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arXiv:2609.30860

Classification of Generalised Triorthogonal Codes through Length 54

Adam Wills, Shubham P. Jain, Shraddha Singh

quant-phSubmitted Sep 25, 2026• Announced Sep 28, 2026• 0 citations
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Abstract

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 $n$ noisy input $\mathrm{T}$ 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 $k$ output $\mathrm{T}$ states with $n+k\leq 38$ (Nezami and Haah, Phys. Rev. A 106, 012437 (2022)). In this work, we significantly expand this classification to all protocols with lengths $n\leq 54$. We restrict to distance $d\geq 3$ to keep the classification to a sensible size, and because efficient searches are well understood at distance $2$ (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 $\mathrm{T}$ states, but only at distance $2$. Under optimality in terms of input count, space footprint, and distance for a given output, we find $74$ optimal generalised triorthogonal protocols in our range, $65$ 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 $38$ to $54$ using a directional derivative method. Using these as the stabiliser spaces, we add logical rows that satisfy the triorthogonality constraints to create full protocols.

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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.
Triorthogonal codes are a class of quantum error-correcting codes used in magic state distillation protocols. We classify all triorthogonal codes with $n+k\ensuremath{\le}38$, where nnn is the number of physical qubits and kkk is the number of logical qubits of the code. We find 38 distinguished triorthogonal subspaces, and we show that every triorthogonal code with $n+k\ensuremath{\le}38$ descends from one of these subspaces through elementary operations such as puncturing and deleting qubits. Specifically, we associate each triorthogonal code with a Reed-Muller polynomial of weight n+kn+kn+k, and we classify the Reed-Muller polynomials of low weight using the results of Kasami, Tokura, and Azumi [IEEE Trans. Inf. Theory 16, 752 (1970); Inf. Contr. 30, 380 (1976)] and an extensive computerized search. In an Appendix independent of the main text, we improve a magic state distillation protocol by reducing the time variance due to stochastic Clifford corrections.
Triorthogonal codes are a basic ingredient in magic-state distillation, as they enable transversal implementation of logical T\mathsf{T}T gates. As a non-Clifford operation, the T\mathsf{T}T gate enables universal fault-tolerant quantum computation when combined with Clifford operations. We derive an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance. The criterion combines triorthogonality constraints with MacWilliams identities via Krawtchouk-polynomial conditions on the dual weight distribution, yielding an integer linear programming formulation for the construction problem. We find novel nontrivial triorthogonal codes (degenerate and non-degenerate) that are not necessarily constructed from triply even binary linear codes. By imposing automorphism-group constraints, the formulation also recovers the known [[49,1,5]] and [[95,1,7]] triorthogonal codes. We also evaluate the performance of the obtained high-distance triorthogonal codes over the dephasing channel using different decoding methods, showing that automorphism ensemble decoding (AED) and guessing random additive-noise decoding (GRAND) are promising options.
Producing high-fidelity magic states using the smallest possible number of physical qubits and operations stands as a very important challenge to achieve fault-tolerant quantum computation at scale. Besides emerging proposals for alternative methods such as cultivation, magic state distillation remains essential for achieving very low error rates. Known distillation protocols are usually built through quantum codes derived from triorthogonal matrices. Here, exploiting the specific noise structure present in magic state distillation protocols, we show that classical error-correcting codes offer a simpler framework for deriving these protocols. This formulation is particularly well suited to systematic numerical and analytical studies of distillation protocols involving a fixed number of qubits. Specifically, we use a SAT solver to derive a series of no-go theorems that relate key figures of merit, including the number of qubits and the factory distance. In particular, considering the standard implementation as a circuit of Z-diagonal Pauli product rotations followed by measurements, we show that no TTT-to-TTT distillation protocol on fewer than eight qubits can exceed distance 3, and no TTT-to-CCZ\mathrm{CC}ZCCZ protocol distance 2. Our results also include new such distillation protocols with the smallest number of qubits for a given distance in the literature, namely distance 4 and 5 TTT-to-TTT protocols supported on 10 and 11 qubits, as well as distance 3 and 4 TTT-to-CCZ\mathrm{CC}ZCCZ distillation protocols supported on 9 and 10 qubits. Finally, going beyond unitary circuits by recycling qubits through mid-circuit measurement and reinitialization, we obtain an implementation of the distance-5 49T49T49T-to-1T1T1T protocol on only 5 active qubits.
Magic-state distillation is a major resource cost in fault-tolerant quantum computing. The cost of a magic-state factory depends strongly on its failure rate, which grows with the number of input magic states. Although symmetry-restricted methods have recently made distance two searches tractable, distance three and above have remained elusive at moderate input counts. We develop symmetry- and AI-assisted methods to search this regime. We present a unified binary-matrix formulation encompassing both triorthogonal-code and direct circuit searches. We show that distance at least three is equivalent to nonzero, pairwise distinct syndromes, separating the choice of syndromes from the search for compatible output gates. We restrict the syndrome search using group symmetry and language-model agents, followed by deterministic solving and independent verification. Our searches yield 699 factory classes, including 564 new ones. These include factories for pure-T states and factories with entangled outputs comprising combinations of T, CS, and CCZ magic states. The pure-T factories [[63, 11, 3]] and [[850, 128, 6]] achieve the lowest overhead exponents we know among protocols with at most 100 and 1000 inputs, respectively, with γ=1.589γ= 1.589γ=1.589 and γ=1.057γ= 1.057γ=1.057. Our [[1715, 287, 6]] factory, with γ=0.998γ= 0.998γ=0.998, is the smallest known pure-T factory with γ<1γ< 1γ<1. We also provide a context directory of search briefs and campaign notes with which readers can train their own agents and tailor the search to their requirements. With these results, we begin constructing an active, open-source repository of magic-state distillation protocols for the quantum community, supplemented by our methods and data, for the practical fault-tolerant quantum computing regime.
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.
In order to circumvent that no universal set of gates can be achieved on a given quantum error-correcting codes, additional gates can be performed through magic state injection. This however requires to have access to high fidelity magic state, which can be obtained by the distillation of low quality states to fewer higher quality ones. In this paper, we provide a family of distillation protocols able to produce diagonal states at every level of the Clifford hierarchy. This family is obtained recursively using code doubling techniques and leveraging qubit recycling, the fact that some idle qubits of the protocols can be measured and re-used. At a given level of the Clifford hierarchy, we show that the protocols we derive can be performed at arbitrary high distance on a fix number of logical qubits. This family recovers many known efficient protocols, and uncovers new ones, such as a 111→1111 \to 1111→1 protocol for ∣T⟩|T\rangle∣T⟩ state distillation at distance 777, which are the most compact in term of volume. Finally, we discuss possible extension of this framework for distillation protocols producing more than one output state.