{
  "abstract": "For squarefree $f\\in\\F_q[x]$, Berlekamp's fixed-point algebra is $B\\cong\\F_q^r$, where $r$ is the number of irreducible factors. We give an exact quantum splitting procedure in a circuit model permitting efficiently computable, input-dependent single-qubit rotations. For a block with $s\\ge2$ factors, a quadratic-character test (odd $q$) or absolute-trace test (characteristic $2$) is nonconstant with an exactly known probability $p_{q,s}\\ge1/2$. Its value depends only on $q,s$, not on the hidden factors. An auxiliary-qubit reduction to success probability $1/4$ makes one Grover iteration exact. Thus factorization takes exactly $r-1$ splitting rounds and $O(n^3\\log q)$ quantum $\\F_q$-operations, with deterministic polynomial classical preprocessing and bookkeeping. The method also extracts primitive idempotents of arbitrary finite-dimensional commutative unital algebras given by structure constants in $O(n^4\\log q)$ quantum $\\F_q$-operations, staying over $\\F_q$ throughout. Computing the radical and passing to the center extends it to the decomposition of $A/\\Rad(A)$ into simple two-sided ideals, for an arbitrary finite-dimensional associative $\\F_q$-algebra $A$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30340",
  "author_total": 1,
  "authors": [
    {
      "name": "Muhammad Imran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dzt94eyhnyst8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30340",
  "fulltext_url": null,
  "id": "dzt94eyhnyst8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30340",
  "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.CC"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7204884615",
  "pdf_url": "https://arxiv.org/pdf/2608.30340",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dzt94eyhnyst8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:52Z",
      "native_id": "2608.30340",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30340"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Exact quantum splitting and the structure of finite algebras",
  "title_text": "Exact quantum splitting and the structure of finite algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:41Z",
  "url": "https://research.draconitic.com/paper/dzt94eyhnyst8",
  "venue": "arXiv",
  "versions": []
}