Here is a list of all quantum codes useful for distilling magic states and characterized by their magic-state distillation scaling exponent.

Name | Magic-state scaling exponent |
---|---|

Ball color code | The 3D ball codes on duals of the truncated octahedron, truncated cuboctahedron, and truncated icosidodecahedron have \(\gamma\) close to one. |

Triorthogonal code | Depends on the matrix. Reference [1] gave a family of \(\frac{k}{3k+8}\) codes with magic-state distillation scaling exponent \(\gamma = \log_2 \frac{3k+8}{k}\). |

\([[15,1,3]]\) quantum Reed-Muller code | Magic-state distillation scaling exponent \( \gamma= \log_d (n/k)\approx 2.46\) [2]. |

\([[k+4,k,2]]\) H code | A total of \(r\) rounds of magic-state distillation yields a magic-state scaling exponent \(\gamma\to 1\) as \(k,r\rightarrow \infty\). This matches a conjectured bound for \(\gamma\) [1]. |

## References

- [1]
- S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
- [2]
- J. Haah et al., “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI