| output gate | \gamma_\rho | N | distance | cited |
|---|---|---|---|---|
| \mathrm{CS}_{01}\cdot \mathrm{CS}_{02}\cdot \mathrm{CS}_{03}\cdot \mathrm{CS}_{13}\cdot \mathrm{CS}_{23}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{123} | — | 10 | 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{CS}_{01}\cdot \mathrm{CS}_{02}\cdot \mathrm{CS}_{03}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{13}\cdot \mathrm{CS}_{23}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{013}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{123} | — | 10 | 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{CS}_{01}\cdot \mathrm{CS}_{23} | — | 10 | 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{CS}_{01} | — | 10 | 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: [\![38,\,4,\,3]\!] · [\![40,\,4,\,3]\!] · [\![36,\,4,\,3]\!] · [\![43,\,4,\,3]\!] · [\![44,\,4,\,3]\!] · [\![31,\,4,\,3]\!] · [\![47,\,4,\,3]\!] · [\![48,\,4,\,3]\!] · [\![50,\,4,\,3]\!] · [\![51,\,4,\,3]\!] · [\![52,\,4,\,3]\!] · [\![54,\,4,\,3]\!]