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arXiv:2510.10852 · Physical Review A

Sublogarithmic distillation in all prime dimensions using punctured Reed-Muller codes

Tanay Saha, Shiroman Prakash

Published Mar 9, 2026• 0 citations
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Abstract

Magic state distillation is a leading but costly approach to fault-tolerant quantum computation, and it is important to explore all possible ways of minimizing its overhead cost. The number of ancillae required to produce a magic state within a target error rate $ε$ is $O(\log^γ (ε^{-1}))$ where $γ$ is known as the yield parameter. Hastings and Haah derived a family of distillation protocols with sublogarithmic overhead (i.e., $γ< 1$) based on punctured Reed-Muller codes. Building on work by Campbell \textit{et al.} and Krishna-Tillich, which suggests that qudits of dimension $p>2$ can significantly reduce overhead, we generalize their construction to qudits of arbitrary prime dimension $p$. We find that, in an analytically tractable puncturing scheme, the number of qudits required to achieve sublogarithmic overhead decreases drastically as $p$ increases, and the asymptotic yield parameter approaches $\frac{1}{\ln p}$ as $p \to \infty$. We also perform a small computational search for optimal puncture locations, which results in several interesting triorthogonal codes, including a $[[519,106,5]]_5$ code with $γ=0.99$.

DOI
10.1103/tzkz-9p2f

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Magic state distillation is a leading but costly approach to fault-tolerant quantum computation, and it is important to explore all possible ways of minimizing its overhead cost. The number of ancillae required to produce a magic state within a target error rate εεε is O(log⁡γ(ε−1))O(\log^γ (ε^{-1}))O(logγ(ε−1)) where γγγ is known as the yield parameter. Hastings and Haah derived a family of distillation protocols with sublogarithmic overhead (i.e., γ<1γ< 1γ<1) based on punctured Reed-Muller codes. Building on work by Campbell \textit{et al.} and Krishna-Tillich, which suggests that qudits of dimension p>2p>2p>2 can significantly reduce overhead, we generalize their construction to qudits of arbitrary prime dimension ppp. We find that, in an analytically tractable puncturing scheme, the number of qudits required to achieve sublogarithmic overhead decreases drastically as ppp increases, and the asymptotic yield parameter approaches 1ln⁡p\frac{1}{\ln p}lnp1​ as p→∞p \to \inftyp→∞. We also perform a small computational search for optimal puncture locations, which results in several interesting triorthogonal codes, including a [[519,106,5]]5[[519,106,5]]_5[[519,106,5]]5​ code with γ=0.99γ=0.99γ=0.99.
We show that using qutrits rather than qubits leads to a substantial reduction in the overhead cost associated with an approach to fault-tolerant quantum computing known as magic state distillation. We construct a family of [[9m−k,k,2]]3 triorthogonal qutrit error-correcting codes for any positive integers m and k with k≤3m−2 that are suitable for magic state distillation. In magic state distillation, the number of ancillae required to produce a magic state with target error rate ϵ is O(logγ⁡ϵ−1) , where the yield parameter γ characterizes the overhead cost. For k=3m−2 , our codes have γ=log2⁡(2+63m−2) , which tends to 1 as m→∞ . Moreover, the [[20,7,2]]3 qutrit code that arises from our construction when m=3 already has a yield parameter of 1.51 which outperforms all known qubit triorthogonal codes of size less than a few hundred qubits.
Qudits offer the potential for low-overhead magic state distillation, although previous results for asymptotically good codes have required qudit dimension q≫100q\gg 100q≫100 or code length N≫100\mathcal{N}\gg 100N≫100. These parameters far exceed experimental demonstrations of qudit platforms, and thus motivate the search for better codes. Using a novel lifting procedure, we construct the first family of good triorthogonal codes on the F22m\mathbb{F}_{2^{2m}}F22m​ alphabet with m≥3m \geq 3m≥3 that lies above the Tsfasman-Vladut-Zink bound. These codes yield a family of asymptotically good quantum codes with transversal CCZ gates, enabling constant space overhead magic state distillation with qudit dimension as small as q=64q=64q=64. Further, we identify a promising code with parameters [[42,14,6]]64[[42,14,6]]_{64}[[42,14,6]]64​. Finally, we show that a distilled ∣CCZ⟩22m|{CCZ}\rangle_{2^{2m}}∣CCZ⟩22m​ can be reduced to a ∣CCZ⟩2n|{CCZ}\rangle_{2^n}∣CCZ⟩2n​ state for arbitrary nnn with a constant-depth Clifford circuit of at most 9 computational basis measurements, 12 single-qudit and 9 two-qudit Clifford gates.
