Magic State Factory Catalog

[\![204,\,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}=1{,}250
\gamma_\rho
1.4830
V_{\mathrm{ex}}
40
wires
N=50
Clifford correction
none with T powers
found by
AI search pure T

The circuit

The 50\times204 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 (73)·1····1···11··111·········1·11·1·1·1······1·····1···1·1·1····1·1·111·11····1··1····1···1·1···1111·111········1···11··11··1·····1···1·11·····1····1····1··1·11··1··11····11111·····1··1··1··1·1··111·111····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 (81)1······111·1··1111··1····1···1··········11···1··11···1111··1······11·1··1·····1···11·1·1·1··1·1··111·····1··1·111·1··11··11··1·1··1···1········11··1·111··1·1111·1···11··111···1·····1·111·1·1·1·1·111·····1
out 3 (59)···1···1··1·1·1···11·······1······1·····111··1···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···1·11··1····11····1···1·······1····111··11··1
out 4 (79)1·····1·11·······1··1·1··11·1··1··11·1·11111·1······1·····11···1·1····1···11······1··1·1·1···1··11·11····1···1·11·········1··111··11·1·····11·111111·1···1·1··1·11·1·11·1···1····11····1·1·1·1··1·11·11····1
out 5 (83)······1···1··1···11·1····1·111·1··1·1··1·11··11·······111·11·1···111··11····1·1··111·11··1·1····1··11········1···1··1·····11·1····1·111····1··1·11·11······11111111··1··111·1·1·····1111111··11···1··11·····
out 6 (59)······1····1··11··1·······1111·1··1····11··1····1···111·1······1·11···1····1·······1······1··11·····1········1····1··11····1···1···1111····1··11··1···1··1111··1···1····11··1·····1·····1·······11····1····1
out 7 (73)······1····1··11··1·······1111·1·1·1·····111····1···111·1······1·11···1····1·······1···111···111111·1········1····1··11····1···1···1111···1·1···111···1··1111··1···1····11··1·····1·····1··111··11111·1····1
out 8 (91)1··1··1··111·1·11·11··1··1··1··1··1····11111··1·11···111···1·1···11··1·1······1··1····11····1·1·1·111··1·1·1·1··111·1·1·11·11·11··1··1·····1··11111·1·11··1111···11·····11·1··1······11···11···1·11·111·1··1
out 9 (77)1··1·····1·1·11····1··1··111·1···1·1···1·1····1··1··1··11··1··11··11·111···1··1··1·1··1·····11·11··1···1·1·1····1·1·11······1·1···111·1·····1·1·1···1··1·1···1·1·1·1·····1111·1···1··11·1·1····11·1··1··1···
out 10 (89)1······1·11··1·····1··1·1·1··1···111···111···11·····1·111·11·1·1··11·111··111····1·1··1··1··11·1···1·1···1····1·11··1·······1·11·1·1··1····11·111··111···1··11·11111··1··1111·1··1111·1·1·1··1·11····1·1···1
out 11 (89)1·······111··1·····1···1·11··1···111···11·1··11·····1·111····1·1··11·1111111·····1·1··1··1··11·1·111·1···1·····111··1·······1··1··11··1····11·11·1·111···1··11·1··11··1··1111·11111···1·1·1··1·11··111·1···1
out 12 (75)1······1·11··1·····1··1······1·····1····1··1··1·····1·111·11·1·1··11·111··1111···1·1··11····11···1·······1····1·11··1·······1·11······1·····1··1··1·11···1··11·11111··1··1111·1··11·1·1·1·11···11··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 (87)···1··1······11··1··1·1··1·111·1·······1······1··1·····11·11·····111·1111·1·1·1···11·11·········111111·1···1·1······11····1··111··1·1111······1·····11·1···1·11111···11·11111·11·1··11·1111·······1111111··1
out 16 (89)1··1··11······1111·1·······1·1····11····1·11·11······1····11·1·1··11···1·1111·····1·1·1·11··1·1····111·111·1·11······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 (83)1·······111··11·1········11··1···111···11··1·11·····1·111····1·1··1··1·11·11·····1·1··1··1··1111·1···1·1·1·····111··11···1·····1··11··1····11111··111····1··11·1··11·····1·1··11·11···1·1·1··1·111·1···11··1
out 18 (91)1··1··11··11···11·11·····11·11·1···1···1··1··1··11·111··1·1··1·1·111······111·1····1·····1···11·1·1111·1·1·1·11··11···1··1·11·····11·11·····1·1··1·1·111·111···11·11··1·111······11·11··1····1··111·11111···
out 19 (59)···1···1······1·1·11·····1········1·····1111·····1····1···1······1·1·11···1·1············1······1111···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 (83)1······11111···111·11····1·1·11·········111··1··11···11·1··1·1····11·11·1·········1··1·1·1··1·1··1·1···1·1····11111···1··11·11·1··1·1·1········111·1··11··1·1·11·11··1···1111··1·······1·1·1·1·1·1·1·1··1··1
out 21 (87)··11··1111·1····1··11····1·····1·1·1···11·11··1·1·1···1·1····1·1···1·1·111·11······1·11·11·····11·1·11·11··1·1111·1······1··11·1··1·········1·11·11·1·1·1··11··1··11······11··111·1·1···111·11····1·1·1111·1
out 22 (81)1·····11·111·111··11··1··1·11··1·11·······1·111·1····1·1··11·1···1·1·1·1··1······1····1·····1·11·11··1···1···11·111·111····11·1···1·11·····1·····1·1111···11·1··111···1·1·11··1··1····1···1····1·1·11··1····
