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 [1]. |

Prime-qudit RM code | An odd-prime-qudit CSS code family constructed from first-order punctured GRM codes can be used for qudit magic-state distillation; see [2; Table I] for yields. |

Prime-qudit RS code | Triorthogonal \(p\)-dimensional prime-qudit RS codes achieve a magic-state distillation scaling exponent \(\gamma = O(1/\log p)\) [3]. |

Quantum Reed-Muller code | The family constructed out of shortened RM codes with parameters \([[\sum_{i=w+1}^m \binom{m}{i}, \sum_{i=0}^{w} \binom{m}{i}, \sum_{i=w+1}^{r+1} \binom{r+1}{i}]]\) for integers \(m > 2r\) and \(r > w \geq 0\) yields protocols with an exponent of \(\gamma < 0.678\), with the fewest resource protocol with \(\gamma < 1\) requiring a code with parameters \(\{r,w,m\} = \{19,14,3r+1\}\) such that \(n \approx 2^{58}\) qubits [4; Corr. 1]. This refutes a conjecture that no protocol could achieve \(\gamma < 1\) [5]. |

Quantum pin code | A family of punctured pin codes admits \(\gamma \approx 1.6\) [6; Table VII]. |

Qubit CSS code | Various self-dual CSS codes yield magic-state distillation protocols whose exponent \(\gamma \to 1^{+}\) [8][7; Thms. 4.1 and 4.2]. |

\([[11,1,5]]_3\) qutrit Golay code | Magic-state distillation scailing exponent \(\gamma=\log_3(1728\times 11) \approx 8.97\), where the \(1728\) factor comes from the fact that one round of distillation succeeds with probability \(\approx 1/1728\) [9]. |

\([[15,1,3]]\) quantum Reed-Muller code | Magic-state distillation scaling exponent \( \gamma= \log_d (n/k)\approx 2.47\) [7][10; Box 2]. |

\([[3k + 8, k, 2]]\) triorthogonal code | The family yields the asymptotic exponent \(\gamma = \log_2 \frac{3k+8}{k} \to \log_2 3 \approx 1.6\) for sufficiently large \(k\) [10; Box 2]; see [8; Table V]. |

\([[4,2,2]]\) CSS code | Various magic-state distillation protocols exist for the \([[4,2,2]]\) qubit code and the \([[6,2,2]]\) \(C_6\) code in what are known as Meier-Eastin-Knill (MEK) protocols [11]. For example, the magic-state scaling exponent is \(\gamma = \log_2 5 \approx 2.322\) for a protocol using the \([[10,2,2]]\) code [10; Box 2]; see also [8; Table IV]. |

\([[49,1,5]]\) triorthogonal code | The code yields an exponent \(\gamma = \log 49 / \log 5 \approx 2.42\). |

\([[6,2,2]]\) \(C_6\) code | Various magic-state distillation protocols exist for the \([[4,2,2]]\) qubit code and the \([[6,2,2]]\) \(C_6\) code in what are known as Meier-Eastin-Knill (MEK) protocols [11]. For example, the magic-state scaling exponent is \(\gamma = \log_2 5 \approx 2.322\) for a protocol using the \([[10,2,2]]\) code [10; Box 2]; see also [8; Table IV]. |

\([[k+4,k,2]]\) H code | A total of \(r\) rounds of magic-state distillation yields a magic-state scaling exponent \(\gamma\to 1^{+}\) as \(k,r\rightarrow \infty\); see [10; Box 2]. This matches a conjectured bound for \(\gamma\) [5]. |

## References

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- M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
- [2]
- E. T. Campbell, H. Anwar, and D. E. Browne, “Magic-State Distillation in All Prime Dimensions Using Quantum Reed-Muller Codes”, Physical Review X 2, (2012) arXiv:1205.3104 DOI
- [3]
- A. Krishna and J.-P. Tillich, “Towards Low Overhead Magic State Distillation”, Physical Review Letters 123, (2019) arXiv:1811.08461 DOI
- [4]
- M. B. Hastings and J. Haah, “Distillation with Sublogarithmic Overhead”, Physical Review Letters 120, (2018) arXiv:1709.03543 DOI
- [5]
- S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
- [6]
- C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
- [7]
- J. Haah et al., “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
- [8]
- Quantum Information and Computation 18, (2018) arXiv:1709.02789 DOI
- [9]
- S. Prakash, “Magic state distillation with the ternary Golay code”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, (2020) arXiv:2003.02717 DOI
- [10]
- E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017) arXiv:1612.07330 DOI
- [11]
- A. M. Meier, B. Eastin, and E. Knill, “Magic-state distillation with the four-qubit code”, (2012) arXiv:1204.4221