2026/07/20 by Xiantao Li
#quant-ph
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 \widetilde\mathcal O(√(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_ℓ2) 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 \widetilde\mathcal O(1+(Tε)-1), with only polylogarithmic dependence on h-1.