Magic State Factory Catalog

[\![44,\,4,\,3]\!]

5 inequivalent factories with n=44 inputs, k=4 outputs and distance d=3.

Inequivalent gates at [[44, 4, 3]]. Click a column header to sort.
output gate \gamma_\rho N distance cited
\mathrm{CS}_{01}\cdot \mathrm{CS}_{02}\cdot \mathrm{CS}_{03}\cdot \mathrm{CS}_{13}\cdot \mathrm{CS}_{23}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{123}—11d = 3Wills et al. (2026); Jain et al. (2026)
\mathrm{CS}_{01}\cdot \mathrm{CS}_{02}\cdot \mathrm{CS}_{13}\cdot \mathrm{CS}_{23}—11d = 3Wills et al. (2026); Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{02}\cdot \mathrm{CS}_{03}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{13}\cdot \mathrm{CS}_{23}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{013}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{123}—11d = 3Wills et al. (2026); Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{23}—11d = 3Wills et al. (2026); Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3} pure T2.18311d = 3Wills et al. (2026); Jain et al. (2026)

These factories share n, k and d but prepare different magic states. No CNOT circuit with S, Z and CZ corrections turns one of these gates into another.

Same n and k at other distances: [\![44,\,4,\,2]\!]
Nearest lengths with the same k and d: [\![43,\,4,\,3]\!] · [\![47,\,4,\,3]\!] · [\![40,\,4,\,3]\!] · [\![48,\,4,\,3]\!] · [\![39,\,4,\,3]\!] · [\![38,\,4,\,3]\!] · [\![50,\,4,\,3]\!] · [\![51,\,4,\,3]\!] · [\![36,\,4,\,3]\!] · [\![52,\,4,\,3]\!] · [\![54,\,4,\,3]\!] · [\![55,\,4,\,3]\!]