| output gate | \gamma_\rho | N | distance | cited |
|---|---|---|---|---|
| \mathrm{T}_{0}\cdot \mathrm{CS}_{01}\cdot \mathrm{CS}_{24}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{124} | — | 12 | d = 3 | Wills et al. (2026) |
| \mathrm{T}_{0}\cdot \mathrm{CS}_{13}\cdot \mathrm{CS}_{24}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{013}\cdot \mathrm{CCZ}_{024} | — | 12 | d = 3 | Wills et al. (2026) |
| \mathrm{T}_{0}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{123} | — | 12 | d = 3 | Wills et al. (2026) |
| \mathrm{T}_{0}\cdot \mathrm{CS}_{13}\cdot \mathrm{CS}_{24}\cdot \mathrm{CCZ}_{012} | — | 12 | d = 3 | Wills et al. (2026) |
| \mathrm{T}_{0}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{34}\cdot \mathrm{CCZ}_{123} | — | 12 | d = 3 | Wills et al. (2026) |
| \mathrm{T}_{0}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{34} | 2.597 | 12 | d = 3 | Wills et al. (2026) |
| \mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3}\cdot \mathrm{T}_{4} pure T | 2.132 | 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: [\![51,\,5,\,3]\!] · [\![53,\,5,\,3]\!] · [\![55,\,5,\,3]\!] · [\![48,\,5,\,3]\!] · [\![47,\,5,\,3]\!] · [\![57,\,5,\,3]\!] · [\![59,\,5,\,3]\!] · [\![44,\,5,\,3]\!] · [\![60,\,5,\,3]\!] · [\![43,\,5,\,3]\!] · [\![63,\,5,\,3]\!] · [\![40,\,5,\,3]\!]