Magic State Factory Catalog

[\![39,\,5,\,3]\!]

4 inequivalent factories with n=39 inputs, k=5 outputs and distance d=3.

Inequivalent gates at [[39, 5, 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}_{04}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{013}\cdot \mathrm{CCZ}_{014}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{024}\cdot \mathrm{CCZ}_{034}—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{T}_{4}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{02}\cdot \mathrm{CS}_{03}\cdot \mathrm{CS}_{04}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{13}\cdot \mathrm{CS}_{14}\cdot \mathrm{CS}_{23}\cdot \mathrm{CS}_{24}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{013}\cdot \mathrm{CCZ}_{014}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{024}\cdot \mathrm{CCZ}_{034}\cdot \mathrm{CCZ}_{123}\cdot \mathrm{CCZ}_{}\cdots 25 terms—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{T}_{4}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{02}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}—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{T}_{4}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{23}—11d = 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: [\![40,\,5,\,3]\!] · [\![43,\,5,\,3]\!] · [\![44,\,5,\,3]\!] · [\![31,\,5,\,3]\!] · [\![47,\,5,\,3]\!] · [\![48,\,5,\,3]\!] · [\![51,\,5,\,3]\!] · [\![52,\,5,\,3]\!] · [\![53,\,5,\,3]\!] · [\![55,\,5,\,3]\!] · [\![57,\,5,\,3]\!] · [\![59,\,5,\,3]\!]