{
  "abstract": "The spectral condition number is a widely adopted measure of worst-case cost for quantum linear system solvers. Yet it can significantly overestimate the actual runtime for a typical problem instance. We present two quantum algorithms that produce the normalized solution $|x\\rangle$ of linear system $Ax=| b \\rangle$ to accuracy $ε$ with complexity independent of the condition number $κ=\\lVert A^{-1}\\rVert$. We focus on the standard input model where $A$ is accessed through a block encoding and $| b \\rangle$ is prepared by a unitary. But we also introduce an affine dilation model that encodes $A$ and $| b \\rangle$ jointly, allowing further refinements of the query complexity. Our truncation-based solver makes an optimal number of queries to $| b \\rangle$ and $\\operatorname{\\mathbf{O}}\\left(κ_{\\mathrm{eff}}\\operatorname{polylog}\\left(\\frac{κ_{\\mathrm{eff}}}ε\\right)\\right)$ queries to $A$. We prove a family of upper bounds on the effective condition number, including $κ_{\\mathrm{eff}}\\leq\\frac{\\lVert(A^\\dagger A)^{-t/2}|x\\rangle\\rVert^{1/t}}{ε^{1/t}}$ for positive even integer $t$ and $κ_{\\mathrm{eff}}\\leq\\frac{\\lVert A^{-1\\dagger}(A^\\dagger A)^{-(t-1)/2}|x\\rangle\\rVert^{1/t}}{ε^{1/t}}$ for positive odd $t$, overcoming the $κ$-barrier. Our filtering-based solver is extremely simple with a favorable runtime prefactor. In particular, the solver has query complexity $3\\frac{\\lVert A^{-1\\dagger}|x\\rangle\\rVert}ε$ to leading order when the solution norm is known. We then present a similarly simple solution norm estimator with the same asymptotic cost up to logarithmic factors. Our quantum linear system solvers thus substantially improve a recent algorithm of Li, enabling faster quantum linear system solving beyond the condition number.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.07691",
  "author_total": 3,
  "authors": [
    {
      "name": "Alexander M. Dalzell"
    },
    {
      "name": "Jianqiang Li"
    },
    {
      "name": "Yuan Su"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/atwv1g523z6jk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.07691",
  "fulltext_url": null,
  "id": "atwv1g523z6jk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.07691",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7167835168",
  "pdf_url": "https://arxiv.org/pdf/2607.07691",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/atwv1g523z6jk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:45Z",
      "native_id": "2607.07691",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.07691"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Faster quantum linear system solver beyond the condition number",
  "title_text": "Faster quantum linear system solver beyond the condition number",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/atwv1g523z6jk",
  "venue": "arXiv",
  "versions": []
}