It has been conjectured that, for any distillation protocol for magic states for the TTT gate, the number of noisy input magic states required per output magic state at output error rate $\ensuremath{\epsilon}$ is $\mathrm{\ensuremath{\Omega}}[\mathrm{log}(1/\ensuremath{\epsilon})]$. We show that this conjecture is false. We find a family of quantum error correcting codes of parameters $⟦\ensuremath{\sum}_{i=w+1}^{m}(\genfrac{}{}{0}{}{m}{i}),\ensuremath{\sum}_{i=0}^{w}(\genfrac{}{}{0}{}{m}{i}),\ensuremath{\sum}_{i=w+1}^{r+1}(\genfrac{}{}{0}{}{r+1}{i})⟧$ for any integers m>2rm>2rm>2r, $r>w\ensuremath{\ge}0$, by puncturing quantum Reed-Muller codes. When $m>\ensuremath{\nu}r$, our code admits a transversal logical gate at the $\ensuremath{\nu}$th level of Clifford hierarchy. In a distillation protocol for magic states at the level $\ensuremath{\nu}=3$ (TTT gate), the ratio of input to output magic states is $O\mathbf{(}{\mathrm{log}}^{\ensuremath{\gamma}}(1/\ensuremath{\epsilon})\mathbf{)}$, where $\ensuremath{\gamma}=\mathrm{log}(n/k)/\mathrm{log}(d)<0.678$ for some mmm, rrr, www. The smallest code in our family for which $\ensuremath{\gamma}<1$ is on $\ensuremath{\approx}{2}^{58}$ qubits.
Magic state distillation is a widely considered primitive in fault-tolerant quantum computation for the preparation of high-fidelity non-Clifford resources. The most commonly considered class of such protocols are generalised triorthogonal codes; these distil nnn noisy input T\mathrm{T}T states into purified third-level diagonal magic states. Extensive prior work has searched this space of protocols, often using heuristic methods that do not guarantee optimality. Existing classification work is limited to protocols distilling kkk output T\mathrm{T}T states with n+k≤38n+k\leq 38n+k≤38 (Nezami and Haah, Phys. Rev. A 106, 012437 (2022)). In this work, we significantly expand this classification to all protocols with lengths n≤54n\leq 54n≤54. We restrict to distance d≥3d\geq 3d≥3 to keep the classification to a sensible size, and because efficient searches are well understood at distance 222 (Singh et al., arXiv:2606.28518). Moreover, our results are complementary to synthillation (Campbell and Howard, Phys. Rev. A 95, 022316 (2017)), which can distil third-level states from T\mathrm{T}T states, but only at distance 222. Under optimality in terms of input count, space footprint, and distance for a given output, we find 747474 optimal generalised triorthogonal protocols in our range, 656565 of which are new to the literature. To achieve our classification, we extend the classification of unital triorthogonal spaces of Nezami and Haah from length 383838 to 545454 using a directional derivative method. Using these as the stabiliser spaces, we add logical rows that satisfy the triorthogonality constraints to create full protocols.
Magic-state distillation is a resource intensive subroutine for quantum computation. The ratio of noisy input states to output states with an error rate at most ε scales as O(log^{γ}(1/ε)) [S. Bravyi and J. Haah, Magic-state distillation with low overhead, Phys. Rev. A 86, 052329 (2012)10.1103/PhysRevA.86.052329]. In a breakthrough paper, Hastings and Haah [Distillation with Sublogarithmic Overhead, Phys. Rev. Lett. 120, 050504 (2018)10.1103/PhysRevLett.120.050504] showed that it is possible to construct distillation routines with a sublogarithmic overhead, achieving γ≈0.6779 and falsifying a conjecture that γ is lower bounded by 1. They then ask whether γ can be made arbitrarily close to 0. We answer this question in the affirmative for magic-state distillation routines using qudits of prime dimension (d dimensional quantum systems for prime d).
Magic state distillation is a crucial yet resource-intensive process in fault-tolerant quantum computation. The protocol's overhead, defined as the number of input magic states required per output magic state with an error rate below εεε, typically grows as O(log⁡γ(1/ε))\mathcal{O}(\log^γ(1/ε))O(logγ(1/ε)). Achieving smaller overheads, i.e., smaller exponents γγγ, is highly desirable; however, all existing protocols require polylogarithmically growing overheads with some γ>0γ> 0γ>0, and identifying the smallest achievable exponent γγγ for distilling magic states of qubits has remained challenging. To address this issue, we develop magic state distillation protocols for qubits with efficient, polynomial-time decoding that achieve an O(1)\mathcal{O}(1)O(1) overhead, meaning the optimal exponent γ=0γ= 0γ=0; this improves over the previous best of γ≈0.678γ\approx 0.678γ≈0.678 due to Hastings and Haah. In our construction, we employ algebraic geometry codes to explicitly present asymptotically good quantum codes for 2102^{10}210-dimensional qudits that support transversally implementable logical gates in the third level of the Clifford hierarchy. The use of asymptotically good codes with non-vanishing rate and relative distance leads to the constant overhead. These codes can be realised by representing each 2102^{10}210-dimensional qudit as a set of 101010 qubits, using stabiliser operations on qubits. The 101010-qubit magic states distilled with these codes can be converted to and from conventional magic states for the controlled-controlled-ZZZ (CCZCCZCCZ) and TTT gates on qubits with only a constant overhead loss, making it possible to achieve constant-overhead distillation of such standard magic states for qubits. These results resolve the fundamental open problem in quantum information theory concerning the construction of magic state distillation protocols with the optimal exponent.