抛物型偏微分方程可观测量量子估计中网格依赖性的指数降低

中文摘要 AI 辅助

量子偏微分方程算法能否避免解析精细空间网格的多项式成本？对于标准固定阶离散化，直接经典方法需要\(h^{-1}\)的多项式工作量。抛物半群的直接量子实现仍具有相干复杂度\(\widetilde{\mathcal O}(\sqrt{T}/h)\)，且与梯度相关的可观测量会引入额外网格依赖性。我们开发了一种多级量子算法，直接估计线性和二次可观测量，并在测量前将精细 - 粗糙抵消置于电路内部。基于轮廓的局部可计算酉（LCU）从相干的移位预解式差族重建每个目标时间校正。通过移位瑞利 - 舒尔分解对精细和粗糙逆的差进行编码，揭示其\(\mathcal O(h_\ell^2)\)双网格归一化。对于傅里叶层次结构，相应的SELECT预言机由量子傅里叶或正弦变换、谱带选择器和可逆对角算术组成。我们还给出了基于一维能量正交二进中点细节的非傅里叶实现，以及在固定秩系数和访问假设下的结构化张量积扩展。对于导数阶数\(0\le\chi\le2\)的读出，优化幅度估计消除了对最精细网格大小的所有多项式依赖性。在线性和二次可观测量都可以在\(\widetilde{\mathcal O}(1+(T\epsilon)^{-1})\)的复杂度下进行估计，仅对\(h^{-1}\)有多项式对数依赖性。

英文摘要

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 $h^{-1}$.

发表机构

• Pennsylvania State University（宾夕法尼亚州立大学）

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