{
  "abstract": "We give a bounded-error quantum polynomial-time algorithm for finding hidden parabolic subgroups of $\\mathrm{GL}_n(R)$ over any effectively presented finite chain ring $R$. We call auxiliary group elements or subgroups used to extract information through oracle queries \\emph{probes}. For chain length at least two, elementary abelian socle probes first recover the complete flag modulo the maximal ideal. When the residue field has size $q\\ge3$, we reconstruct the flag recursively, lifting rank-one terminal members one digit at a time. The intermediate digits reduce to abelian hidden subgroup problems, while the final digit is a finite-field multiple-shift problem solved using learning from subset and Fourier sampling. For $q=2$, where distinct flags can have the same stabilizer, we instead lift the conjugate parabolic subgroup through the congruence filtration. A final linear system determines it modulo normalizer directions. The algorithm outputs generators with probability at least $1-ε$ in time $\\mathrm{poly}(n,\\log|R|,\\log(1/ε))$. Consequences include $\\mathrm{GL}_n(\\mathbb{Z}/N\\mathbb{Z})$ for every $N\\ge2$ and, with an explicit maximal-order presentation, $\\mathrm{GL}_n(\\mathcal{O}_K/N\\mathcal{O}_K)$ for number fields $K$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28368",
  "author_total": 1,
  "authors": [
    {
      "name": "Muhammad Imran"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ckk97dbx5yaec.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28368",
  "fulltext_url": null,
  "id": "ckk97dbx5yaec",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28368",
  "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": "W7223158982",
  "pdf_url": "https://arxiv.org/pdf/2610.28368",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ckk97dbx5yaec.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:18Z",
      "native_id": "2610.28368",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28368"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Hidden parabolic subgroups of $\\mathrm{GL}_n$ over finite chain rings",
  "title_text": "Hidden parabolic subgroups of GL_n over finite chain rings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:18Z",
  "url": "https://research.draconitic.com/paper/ckk97dbx5yaec",
  "venue": "arXiv",
  "versions": []
}