{
  "abstract": "Resource analyses for quantum programs commonly provide safe upper bounds on qubits, gates, or non-Clifford operations. We study a single-qubit dyadic fragment in which the deepest exact resource layer can instead be inferred exactly. Starting from the normalized-midpoint misty-state syntax, we define dyadic program states and the real cyclotomic tower $K_D=\\mathbb Q(2\\cos(π/2^D))$. For the projective single-qubit Clifford-cyclotomic tower $\\widehat G_D\\subseteq SO(3)$, we prove the relative intersection law $\\widehat G_D\\cap SO(3,K_d)=\\widehat G_d$ for $d\\le D$. Combining this descent theorem with an arithmetic-closure invariant for adaptive stabilizer processing yields $\\mathrm{Fdepth}(U)=\\mathrm{Adepth}(U)=\\mathrm{Rdepth}(U)$: the smallest real cyclotomic field containing the Bloch matrix of an exactly synthesizable channel, the shallowest dyadic gate alphabet realizing it up to global phase, and the shallowest dyadic program-state layer sufficient for exact stabilizer-assisted implementation coincide. At that common minimum, known fixed-alphabet synthesis gives minimum primitive-rotation count. We package the result as an exact static resource effect and outline a certificate-producing compiler.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06909",
  "author_total": 2,
  "authors": [
    {
      "name": "Tony Newton"
    },
    {
      "name": "Dan-Adrian German"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/131x28t59rgw6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06909",
  "fulltext_url": null,
  "id": "131x28t59rgw6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06909",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06909",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/131x28t59rgw6.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:20:56Z",
      "native_id": "2611.06909",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06909"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Arithmetic Depth, Minimal Gate Alphabets, and Exact Program-State Resources for Single-Qubit Dyadic Programs",
  "title_text": "Arithmetic Depth, Minimal Gate Alphabets, and Exact Program-State Resources for Single-Qubit Dyadic Programs",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:21Z",
  "url": "https://research.draconitic.com/paper/131x28t59rgw6",
  "venue": "arXiv",
  "versions": []
}