Classification of Generalised Triorthogonal Codes through Length 54
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 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.
Related
- Asymptotically good binary triorthogonal codes and higher-level transversal gates
- On Constructing and Decoding Quantum Triorthogonal Codes
- Cyclic triorthogonal codes in prime dimension
- Short qudit triorthogonal codes from doubling
- Finite Classification of Transversal Diagonal Gates Beyond Stabilizer Codes
BibTeX
@misc{wills2026classification,
title = {{Classification of Generalised Triorthogonal Codes through Length 54}},
author = {Adam Wills and Shubham P. Jain and Shraddha Singh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30860},
eprint = {2609.30860},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30860},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classification of Generalised Triorthogonal Codes through Length 54
AU - Adam Wills
AU - Shubham P. Jain
AU - Shraddha Singh
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30860
UR - https://arxiv.org/abs/2609.30860
AB - 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.
ER -