{
  "abstract": "Equational reasoning about circuits underpins quantum-circuit optimisation and verification. The qubit CNOT-dihedral fragment achieves this through phase polynomials, layered normal forms, and a complete equational theory; we develop the corresponding theory for prime-dimensional qudits, where basis labels, value controls, and phase exponents share prime-field arithmetic. We first describe reversible affine circuits over Fd as transformations x->Ax+b, with an affine normal form extending Lafont's linear normal form by translations. Adjoining finite-angle diagonal phases by polynomial degree yields linear, quadratic (odd prime), and cubic (prime greater than 3) calculi whose binomial-basis identities expose the mixed diagonal gates forced by affine transport. These calculi have unique phase-affine normal forms and are complete: semantic equality coincides with derivable equality, giving a prime-dimensional phase-polynomial analogue of the CNOT-dihedral equational theory.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2603.06466",
  "author_total": 1,
  "authors": [
    {
      "name": "Colin Blake"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/289dwce7zy8x0.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "289dwce7zy8x0",
  "issns": [],
  "landing_url": "https://hal.science/hal-05540565v2",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.info"
    ],
    [
      "hal",
      "0.phys"
    ],
    [
      "hal",
      "1.info.info-lo"
    ],
    [
      "hal",
      "1.phys.qphy"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05540565/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/289dwce7zy8x0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:01Z",
      "native_id": "hal-05540565",
      "source": "hal",
      "url": "https://hal.science/hal-05540565v2"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "hal:1.phys.qphy",
      "subfield": "quantum"
    }
  ],
  "title": "Completeness for Prime-Dimensional Phase-Affine Circuits",
  "title_text": "Completeness for Prime-Dimensional Phase-Affine Circuits",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:01Z",
  "url": "https://research.draconitic.com/paper/289dwce7zy8x0",
  "venue": "HAL",
  "versions": []
}