{
  "abstract": "In the quantum linear systems problem (QLSP), we are given query access to a $d$-sparse $N\\times N$ matrix $A$ with condition number $κ$, and the ability to prepare a quantum state proportional to a vector $\\vec b$. The goal is to output an $ε$-approximation to the quantum state proportional to the solution $\\vec x$ of $A\\vec{x}=\\vec{b}$. Following a long line of work, the best previously known quantum algorithms for the QLSP had query complexities $O(κd\\log(1/ε))$ and $κ\\sqrt d(κd/ε)^{o(1)}$, while the best known lower bounds were $Ω(κ\\log(1/ε))$ and $Ω(κ\\sqrt d)$. We improve these bounds and show that the complexity of the QLSP is $Θ(κ\\sqrt d\\log(1/ε))$. We also resolve an open problem of Berry and Childs by showing that any $N\\times N$ unitary can be implemented with bounded error using $O(\\sqrt N)$ queries to its matrix entries.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35660",
  "author_total": 3,
  "authors": [
    {
      "name": "Carlos Bravo-Prieto"
    },
    {
      "name": "Aram W. Harrow"
    },
    {
      "name": "Robin Kothari"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ef956bxczb7bt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35660",
  "fulltext_url": null,
  "id": "ef956bxczb7bt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35660",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7214708940",
  "pdf_url": "https://arxiv.org/pdf/2609.35660",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ef956bxczb7bt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2609.35660",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35660"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "An Optimal Quantum Linear Systems Algorithm",
  "title_text": "An Optimal Quantum Linear Systems Algorithm",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:12Z",
  "url": "https://research.draconitic.com/paper/ef956bxczb7bt",
  "venue": "arXiv",
  "versions": []
}