{
  "abstract": "Bu, Gu, and Jaffe introduced the quantum uniformity measures to analytically characterize the Clifford hierarchy. We prove that the quantum uniformity measures are the pullbacks of the matrix-valued uniformity functionals of Gowers and Hatami under the map $U\\mapsto(a\\mapsto W_aUW_a^*)$, which we call the Weyl-orbit embedding. We study the structure of the Weyl-orbit embedding, and among other results, establish a canonical decomposition of the ambient matrix-valued uniformity space into uniformity preserving twisted copies of the quantum uniformity space. The Weyl-orbit embedding identifies the Clifford hierarchy with Leibman polynomial maps. This identification yields a representation theoretic characterization of the Clifford hierarchy in terms of unitary representations of the universal polynomial groups introduced by the first author and Thom.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.16862",
  "author_total": 2,
  "authors": [
    {
      "name": "Asgar Jamneshan"
    },
    {
      "name": "Berke Ünal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cjhpp8khqtjec.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.16862",
  "fulltext_url": null,
  "id": "cjhpp8khqtjec",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.16862",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7164892791",
  "pdf_url": "https://arxiv.org/pdf/2606.16862",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cjhpp8khqtjec.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:34Z",
      "native_id": "2606.16862",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.16862"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "The Weyl-orbit embedding and a representation-theoretic characterization of the Clifford hierarchy",
  "title_text": "The Weyl-orbit embedding and a representation-theoretic characterization of the Clifford hierarchy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/cjhpp8khqtjec",
  "venue": "arXiv",
  "versions": []
}