{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30860",
  "author_total": 3,
  "authors": [
    {
      "name": "Adam Wills"
    },
    {
      "name": "Shubham P. Jain"
    },
    {
      "name": "Shraddha Singh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fyqwy728gvhjk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30860",
  "fulltext_url": null,
  "id": "fyqwy728gvhjk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30860",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7214680542",
  "pdf_url": "https://arxiv.org/pdf/2609.30860",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fyqwy728gvhjk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:52Z",
      "native_id": "2609.30860",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30860"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Classification of Generalised Triorthogonal Codes through Length 54",
  "title_text": "Classification of Generalised Triorthogonal Codes through Length 54",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:52Z",
  "url": "https://research.draconitic.com/paper/fyqwy728gvhjk",
  "venue": "arXiv",
  "versions": []
}