{
  "abstract": "We introduce a framework based on classical quasi group codes to define a class of quantum CSS codes, called quantum group codes, supporting transversal multi-control-$Z$ gates which are both addressable and parallelizable, thus allowing to efficiently implement circuits composed of non-Clifford gates at the logical level. Building on this, we use a lifting procedure of classical AG codes established from class field theory to construct good quantum group codes with improved decoding complexity and logical multi-control-$Z$ gate parallelizability. More precisely, on input a good quantum AG code over the alphabet $\\mathbb F_q$ with transversal $\\mathsf{C}^m\\mathsf Z$ gate, we apply this lifting procedure to its underlying classical AG code and obtain a quantum group code over the alphabet $\\mathbb F_{q^2}$ supporting a transversal $\\mathsf{C}^m\\mathsf Z$ gate as well as addressable and parallelizable $\\mathsf{C}^{m-1}\\mathsf Z$ gates. In addition, this quantum code admits a quasi-quadratic time decoder with a linear decoding radius. This is to be compared with the previous quantum AG codes which have a cubic-time decoder. Hence, our work implies a decrease of the time complexity of state-of-the-art magic-state distillation protocols by an almost linear factor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.27211",
  "author_total": 2,
  "authors": [
    {
      "name": "Jean Gasnier"
    },
    {
      "name": "Virgile Guémard"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0p7w1dyg4hpx1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.27211",
  "fulltext_url": null,
  "id": "0p7w1dyg4hpx1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.27211",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7166053676",
  "pdf_url": "https://arxiv.org/pdf/2606.27211",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0p7w1dyg4hpx1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:45Z",
      "native_id": "2606.27211",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.27211"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Quantum group codes for non-Clifford logic: enhanced decoding, addressability and parallelizability",
  "title_text": "Quantum group codes for non-Clifford logic: enhanced decoding, addressability and parallelizability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/0p7w1dyg4hpx1",
  "venue": "arXiv",
  "versions": []
}