30, check 51, \mathrm{CZ} on output 31, output 32, \mathrm{CZ} on output 31, output 34, \mathrm{CZ} on output 31, output 37, \mathrm{CZ} on output 31, output 39, \mathrm{CZ} on output 31, check 40, \mathrm{CZ} on output 31, check 43, \mathrm{CZ} on output 31, check 44, \mathrm{CZ} on output 31, check 45, \mathrm{CZ} on output 31, check 46, \mathrm{CZ} on output 31, check 47, \mathrm{CZ} on output 31, check 48, \mathrm{CZ} on output 31, check 51, \mathrm{CZ} on output 32, output 37, \mathrm{CZ} on output 32, check 41, \mathrm{CZ} on output 32, check 42, \mathrm{CZ} on output 32, check 43, \mathrm{CZ} on output 32, check 44, \mathrm{CZ} on output 32, check 45, \mathrm{CZ} on output 32, check 46, \mathrm{CZ} on output 32, check 48, \mathrm{CZ} on output 32, check 50, \mathrm{CZ} on output 34, output 39, \mathrm{CZ} on output 34, check 40, \mathrm{CZ} on output 34, check 41, \mathrm{CZ} on output 34, check 42, \mathrm{CZ} on output 34, check 47, \mathrm{CZ} on output 34, check 50, \mathrm{CZ} on output 34, check 51, \mathrm{CZ} on output 35, check 41, \mathrm{CZ} on output 35, check 42, \mathrm{CZ} on output 35, check 44, \mathrm{CZ} on output 35, check 45, \mathrm{CZ} on output 35, check 47, \mathrm{CZ} on output 35, check 48, \mathrm{CZ} on output 35, check 51, \mathrm{CZ} on output 37, check 41, \mathrm{CZ} on output 37, check 42, \mathrm{CZ} on output 37, check 43, \mathrm{CZ} on output 37, check 44, \mathrm{CZ} on output 37, check 45, \mathrm{CZ} on output 37, check 46, \mathrm{CZ} on output 37, check 48, \mathrm{CZ} on output 37, check 50, \mathrm{CZ} on output 39, check 40, \mathrm{CZ} on output 39, check 44, \mathrm{CZ} on output 39, check 45, \mathrm{CZ} on output 39, check 48, \mathrm{CZ} on output 39, check 50, \mathrm{CZ} on check 40, check 44, \mathrm{CZ} on check 40, check 45, \mathrm{CZ} on check 40, check 48, \mathrm{CZ} on check 40, check 50, \mathrm{CZ} on check 41, check 43, \mathrm{CZ} on check 41, check 44, \mathrm{CZ} on check 41, check 47, \mathrm{CZ} on check 41, check 49, \mathrm{CZ} on check 41, check 50, \mathrm{CZ} on check 42, check 43, \mathrm{CZ} on check 42, check 45, \mathrm{CZ} on check 42, check 48, \mathrm{CZ} on check 42, check 49, \mathrm{CZ} on check 42, check 50, \mathrm{CZ} on check 42, check 51, \mathrm{CZ} on check 43, check 44, \mathrm{CZ} on check 43, check 45, \mathrm{CZ} on check 43, check 46, \mathrm{CZ} on check 43, check 48, \mathrm{CZ} on check 43, check 50, \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 45, check 49, \mathrm{CZ} on check 45, check 51, \mathrm{CZ} on check 46, check 47, \mathrm{CZ} on check 46, check 50, \mathrm{CZ} on check 46, check 51, \mathrm{CZ} on check 47, check 49, \mathrm{CZ} on check 47, check 51, \mathrm{CZ} on check 48, check 49, \mathrm{CZ} on check 48, check 51, \mathrm{CZ} on check 49, check 51

Or no Clifford gate at all: run \mathrm{T}^3 on rotations 0, 1, 5, 10, 11, 13, 23, 27, 28, 30, 37, 38, 52, 60, 62, 66, 68, 70, 71, 73, 74, 75, 76, 81 and 61 more (listed below); \mathrm{T}^5 on rotations 3, 4, 8, 9, 20, 21, 26, 29, 34, 39, 109; \mathrm{T}^\dagger on rotations 2, 6, 14, 17, 18, 22, 25, 31, 32, 33, 35, 45, 54, 78, 111, 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 85 rotations at T³

