{
  "abstract": "Spatial derivatives pose a basic challenge for quantum algorithms for partial differential equations: their standard discretizations have operator norms that grow as the mesh is refined, leading to large block-encoding normalization factors. Even when preconditioning improves access to the solution, estimating derivative-dependent physical observables can reintroduce polynomial dependence on the inverse mesh width. We develop a multilevel quantum algorithm for estimating linear and quadratic observables of elliptic equations that addresses this readout cost. The algorithm decomposes the output into corrections across nested discretizations and realizes fine--coarse cancellation coherently, before measurement. A Ritz-complement factorization connects the $O(h^2)$ size of the corrected response with a block encoding at the same scale, allowing this decay to offset growing readout normalizations. Under efficient quantum access to the corrected coordinates, input data, and readouts, observables with readout scale $O(h^{-χ})$, $0\\leχ\\le2$, can be estimated to additive accuracy $ε$ at cost $\\widetilde O(ε^{-1})$ using amplitude estimation or $\\widetilde O(ε^{-2})$ using direct sampling, with only polylogarithmic dependence on the inverse finest mesh width $h_L^{-1}$. We give explicit constructions of the required operator oracles for one-dimensional piecewise linear finite elements with diffusion piecewise constant on a fixed coarsest partition, and for Fourier spectral hierarchies. Output-specific bias estimates connect the discrete results to continuum accuracy.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.01270",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiantao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3h8k4rkrcjhny.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.01270",
  "fulltext_url": null,
  "id": "3h8k4rkrcjhny",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.01270",
  "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",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Computing Algorithms and Architecture"
    ]
  ],
  "openalex": "W7163203620",
  "pdf_url": "https://arxiv.org/pdf/2606.01270",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3h8k4rkrcjhny.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:39Z",
      "native_id": "2606.01270",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.01270"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Toward Efficient End-to-End Quantum Elliptic PDE Solvers: a Multilevel Correction Algorithm for Direct Observable Estimation",
  "title_text": "Toward Efficient End-to-End Quantum Elliptic PDE Solvers: a Multilevel Correction Algorithm for Direct Observable Estimation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:39Z",
  "url": "https://research.draconitic.com/paper/3h8k4rkrcjhny",
  "venue": "arXiv",
  "versions": []
}