| output gate | \gamma_\rho | N | distance | cited |
|---|---|---|---|---|
| \mathrm{CS}_{01}\cdot \mathrm{CS}_{02}\cdot \mathrm{CS}_{12} | — | 9 | d = 3 | Wills et al. (2026); Jain et al. (2026) |
| \mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{02}\cdot \mathrm{CS}_{12}\cdot \mathrm{CCZ}_{012} | — | 9 | d = 3 | Wills et al. (2026); Jain et al. (2026) |
| \mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{CS}_{01} | — | 9 | d = 3 | Wills et al. (2026); Jain et al. (2026) |
| \mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2} pure T | 2.335 | 9 | 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,\,3,\,3]\!] · [\![40,\,3,\,3]\!] · [\![37,\,3,\,3]\!] · [\![36,\,3,\,3]\!] · [\![35,\,3,\,3]\!] · [\![43,\,3,\,3]\!] · [\![44,\,3,\,3]\!] · [\![45,\,3,\,3]\!] · [\![31,\,3,\,3]\!] · [\![47,\,3,\,3]\!] · [\![48,\,3,\,3]\!] · [\![49,\,3,\,3]\!]