Exponential Reduction of Mesh Dependence in Quantum Estimation of Parabolic PDE Observables
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 Nh=Θ(h−d). Direct quantum implementations of a parabolic semigroup still have coherent complexity 𝒪̃(√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 𝒪(hℓ2) 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≤χ≤2, 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 𝒪̃(1+(Tε)−1), with only polylogarithmic dependence on h−1.
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BibTeX
@misc{li2026exponential,
title = {{Exponential Reduction of Mesh Dependence in Quantum Estimation of Parabolic PDE Observables}},
author = {Xiantao Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.18113},
eprint = {2607.18113},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.18113},
}
RIS
TY - UNPB
TI - Exponential Reduction of Mesh Dependence in Quantum Estimation of Parabolic PDE Observables
AU - Xiantao Li
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2607.18113
UR - https://arxiv.org/abs/2607.18113
AB - 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 𝒪̃(√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 𝒪(h_ℓ^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≤χ≤2, 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 𝒪̃(1+(Tε)^−1), with only polylogarithmic dependence on h^−1.
ER -