Classification of Generalised Triorthogonal Codes through Length 54

Quantum Physics arXiv:2609.30860 (quant-ph) [Submitted on 25 Sep 2026] Title:Classification of Generalised Triorthogonal Codes through Length 54 Authors:Adam Wills, Shubham P. Jain, Shraddha Singh View a PDF of the paper titled Classification of Generalised Triorthogonal Codes through Length 54, by Adam Wills and 2 other authors View PDF HTML (experimental) 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. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2609.30860 [quant-ph] (or arXiv:2609.30860v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2609.30860 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Adam Wills [view email] [v1] Fri, 25 Sep 2026 06:10:26 UTC (73 KB) Full-text links: Access Paper: View a PDF of the paper titled Classification of Generalised Triorthogonal Codes through Length 54, by Adam Wills and 2 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-09 References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)