| output gate | \gamma_\rho | N | distance | cited |
|---|---|---|---|---|
| \mathrm{CS}_{01}\cdot \mathrm{CS}_{35}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{034}\cdot \mathrm{CCZ}_{145}\cdot \mathrm{CCZ}_{235} | — | 12 | d = 3 | Wills et al. (2026) |
| \mathrm{CS}_{01}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{034}\cdot \mathrm{CCZ}_{135} | — | 12 | d = 3 | Wills et al. (2026) |
| \mathrm{CS}_{01}\cdot \mathrm{CS}_{35}\cdot \mathrm{CCZ}_{012}\cdot \mathrm{CCZ}_{034}\cdot \mathrm{CCZ}_{235} | — | 12 | d = 3 | Wills et al. (2026) |
| \mathrm{CS}_{01}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{024}\cdot \mathrm{CCZ}_{035}\cdot \mathrm{CCZ}_{123}\cdot \mathrm{CCZ}_{145} | — | 12 | d = 3 | Gong et al. (2026) |
| \mathrm{CS}_{01}\cdot \mathrm{CS}_{24}\cdot \mathrm{CCZ}_{023}\cdot \mathrm{CCZ}_{145} | — | 12 | d = 3 | Wills 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: [\![47,\,6,\,3]\!] · [\![51,\,6,\,3]\!] · [\![44,\,6,\,3]\!] · [\![52,\,6,\,3]\!] · [\![43,\,6,\,3]\!] · [\![55,\,6,\,3]\!] · [\![56,\,6,\,3]\!] · [\![39,\,6,\,3]\!] · [\![58,\,6,\,3]\!] · [\![59,\,6,\,3]\!] · [\![60,\,6,\,3]\!] · [\![62,\,6,\,3]\!]