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
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
T states with
n+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≤54. We restrict to distance
d≥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
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.
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.
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,
[[37,1,5]]3,
[[72,1,6]]3,
[[118,1,8]]3,
[[33,1,5]]5 and
[[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 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
11 for
p=5 and length
7 for
p=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.
We propose a new systematic construction of CSS-T codes from any given CSS code using a map ϕ. When ϕ is the identity map I, we retrieve the construction of hu2021mitigating and use it to prove the existence of asymptotically good binary CSS-T codes, resolving a previously open problem in the literature, and of asymptotically good quantum LDPC CSS-T codes. We analyze the structure of the logical operators corresponding to certain non-Clifford gates supported by the quantum codes obtained from this construction (ϕ=I), concluding that they always result in the logical identity. An immediate application of these codes in dealing with coherent noise is discussed. We then develop a new doubling transformation for obtaining triorthogonal codes, which generalizes the doubling construction presented in jain2024. Our approach permits using self-orthogonal codes, instead of only doubly-even codes, as building blocks for triorthogonal codes. This broadens the range of codes available for magic state distillation.
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⌊p/6⌋+1]]p codes.
The [[7,1,3]] Steane code and [[23,1,7]] quantum Golay code have been identified as good candidates for fault-tolerant quantum computing via code concatenation. These two codes have transversal implementations of all Clifford gates but require some other scheme for fault-tolerant
T gates. Using magic states, Clifford operations, and measurements is one common scheme, but magic-state distillation can have a large overhead. Code conversion is one avenue for implementing a universal gate set fault tolerantly without the use of magic-state distillation. Analogously to how the [[7,1,3]] Steane code can be fault tolerantly converted to and from the [[15,1,3]] Reed-Muller code which has a transversal
T gate, the [[23,1,7]] Golay code can be converted to a [[95,1,7]] triorthogonal code with a transversal
T gate. A crucial ingredient of this procedure is the [[49,1,5]] triorthogonal code, which can itself be seen as being related to the self-dual [[17,1,5]] two-dimensional color code. Additionally, a method for code conversion based on a transversal cnot between the codes, rather than stabilizer measurements, is described.