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arXiv:2610.08012

Cyclic triorthogonal codes in prime dimension

Shiroman Prakash

quant-phcs.ITSubmitted Oct 6, 2026• Announced Oct 7, 2026• 0 citations
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Abstract

We study cyclic triorthogonal codes in prime dimension $p$, on which a transversal third-level diagonal gate acts as a logical non-Clifford gate. For odd $p$, we use a definition of triorthogonality that does not require self-orthogonality. For cyclic codes of length coprime to $p$, triorthogonality reduces to a sumset condition on the spectral support, which makes exhaustive search feasible: for two natural constructions, we enumerate all maximal codes of this kind for $p=3$ up to length $121$ and $p=5,7$ up to length $96$. For qubits, we prove that a cyclic code of odd length is triply even if and only if it is classically triorthogonal, giving a complete description of triply-even cyclic codes, and we determine all such codes up to length $763$. For qudits, we find codes such as $[[19,1,3]]_3$ and $[[7,1,3]]_7$, which improve on the shortest Reed-Muller and Reed-Solomon codes in both overhead exponent and distillation threshold. The latter is the first member of an infinite family of $[[p,1,2\lfloor p/6\rfloor+1]]_p$ codes.

Comments
24 pages, 2 tables. Code and data: https://github.com/spdei/cyclic-triorthogonal-codes

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We study cyclic triorthogonal codes in prime dimension ppp, on which a transversal third-level diagonal gate acts as a logical non-Clifford gate. For odd ppp, we use a definition of triorthogonality that does not require self-orthogonality. For cyclic codes of length coprime to ppp, triorthogonality reduces to a sumset condition on the spectral support, which makes exhaustive search feasible: for two natural constructions, we enumerate all maximal codes of this kind for p=3p=3p=3 up to length 121121121 and p=5,7p=5,7p=5,7 up to length 969696. For qubits, we prove that a cyclic code of odd length is triply even if and only if it is classically triorthogonal, giving a complete description of triply-even cyclic codes, and we determine all such codes up to length 763763763. For qudits, we find codes such as [[19,1,3]]3[[19,1,3]]_3[[19,1,3]]3​ and [[7,1,3]]7[[7,1,3]]_7[[7,1,3]]7​, which improve on the shortest Reed-Muller and Reed-Solomon codes in both overhead exponent and distillation threshold. The latter is the first member of an infinite family of [[p,1,2⌊p/6⌋+1]]p[[p,1,2\lfloor p/6\rfloor+1]]_p[[p,1,2⌊p/6⌋+1]]p​ codes.
We construct short triorthogonal codes for qudits of odd prime dimension by generalizing doubling, which combines a triorthogonal code with a code admitting a transversal phase gate, and raises the distance by up to two. This gives codes with parameters [[28,1,4]]3[[28,1,4]]_3[[28,1,4]]3​, [[37,1,5]]3[[37,1,5]]_3[[37,1,5]]3​, [[72,1,6]]3[[72,1,6]]_3[[72,1,6]]3​, [[118,1,8]]3[[118,1,8]]_3[[118,1,8]]3​, [[33,1,5]]5[[33,1,5]]_5[[33,1,5]]5​ and [[20,1,4]]7[[20,1,4]]_7[[20,1,4]]7​, each shorter than the cyclic, concatenated and Reed-Muller codes of the same distance, and a doubling construction of the [[15,1,3]]3[[15,1,3]]_3[[15,1,3]]3​ code. When the columns of a generator matrix are regarded as charged points, triorthogonality becomes the vanishing of multipole moments up to octupole order, and doubling becomes a double layer of opposite charges; this picture explains why inputs with nontrivial coefficients can give shorter codes. We also show by exhaustive search that the shortest codes of distance three have length 111111 for p=5p=5p=5 and length 777 for p=7p=7p=7. Combined with high-rate distance-two codes, the new codes reduce the cost of qutrit magic state distillation by up to an order of magnitude in regimes with high input noise.
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.
We construct quantum codes that support transversal CCZCCZCCZ gates over qudits of arbitrary prime power dimension qqq (including q=2q=2q=2) such that the code dimension and distance grow linearly in the block length. The only previously known construction with such linear dimension and distance required a growing alphabet size qqq (Krishna & Tillich, 2019). Our codes imply protocols for magic state distillation with overhead exponent γ=log⁡(n/k)/log⁡(d)→0γ=\log(n/k)/\log(d)\rightarrow 0γ=log(n/k)/log(d)→0 as the block length n→∞n\rightarrow\inftyn→∞, where kkk and ddd denote the code dimension and distance respectively. It was previously an open question to obtain such a protocol with a contant alphabet size qqq. We construct our codes by combining two modular components, namely, (i) a transformation from classical codes satisfying certain properties to quantum codes supporting transversal CCZCCZCCZ gates, and (ii) a concatenation scheme for reducing the alphabet size of codes supporting transversal CCZCCZCCZ gates. For this scheme we introduce a quantum analogue of multiplication-friendly codes, which provide a way to express multiplication over a field in terms of a subfield. We obtain our asymptotically good construction by instantiating (i) with algebraic-geometric codes, and applying a constant number of iterations of (ii). We also give an alternative construction with nearly asymptotically good parameters (k,d=n/2O(log⁡∗n)k,d=n/2^{O(\log^*n)}k,d=n/2O(log∗n)) by instantiating (i) with Reed-Solomon codes and then performing a superconstant number of iterations of (ii).
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.
The non-local interactions in several quantum device architectures allow for the realization of more compact quantum encodings while retaining the same degree of protection against noise. Anticipating that short to medium-length codes will soon be realizable, it is important to construct stabilizer codes that, for a given code distance, admit fault-tolerant implementations of logical gates with the fewest number of physical qubits. To this aim, we construct three kinds of codes encoding a single logical qubit for distances up to 31. First, we construct the smallest known doubly even codes, all of which admit a transversal implementation of the Clifford group. Applying a doubling procedure arXiv:1509.03239 to such codes yields the smallest known weak triply even codes for the same distances and number of encoded qubits. This second family of codes admit a transversal implementation of the logical T-gate. Relaxing the triply even property, we obtain our third family of triorthogonal codes with an even lower overhead at the cost of requiring additional Clifford gates to achieve the same logical operation. To our knowledge, these are the smallest known triorthogonal codes for their respective distances. While not qLDPC, the stabilizer generator weights of the code families with transversal T-gates scale roughly as the square root of their lengths.
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.