Magic State Factory Catalog

[\![56,\,6,\,3]\!]

3 inequivalent factories with n=56 inputs, k=6 outputs and distance d=3.

Inequivalent gates at [[56, 6, 3]]. Click a column header to sort.
output gate \gamma_\rho N distance cited
\mathrm{T}_{0}\cdot \mathrm{T}_{3}\cdot \mathrm{T}_{4}\cdot \mathrm{T}_{5}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{02}\cdot \mathrm{CS}_{23}\cdot \mathrm{CS}_{24}\cdot \mathrm{CS}_{25}\cdot \mathrm{CS}_{34}\cdot \mathrm{CS}_{35}\cdot \mathrm{CS}_{45}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{013}\cdot \mathrm{CCZ}_{014}\cdot \mathrm{CCZ}_{015}\cdot \mathrm{CCZ}_{034}\cdot \mathrm{CCZ}_{134}\cdot \mathrm{CCZ}_{234}\cdot \mathrm{CCZ}_{235}\cdot \mathrm{CCZ}_{245}—13d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3}\cdot \mathrm{T}_{4}\cdot \mathrm{T}_{5}\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}_{15}\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}_{025}\cdot \mathrm{CCZ}_{035}\cdot \mathrm{CCZ}_{045}\cdot \mathrm{CCZ}_{12}\cdots 28 terms—13d = 3Jain et al. (2026)
\mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3}\cdot \mathrm{T}_{4}\cdot \mathrm{T}_{5} pure T2.03313d = 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.

Same n and k at other distances: [\![56,\,6,\,2]\!]
Nearest lengths with the same k and d: [\![55,\,6,\,3]\!] · [\![58,\,6,\,3]\!] · [\![59,\,6,\,3]\!] · [\![52,\,6,\,3]\!] · [\![60,\,6,\,3]\!] · [\![51,\,6,\,3]\!] · [\![62,\,6,\,3]\!] · [\![63,\,6,\,3]\!] · [\![48,\,6,\,3]\!] · [\![47,\,6,\,3]\!] · [\![66,\,6,\,3]\!] · [\![67,\,6,\,3]\!]