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

Abstract

A central test case for quantum linear system algorithms (QLSA) is elliptic PDEs after a finite element discretization. Most existing analyses focus on preparing a normalized solution state. But an end-to-end quantum PDE solver must also extract physical quantities of interest, such as fluxes, currents, tractions, and energy. These outputs require quantum measurement, and their observable norms may grow like $h^{-\ddot{I}‡}$ with mesh size $h$, creating a readout bottleneck even when a quantum preconditioner reduces the condition-number dependence on $h$. We present a multilevel framework for this readout problem, motivated by the variance-reduction mechanism of multilevel Monte Carlo (MLMC), which is naturally compatible with a multi-level finite element discretization. Instead of estimating the full fine-grid observable directly, the method estimates a telescoping sum of interlevel corrections, so that the fine-coarse cancellation is exposed before quantum measurement. Our algorithm is based on Schur-complement factorization of the corrected Green's operator through a Ritz-complement map. For quantities of interest with readout order $\ddot{I}‡\le 2$, the multilevel estimator removes the polynomial $h$-dependent readout overhead. With amplitude estimation, the remaining statistical dependence is $\tilde{O}(1/\epsilon )$, i.e., Heisenberg scaling in the inference precision up to logarithmic factors and with direct sampling, the complexity is reduced to standard Monte Carlo scaling $\tilde{O}(1/\hat{I}{\mu }^2)$.