{
  "abstract": "This paper presents an efficient quantum spectral-Galerkin method for constant-coefficient second-order linear elliptic equations. The method employs a boundary-adapted Jacobi tensor-product basis that incorporates the boundary condition into the approximation space and produces Galerkin systems with polynomially bounded condition numbers and sparsity. For analytic solutions satisfying a suitable uniform regularity condition, spectral convergence allows the normalized Galerkin coefficient state to be prepared with complexity $\\mathrm{poly}(d,\\log(1/ε))$, where $d$ is the spatial dimension and $ε$ is the error tolerance, under standard sparse-access and state-preparation assumptions. At the state-preparation level, this improves the precision dependence over fixed-order finite-difference and finite-element discretizations. We further show how to estimate prescribed linear functionals of the solution, with an additional $O(1/ε)$ factor arising from norm and inner-product estimation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33095",
  "author_total": 2,
  "authors": [
    {
      "name": "Dongjie Wang"
    },
    {
      "name": "Sheng Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/96pqqkxcacfjv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33095",
  "fulltext_url": null,
  "id": "96pqqkxcacfjv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33095",
  "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",
      "physics.comp-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7223838721",
  "pdf_url": "https://arxiv.org/pdf/2610.33095",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/96pqqkxcacfjv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2610.33095",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33095"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.comp-ph",
      "subfield": "computational"
    }
  ],
  "title": "An Efficient Quantum Spectral-Galerkin Method for Linear Elliptic Equations",
  "title_text": "An Efficient Quantum Spectral-Galerkin Method for Linear Elliptic Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:35Z",
  "url": "https://research.draconitic.com/paper/96pqqkxcacfjv",
  "venue": "arXiv",
  "versions": []
}