{
  "abstract": "Quantum algorithms for linear algebra that access a matrix $A$ through quantum random access memory (QRAM) prepare quantum states proportional to the solutions of linear systems and related problems. Dequantized classical algorithms, when given sampling and query access to $A$, return sampling and query access to an approximate solution. Both kinds of algorithms use a number of queries that is polynomial in the normalized Frobenius norm $ρ=\\|A\\|_F/\\|A\\|$ and does not directly depend on the dimension. For linear systems at fixed condition number and accuracy, quantum algorithms use $\\widetilde O(ρ)$ queries, whereas the best dequantized algorithms use $\\widetilde O(ρ^4)$. We prove that this quartic gap is tight in the worst case, up to a logarithmic factor. For every constant $κ_0>1$, we construct dense linear systems with condition number at most $κ_0$ for which sampling from the solution within a sufficiently small constant total variation distance requires $Ω(ρ^4/\\logρ)$ classical queries but only $Θ(ρ)$ quantum queries. The same separation holds, for errors below smaller thresholds, for positive-definite systems, least squares, ridge regression, matrix powers and square roots, imaginary-time and real-time Hamiltonian evolution, and sampling from the top principal component. Our results therefore show that many quantum algorithms for linear algebra, achieve a near-quartic speedup in the worst case, over classical algorithms with sampling and query access. All of these classical lower bounds follow from the same framework, which is based on a lower bound for sampling from distributions that depend on a fixed set of Fourier coefficients of an unknown Boolean function.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.42069",
  "author_total": 4,
  "authors": [
    {
      "name": "Shouvanik Chakrabarti"
    },
    {
      "name": "Dylan Herman"
    },
    {
      "name": "Junhyung Lyle Kim"
    },
    {
      "name": "Guneykan Ozgul"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9fj2ahqqp63be.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.42069",
  "fulltext_url": null,
  "id": "9fj2ahqqp63be",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.42069",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.42069",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9fj2ahqqp63be.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:15Z",
      "native_id": "2610.42069",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.42069"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "The Quartic Gap Between Quantum and Dequantized Algorithms for Linear Algebra Is Tight",
  "title_text": "The Quartic Gap Between Quantum and Dequantized Algorithms for Linear Algebra Is Tight",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/9fj2ahqqp63be",
  "venue": "arXiv",
  "versions": []
}