Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Wills, Adam, Jain, Shubham P., Singh, Shraddha
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G122673668, G1286236842, G1677143136, G1863003419, G2598886323, Partnership to Advance Throughput Computing (PATh), AI Institute: AI Research Institute for Fundamental Interactions, G419881185, G42916606, G4501827968, G498139845, G5076365615, Institute for Research and Innovation in Software for High Energy Physics (IRIS-HEP), G6652958823
OpenAlex ID: W7214680542