{
  "abstract": "We study quantum random self-reductions for linear problems over finite fields. Let $M\\in\\mathbb{F}^{n\\times n}$ be an arbitrary matrix, and let $\\mathcal{O}$ be an oracle that agrees with the linear map $x\\mapsto Mx$ on an $\\varepsilon$-fraction of inputs $x\\sim\\mathbb{F}^n$. Given coherent access to $\\mathcal{O}$ and coherent entry access to $M$, we give a uniform quantum reduction that computes $Mx$ on any prescribed input $x$ with probability at least $2/3$ in time $\\widetilde{O}(nT^{1/3})$, for $n\\le T\\le n^{3/2}$ and constant field size and $\\varepsilon$, where $T$ is the cost of one coherent query to $\\mathcal{O}$. In particular, when $T=\\widetilde{O}(n)$, the reduction runs in time $\\widetilde{O}(n^{4/3})$, improving the $\\widetilde{O}(n^{3/2}+T)$ reduction of Asadi, Golovnev, Gur, Shinkar, and Subramanian (SODA 2024). Our reduction uses the Bogolyubov--Ruzsa subspace guaranteed by additive combinatorics, but it avoids learning this subspace explicitly, which was computationally expensive for the previous reduction; in particular, it does not recover a basis for its orthogonal complement. The main technical step is to decompose the inputs into sparse pieces and find a vector that lies outside the Bogolyubov--Ruzsa subspace via a quantum search based on amplitude amplification. This yields a tunable tradeoff between the cost of querying the average-case oracle and the cost of verifying matrix-vector products.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39823",
  "author_total": 3,
  "authors": [
    {
      "name": "Vahid R. Asadi"
    },
    {
      "name": "Shuichi Hirahara"
    },
    {
      "name": "Nobutaka Shimizu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/44tyn6d9jwg54.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39823",
  "fulltext_url": null,
  "id": "44tyn6d9jwg54",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39823",
  "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": "W7215098040",
  "pdf_url": "https://arxiv.org/pdf/2609.39823",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/44tyn6d9jwg54.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39823",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39823"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Improved Quantum Random Self-Reduction for Linear Problems",
  "title_text": "Improved Quantum Random Self-Reduction for Linear Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/44tyn6d9jwg54",
  "venue": "arXiv",
  "versions": []
}