out 23 (89)···1····1·11·111111·1····111·····111···1···1·11·····1111··1··1·1·····1·1···11····11··111·····111·111·······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 (73)1··1··1111······111·1·1···1·1··1···1···1··1······1··1·1····1···1·1·1·11·····1·····1··1·1····11······11···1·1·1111········111·11·1··1·1······1·1··1···1·1·1·11·1··1·1·11·1·111·······1··1·1·1···11·····11····
out 25 (81)···1·11···1··1··11111·1··1··11·1·······1·111··1··1·····1··11·1···11····11·1·1····111·11·········1···11·····1·1···1··1····1111111··1··11·······1·111·1··1···1·11·111··1··11····11·1··1·11111·······1···11···1
out 26 (75)1······1·····1····1···1···11·1·····1····1111··1·····1·111·111··1··11···1··111·1··1·1··11····11····1······1····1·····1······1··11···11·1·····1··1111·11···1··11·111·1··1··11···1··11·111·1·11···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 (85)···1··111····11··1111·1·····11·1·11·······11·11··1·····11········111···11·1······1·1·111·1·····111··1··1···1·111····11····111111·····11·1··1·····11111·1···1·111·····11·111···11·1····1·1111·1····11··1·1··1
out 29 (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··11····1··1····1·1··1···1
out 30 (81)1·····11·111·11111····1··1·11··1·11······1·1·11·1····1·1··11·1···1··1·11··1······1····1·····1·11···111···1···11·111·111··11···1···1·11·····1····1·11111···11·1··111···1·1···1·1··1····1···1····1·1···111····
out 31 (41)···1···1······1····11······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 (43)···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 (87)···1···11·1··1··11111·1····111·1·11·······1··11··1·····111···1···11····11·1···1··111·111·1·····111·11······1··11·1··1····1111111····111····1·····1·11··1···1·111··1··1··11····11·1···1111111·1····11·11····1
out 35 (79)1·····1111·1·111···1··1·····1··111······1··1··1·11···1·1··1······1·1···1····1·1··1···11··1··1·1111·11····1···1111·1·111·····1·11·····1·········1··1·1111··11·1··1·····1·1·1···1·····111··11··1·1·111·11····1
out 36 (45)···1······1·····1········1·11·····1·····1··1·1···1····1······1·····1······1·1·11·········1······1··1·1·····1·····1·······1·····1··1·11·····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 (69)···1···11·1·····1·11·····1·1·1····1·····1111·1···1····1···11·1·····1·11···1·1·1·1········1······1111·1·····1··11·1·······1·11··1··1·1·1····1···11111···1····1···111·······111····1··11·······1····1111·1···1
out 39 (83)1·····11·111·111··11··1·····1··1·······1·11··1··1····1·1··11·1···1·1·1·1··1·1·1··1····11·1··1·1···1·111··1···11·111·111····11·1······1········1·11·1·11···11·1··111···1·1·11··1··1··111···11·1·1·1··1·11····
check 40 (120)1·····1··1·1··11·11·1···1·1111·1·1·1······11··1·1·1·11111·11·1·11111··1111·11····111·11·11·1111111··11·111···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 (90)···1··11··1····1··1········111·1········1111·1··11·····11··1·1··1····1··1111··1··111·11····1111·1111·1··1··1·11··1····1····1········111········11111·111···1·111·11·111····1···1111··111111···11111111·1·1··
check 42 (80)···1·····111·1··111···1·1·11···1·11····1·····11··1·····1·······1·11··1···1·11······1·111·1·11111···11······1····111·1····111··11·1·11······1··1····11··1···1·11····1·1··11·1····1·1·1···1111·11111···11····1
check 43 (122)11··111··111111·11···1·1·11·1111111111111·111111··11·1··11··1111···11··1·1··11·11··111·1·11111·11·1111111·111·11···1··111··1111111··1··1111··1··1·····11·1·111··11··1··111·111·1·111·111··1·1····1·1······11
check 44 (106)·1··1··1·11·111···1111·1·11·11···11··1··111·1111···111···1···1···1··11·11111····1·1·11···11··1·111····1111·1·1111····11·1·11·1·11··111··111111··1··1···1·1·11·1··1·1·1·111·11··1·1··11···11·11·111·11·1·1···
check 45 (106)11·11·11111·1·111111·11··1111··11·11··111·1·····1·····11··1··111·1··11····1··1···1···111··1·111·111·1·1··11111··111········111111·1··1·111···1111··111·1·1111····11··1·11·11·11···1··1·1···1·1·1··1···11··11
check 46 (98)··111·1·1···11··11··1·11···111·1········1111··1111····1·1·1·111····11111··11··111··1·111111·····111·11·1····11·1111·1··11····1·11···1··11·111·1···1·11·······1·11··1·1··11·11·111····111111···········11·111
check 47 (98)1····1····1·1111··11·111·1·11·1·1··1······1·11·11····1·111·11111·1·1·1···11······1·11111·11·1·11·11··11·11···1111·1·111······1·11·1·11···111·····11·1··1··11·1··1·····1·1·1·1111··1···1·····1··1·1·1111·1·11
check 48 (100)1·····11·111·1·1·1···1·1····1··1····111·11·11··111···1·1··11·1···1·11·1·11··1·1··1····11·1·1·1111··1··1111···11·1111·1·111·1111·1····1·······11··1·111····11·1··111·1111·1·1·11··11·111···11·1·1···11·11··1·
check 49 (100)···1··1···11··111·1········1··1·1·1·1111·1··1·1111··11··1····1·1·111··1···11·1·11···11····111·1·11·1··1·1··1·1···11··11··1·1·····111·111111·11·1·111·1···111···1··11····111·1····11111111·1111··1···1·111·11

