Magic State Factory Catalog

[\![51,\,5,\,3]\!]

10 inequivalent factories with n=51 inputs, k=5 outputs and distance d=3.

Inequivalent gates at [[51, 5, 3]]. Click a column header to sort.
output gate \gamma_\rho N distance cited
\mathrm{CS}_{01}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}—12d = 3Wills et al. (2026)
\mathrm{CS}_{01}\cdot \mathrm{CCZ}_{234}2.57912d = 3Wills et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{24}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{013}\cdot \mathrm{CCZ}_{024}\cdot \mathrm{CCZ}_{134}—12d = 3Wills et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{034}—12d = 3Wills et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CCZ}_{234}—12d = 3Wills et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{013}\cdot \mathrm{CCZ}_{123}—12d = 3Wills et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{12}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{034}—12d = 3Wills et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{12}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{034}\cdot \mathrm{CCZ}_{134}—12d = 3Wills et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}—12d = 3Wills et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3}\cdot \mathrm{T}_{4} pure T2.11412d = 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.

Nearest lengths with the same k and d: [\![52,\,5,\,3]\!] · [\![53,\,5,\,3]\!] · [\![48,\,5,\,3]\!] · [\![47,\,5,\,3]\!] · [\![55,\,5,\,3]\!] · [\![57,\,5,\,3]\!] · [\![44,\,5,\,3]\!] · [\![43,\,5,\,3]\!] · [\![59,\,5,\,3]\!] · [\![60,\,5,\,3]\!] · [\![40,\,5,\,3]\!] · [\![39,\,5,\,3]\!]