| output gate | \gamma_\rho | N | distance | cited |
|---|---|---|---|---|
| \mathrm{CS}_{01}\cdot \mathrm{CCZ}_{234} | 2.771 | 11 | d = 3 | Jain et al. (2026) |
| \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{034} | — | 11 | d = 3 | Jain et al. (2026) |
| \mathrm{T}_{0}\cdot \mathrm{CS}_{12}\cdot \mathrm{CS}_{34} | 2.771 | 11 | d = 3 | Jain et al. (2026) |
| \mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{CS}_{34} | 2.509 | 11 | d = 3 | Jain et al. (2026) |
| \mathrm{T}_{0}\cdot \mathrm{T}_{1}\cdot \mathrm{T}_{2}\cdot \mathrm{T}_{3}\cdot \mathrm{T}_{4} pure T | 2.306 | 11 | d = 3 | Shi et al. (2024) |
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: [\![65,\,5,\,3]\!] · [\![60,\,5,\,3]\!] · [\![59,\,5,\,3]\!] · [\![67,\,5,\,3]\!] · [\![68,\,5,\,3]\!] · [\![57,\,5,\,3]\!] · [\![55,\,5,\,3]\!] · [\![71,\,5,\,3]\!] · [\![72,\,5,\,3]\!] · [\![53,\,5,\,3]\!] · [\![52,\,5,\,3]\!] · [\![51,\,5,\,3]\!]