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 n204Noisy T states per run.
outputs k40
distance d3Exact, proved and witnessed here.
error coefficient A_{3}1,250Faults 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 N50
checks r10
V_{\mathrm{ex}}40T states per run. CCZ counts as two.
\gamma_\rho = \log(n/V_{\mathrm{ex}})/\log d1.4830Overhead per extractable T state. Lower is better.
\gamma = \log(n/k)/\log d1.4830Overhead 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 36 single-wire gates and 604 \mathrm{CZ} gates: \mathrm{S}^\dagger on output 1, \mathrm{S}^\dagger on output 3, \mathrm{S} on output 4, \mathrm{S}^\dagger on output 5, \mathrm{S}^\dagger on output 6, \mathrm{S}^\dagger on output 8, \mathrm{Z} on output 9, \mathrm{S}^\dagger on output 12, \mathrm{S} on output 13, \mathrm{Z} on output 14, \mathrm{S} on output 15, \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} on output 21, \mathrm{S}^\dagger on output 26, \mathrm{Z} on output 27, \mathrm{Z} on output 28, \mathrm{Z} on output 29, \mathrm{S}^\dagger on output 32, \mathrm{S} on output 33, \mathrm{S} on output 34, \mathrm{S} on output 35 and 616 more (listed below). The circuit's action is then exactly the gate above, and an error-free run passes every check with certainty.

