Classification of Generalised Triorthogonal Codes through Length 54

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 at most 38 (Nezami and Haah, Phys. Rev. A 106, 012437 (2022)). In this work, we significantly expand this classification to all protocols with lengths n at most 54. We restrict to distance d at least 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.