| output gate | \gamma_\rho | N | distance | cited |
|---|---|---|---|---|
| \mathrm{T}_{0}\cdot \mathrm{T}_{1}\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}_{05}\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}_{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}_{}\cdots 41 terms | — | 12 | d = 3 | Wills 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{T}_{5}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{23}\cdot \mathrm{CS}_{45} | — | 12 | d = 3 | Wills 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: [\![43,\,6,\,3]\!] · [\![44,\,6,\,3]\!] · [\![47,\,6,\,3]\!] · [\![48,\,6,\,3]\!] · [\![51,\,6,\,3]\!] · [\![52,\,6,\,3]\!] · [\![55,\,6,\,3]\!] · [\![56,\,6,\,3]\!] · [\![58,\,6,\,3]\!] · [\![59,\,6,\,3]\!] · [\![60,\,6,\,3]\!] · [\![62,\,6,\,3]\!]