Magic State Factory Catalog

[\![60,\,5,\,3]\!]

27 inequivalent factories with n=60 inputs, k=5 outputs and distance d=3.

Inequivalent gates at [[60, 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}\cdot \mathrm{CCZ}_{034}—12d = 3Jain et al. (2026)
\mathrm{CS}_{01}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{234}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{12}\cdot \mathrm{CCZ}_{134}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{013}\cdot \mathrm{CCZ}_{124}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{124}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{234}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{034}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{123}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{23}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{134}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{134}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{234}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{124}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{124}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{24}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{234}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{24}\cdot \mathrm{CCZ}_{123}—12d = 3Jain 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}_{024}\cdot \mathrm{CCZ}_{034}—12d = 3Jain 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}_{024}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{013}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{013}\cdot \mathrm{CCZ}_{124}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{13}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{034}\cdot \mathrm{CCZ}_{123}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{13}\cdot \mathrm{CS}_{24}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{034}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{13}\cdot \mathrm{CS}_{24}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{123}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{123}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{13}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{234}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{134}—12d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}—12d = 3Jain 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: [\![59,\,5,\,3]\!] · [\![57,\,5,\,3]\!] · [\![63,\,5,\,3]\!] · [\![55,\,5,\,3]\!] · [\![65,\,5,\,3]\!] · [\![53,\,5,\,3]\!] · [\![67,\,5,\,3]\!] · [\![52,\,5,\,3]\!] · [\![68,\,5,\,3]\!] · [\![51,\,5,\,3]\!] · [\![71,\,5,\,3]\!] · [\![48,\,5,\,3]\!]