{
  "abstract": "We present a structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of constant-coefficient reaction--diffusion equations on the unit cube, including the Poisson equation. Building on existing quantum finite element and finite difference methods, we express the preconditioned operator through explicit one-dimensional maps, tensor products, and norm-preserving interlevel transfers. Symmetric preconditioning gives multilevel coefficients, and its favorable conditioning alone does not ensure efficient recovery of a state in the original finite element nodal basis. We instead use the local $L^2$-orthonormal basis underlying the construction as the output representation of the same finite element function. Its coefficient norm equals the continuous $L^2$ norm. Assuming direct access to the preconditioned right-hand side, we prepare this orthonormal coefficient state with a represented function error at most $\\varepsilon$. The expected query complexity is \\[ \\mathcal{O}\\!\\Big(dL^3\\frac{\\|f\\|_{L^2}}{\\|u\\|_{L^2}} \\log\\!\\Big(2+\\frac{L\\|f\\|_{L^2}}{\\varepsilon}\\Big)\\Big), \\] where $u$ is the exact solution, $f$ is the right-hand side, $d$ is the spatial dimension, and $L=\\log_2(1/h)$ for mesh width $h$. The bound holds for $0<\\varepsilon\\le\\|u\\|_{L^2}/2$ and depends only polylogarithmically on mesh refinement for fixed continuous data. We also analyze linear-functional estimation and illustrate the construction and continuous error bounds with a one-dimensional UnitaryLab experiment.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06987",
  "author_total": 3,
  "authors": [
    {
      "name": "Yin Yang"
    },
    {
      "name": "Yue Yu"
    },
    {
      "name": "Long Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2y1rypmt42ydb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06987",
  "fulltext_url": null,
  "id": "2y1rypmt42ydb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06987",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7220919645",
  "pdf_url": "https://arxiv.org/pdf/2610.06987",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2y1rypmt42ydb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.06987",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06987"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of reaction-diffusion problems",
  "title_text": "A structure-aware construction of quantum BPX preconditioners for Q1 finite element discretizations of reaction-diffusion problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/2y1rypmt42ydb",
  "venue": "arXiv",
  "versions": []
}