0, 1, 5, 10, 11, 13, 23, 27, 28, 30, 37, 38, 52, 60, 62, 66, 68, 70, 71, 73, 74, 75, 76, 81, 83, 85, 90, 91, 92, 95, 96, 98, 100, 101, 102, 104, 113, 114, 116, 118, 119, 121, 123, 126, 128, 129, 130, 132, 137, 140, 141, 143, 144, 145, 147, 148, 150, 151, 153, 154, 156, 159, 160, 161, 162, 166, 168, 169, 170, 171, 174, 175, 176, 178, 179, 183, 188, 189, 195, 197, 199, 200, 201, 203, 205

Rotations are numbered by matrix column, from 0. The correction comes from the columns by the weight expansion of Bravyi and Haah (2012). verify_catalog.py re-derives it, checks the corrected circuit's phase input by input against the gate, and on small circuits simulates it gate by gate.

Convert the output with CNOT and S

A CNOT circuit on the outputs, followed by S, Z and CZ gates, turns this factory's gate into other gates that prepare the same magic state. Type a target gate to check whether it is reachable. If it is, you get the circuit, checked on every basis state.

Or list every gate it reaches. The list is built in your browser and saved as a text file, one gate per line.

Python code that writes the same list
import json, urllib.request

NAMES = {1: "T", 2: "CS", 3: "CCZ"}


def equivalent_gates(k, terms, limit=1_000_000):
    """Every gate a CNOT circuit and S, Z, CZ gates reach from this one.

    A gate is its set of terms mod Clifford: {a} is T, {a,b} is CS and
    {a,b,c} is CCZ on those outputs. CNOT c->t sends x_t to
    x_c + x_t - 2 x_c x_t, so a term on t but not c adds the term with t
    replaced by c, and, below CCZ, the term with c added.
    """
    start = frozenset(frozenset(m) for m in terms)
    seen, queue = {start}, [start]
    for gate in queue:
        for c in range(k):
            for t in range(k):
                if c == t:
                    continue
                new = set(gate)
                for m in gate:
                    if t in m and c not in m:
                        new ^= {(m - {t}) | {c}}
                        if len(m) < 3:
                            new ^= {m | {c}}
                new = frozenset(new)
                if new not in seen:
                    seen.add(new)
                    queue.append(new)
                    if len(seen) > limit:
                        raise RuntimeError(f"more than {limit} equivalent gates")
    return seen


def name(gate, k):
    sep = "," if k > 10 else ""
    terms = sorted((sorted(m) for m in gate), key=lambda m: (len(m), m))
    return "·".join(NAMES[len(m)] + sep.join(map(str, m)) for m in terms) or "I"


url = "https://shubhamj810.github.io/magic-state-factory-repo/data/factories/208.40.3.a.json"
record = json.load(urllib.request.urlopen(url))
k = record["parameters"]["k"]
gates = equivalent_gates(k, record["gate"]["monomials"])
lines = sorted(name(g, k) for g in gates)
print(len(lines), "equivalent gates")
with open("equivalent-gates-208.40.3.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 398 faults of weight 3 are undetectable and damaging, so A_{3}=398. If each rotation fails independently with probability p, an accepted run has a damaged output with probability A_{3}\,p^{3}+O(p^{4}).

A damaging fault of weight 3 sits on columns [0,21,73].

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_208_40_3_2d3cd988f8bb.json",
  "label": "paired_column_contraction_208_40_3_2d3cd988f8bb",
  "N": 52,
  "d": 3,
  "d_is_exact": true,
  "origin": "Existing triorthogonal code with physical paired-column contraction",
  "provenance": "gamma frontier release record paired_column_contraction_208_40_3_2d3cd988f8bb (source catalogue id paired_column_contraction_208_40_3_2d3cd988f8bb); construction: Existing triorthogonal code with physical paired-column contraction; matrix 2d3cd988f8bb6d47...; the release states d=3 (exact), gamma=1.5006737523263227 (exact)",
  "notes": "matrix gamma_frontier_release/matrices/2d3cd988f8bb6d4745649dec2dfec050acf0a0d6251800c6bbf958142e921de5.matrix"
 }
]

Use it in code

Load this factory and re-derive every number from its circuit with the catalogue's own verifier.

# git clone https://github.com/shubhamj810/magic-state-factory-repo
# cd magic-state-factory-repo/master_catalog
import json, urllib.request
import verify_catalog as VC

url = "https://shubhamj810.github.io/magic-state-factory-repo/data/factories/208.40.3.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[846].