{
  "abstract": "Can a quantum PDE algorithm avoid the polynomial cost of resolving a fine spatial mesh? For standard fixed-order discretizations, direct classical methods require work polynomial in $h^{-1}$, or equivalently in the number of spatial degrees of freedom $N_h=Θ(h^{-d})$. Direct quantum implementations of a parabolic semigroup still have coherent complexity $\\widetilde{\\mathcal O}(\\sqrt{T}/h)$, and gradient-dependent observables such as heat flux and dissipation introduce additional mesh dependence. Decay of the solution norm will further suppress the postselection probability for preparing a normalized final state. We develop a multilevel quantum algorithm that estimates linear and quadratic observables $directly$ and places the fine--coarse cancellation inside the circuit before measurement. A contour-based LCU reconstructs each target-time correction from a coherent family of shifted resolvent differences. Rather than block encoding the fine and coarse inverses separately, we encode their difference through a shifted Ritz--Schur factorization, exposing its $\\mathcal O(h_\\ell^2)$ two-grid normalization. For Fourier hierarchies, the corresponding SELECT oracle consists of a quantum Fourier or sine transform, a spectral-band selector, and reversible diagonal arithmetic. We also give a non-Fourier realization based on energy-orthogonal dyadic midpoint details in one dimension, together with structured tensor-product extensions under fixed-rank coefficient and access assumptions. For readouts with derivative order $0\\leχ\\le2$, optimized amplitude estimation removes $all$ polynomial dependence on the finest mesh size. Under the stated access assumptions, both linear and quadratic observables can be estimated with complexity $\\widetilde{\\mathcal O}(1+(Tε)^{-1})$, with only polylogarithmic dependence on $h^{-1}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.18113",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiantao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8xdq0n1dh261m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.18113",
  "fulltext_url": null,
  "id": "8xdq0n1dh261m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.18113",
  "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": "W7169849107",
  "pdf_url": "https://arxiv.org/pdf/2607.18113",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8xdq0n1dh261m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:22Z",
      "native_id": "2607.18113",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.18113"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Exponential Reduction of Mesh Dependence in Quantum Estimation of Parabolic PDE Observables",
  "title_text": "Exponential Reduction of Mesh Dependence in Quantum Estimation of Parabolic PDE Observables",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:22Z",
  "url": "https://research.draconitic.com/paper/8xdq0n1dh261m",
  "venue": "arXiv",
  "versions": []
}