Toward Efficient End-to-End Quantum Elliptic PDE Solvers: a Multilevel Correction Algorithm for Direct Observable Estimation

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 $O(h^{-\chi })$, $0\le \chi \le 2$, can be estimated to additive accuracy $\epsilon$ at cost $\tilde{O}({\epsilon }^{-1})$ using amplitude estimation or $\tilde{O}({\epsilon }^{-2})$ 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.