All 640 correction gates

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

Or no Clifford gate at all: run \mathrm{T}^3 on rotations 1, 6, 7, 9, 11, 12, 20, 21, 22, 25, 26, 28, 29, 32, 33, 38, 39, 58, 62, 63, 64, 65, 66, 67 and 65 more (listed below); \mathrm{T}^5 on rotations 3, 8, 14, 15, 16, 50, 52; \mathrm{T}^\dagger on rotations 4, 10, 13, 18, 19, 41, 56, 106, 107, 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 89 rotations at T³

1, 6, 7, 9, 11, 12, 20, 21, 22, 25, 26, 28, 29, 32, 33, 38, 39, 58, 62, 63, 64, 65, 66, 67, 68, 71, 73, 75, 77, 81, 82, 87, 88, 89, 90, 91, 92, 93, 94, 96, 97, 101, 102, 105, 108, 110, 114, 125, 128, 129, 131, 135, 136, 137, 141, 143, 144, 146, 149, 152, 155, 156, 160, 162, 163, 166, 167, 170, 171, 172, 173, 174, 175, 178, 180, 182, 183, 184, 189, 192, 193, 195, 197, 198, 199, 200, 201, 202, 203

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/204.40.3.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-204.40.3.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 1,250 faults of weight 3 are undetectable and damaging, so A_{3}=1{,}250. 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 [0,8,160].

A fault is a set of faulty columns. It is undetectable when their check parts cancel, and damaging when their output parts do not.

Original source

S. P. Jain, A. Wills, and S. Singh, "Symmetry and AI-assisted discovery of magic-state factories," arXiv:2610.06535 (2026). arxiv.org/abs/2610.06535

Provenance

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

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

Discovery: AI search

1 recorded source
[
 {
  "regime": "AI search (gamma frontier): from Wills parent codes",
  "file": "gamma_frontier_release/protocols/paired_column_contraction_204_40_3_ef11821ff767.json",
  "label": "paired_column_contraction_204_40_3_ef11821ff767",
  "N": 50,
  "d": 3,
  "d_is_exact": true,
  "origin": "Wills weighted Reed\u2013Muller parent (weights 1^6,2^2, cutoff 3), with searched graph punctures; 6 paired-column contractions",
  "provenance": "gamma frontier release record paired_column_contraction_204_40_3_ef11821ff767 (source catalogue id paired_column_contraction_204_40_3_ef11821ff767); construction: Wills weighted Reed\u2013Muller parent (weights 1^6,2^2, cutoff 3), with searched graph punctures; 6 paired-column contractions; matrix ef11821ff76751c6...; the release states d=3 (exact), gamma=1.4829986488731812 (exact)",
  "notes": "matrix gamma_frontier_release/matrices/ef11821ff76751c686141ae9d344707ca58571e19eadc048ffa5c540b998e454.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/204.40.3.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[837].