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

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Name Magic-state yield parameter
Ball color code The 3D ball codes on duals of the truncated octahedron, truncated cuboctahedron, and truncated icosidodecahedron have \(\gamma\) close to one [1].
Galois-qudit expander code Hypergraph products of expander codes with RS inner codes yield \([[n,k\geq n^{1-\epsilon},d\geq n^{1/r}/\text{poly}(\log n)]]_q\) QLDPC Galois-qudit quantum expander codes with transversal \(C^{r-1} Z\) gates [2]. This construction allows for arbitrarily small magic-state yield parameter \(\gamma\).
Modular-qudit stabilizer code The magic-state yield parameter \(\gamma = \log_d(n/k)\) quantifies the overhead cost of magic-state distillation per the original protocol [3,4].
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 [5; Table I] for yields.
Prime-qudit RS code Triorthogonal \(p\)-dimensional prime-qudit RS codes achieve a magic-state yield parameter \(\gamma = O(1/\log p)\) [6].
Quantum AG code By defining a generalization of triorthogonal matrices to Galois qudits of dimension \(q=2^m\), one can construct an asymptotically good family of quantum AG codes that admits a diagonal transversal gate at the third level of the Clifford hierarchy and attains a zero magic-state yield parameter, \(\gamma = 0\) [7]. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of \(m\) qubits; see [8,10–12][9; Sec. 5.3]. Two other such asymptotically good families exist [11,12], admitting a different diagonal gate at the third level of the Clifford hierarchy.
Quantum Reed-Muller (RM) 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 [13; Corr. 1]. This refutes a conjecture that no protocol could achieve \(\gamma < 1\) [4].
Quantum pin code A family of punctured pin codes admits \(\gamma \approx 1.6\) [14; Table VII].
Quantum rainbow code Hypergraph products of color codes yield quantum rainbow codes with growing distance and transversal gates in the Clifford hierarchy. In particular, utilizing this construction for quasi-hyperbolic color codes [15] yields an \([[n,O(n),O(\log n)]]\) triorthogonal code family with magic-state yield parameter \(\gamma\to 0\) [16].
Qubit CSS code Various self-dual CSS codes yield magic-state distillation protocols whose yield parameter \(\gamma \to 1^{+}\) [18][17; 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 \(\approx 1/1728\) [19].
\([[15,1,3]]\) quantum RM code Magic-state yield parameter \( \gamma= \log_d (n/k)\approx 2.47\) [17][20; Box 2].
\([[23, 1, 7]]\) Quantum Golay code Magic-state distillation scailing exponent \(\gamma=\log_2 23 \approx 4.52\)[19]. Various magic-state distillation protocols have been developed for this code [17,21] and its \([[21,3,5]]\) doubly punctured version [17].
\([[3k + 8, k, 2]]\) 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 \(k\) [20; Box 2]; see [18; Table V].
\([[4,2,2]]\) Four-qubit code Various magic-state distillation protocols exist for the \([[4,2,2]]\) qubit code and the \(C_6\) code in what are known as Meier-Eastin-Knill (MEK) protocols [22]. For example, the magic-state yield parameter is \(\gamma = \log_2 5 \approx 2.322\) for a protocol using the \([[10,2,2]]\) code [20; Box 2]; see also [18; 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 \(C_6\) code in what are known as Meier-Eastin-Knill (MEK) protocols [22]. For example, the magic-state yield parameter is \(\gamma = \log_2 5 \approx 2.322\) for a protocol using the \([[10,2,2]]\) code [20; Box 2]; see also [18; Table IV].
\([[6k+2,3k,2]]\) Campbell-Howard code A total of \(r\) rounds of magic-state distillation yields a magic-state yield parameter \(\gamma\to 1^{+}\) as \(k,r\rightarrow \infty\). This matches a conjectured bound for \(\gamma\) [4].
\([[9m-k,k,2]]_3\) triorthogonal code For \(k = 3m-2\), the family yields the magic-state yield parameter \(\gamma = \log_2 (2+\frac{6}{3m-2}) \to 1\) as \(m\to\infty\) [23].
\([[k+4,k,2]]\) H code A total of \(r\) rounds of magic-state distillation yields a magic-state yield parameter \(\gamma\to 1^{+}\) as \(k,r\rightarrow \infty\); see [20; Box 2]. This matches a conjectured bound for \(\gamma\) [4].

References

[1]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[2]
L. Golowich and T.-C. Lin, “Quantum LDPC Codes with Transversal Non-Clifford Gates via Products of Algebraic Codes”, (2024) arXiv:2410.14662
[3]
S. Bravyi and A. Kitaev, “Universal quantum computation with ideal Clifford gates and noisy ancillas”, Physical Review A 71, (2005) arXiv:quant-ph/0403025 DOI
[4]
S. Bravyi and J. Haah, “Magic-state distillation with low overhead”, Physical Review A 86, (2012) arXiv:1209.2426 DOI
[5]
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
[6]
A. Krishna and J.-P. Tillich, “Towards Low Overhead Magic State Distillation”, Physical Review Letters 123, (2019) arXiv:1811.08461 DOI
[7]
A. Wills, M.-H. Hsieh, and H. Yamasaki, “Constant-Overhead Magic State Distillation”, (2024) arXiv:2408.07764
[8]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[9]
A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
[10]
D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
[11]
Q. T. Nguyen, “Good binary quantum codes with transversal CCZ gate”, (2024) arXiv:2408.10140
[12]
L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
[13]
M. B. Hastings and J. Haah, “Distillation with Sublogarithmic Overhead”, Physical Review Letters 120, (2018) arXiv:1709.03543 DOI
[14]
C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
[15]
G. Zhu, S. Sikander, E. Portnoy, A. W. Cross, and B. J. Brown, “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2024) arXiv:2310.16982
[16]
T. R. Scruby, A. Pesah, and M. Webster, “Quantum Rainbow Codes”, (2024) arXiv:2408.13130
[17]
J. Haah, M. B. Hastings, D. Poulin, and D. Wecker, “Magic state distillation with low space overhead and optimal asymptotic input count”, Quantum 1, 31 (2017) arXiv:1703.07847 DOI
[18]
Quantum Information and Computation 18, (2018) arXiv:1709.02789 DOI
[19]
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
[20]
E. T. Campbell, B. M. Terhal, and C. Vuillot, “Roads towards fault-tolerant universal quantum computation”, Nature 549, 172 (2017) arXiv:1612.07330 DOI
[21]
B. W. Reichardt, “Quantum Universality from Magic States Distillation Applied to CSS Codes”, Quantum Information Processing 4, 251 (2005) arXiv:quant-ph/0411036 DOI
[22]
A. M. Meier, B. Eastin, and E. Knill, “Magic-state distillation with the four-qubit code”, (2012) arXiv:1204.4221
[23]
S. Prakash and T. Saha, “Low Overhead Qutrit Magic State Distillation”, Quantum 9, 1768 (2025) arXiv:2403.06228 DOI