Spatial derivatives can introduce substantial mesh dependence in both the solver and the readout, even when only a few quantities of interest are needed. This talk presents multilevel quantum algorithms that estimate these quantities directly for elliptic and parabolic PDEs. The key idea is to encode fine–coarse cancellation before measurement, allowing small corrections to offset mesh-dependent costs. Under suitable approximation and quantum-access assumptions, the resulting algorithms reduce polynomial dependence on spatial resolution to polylogarithmic dependence. I will discuss connections with multigrid, hierarchical bases, and quantum preconditioning, and outline opportunities in stochastic dynamics, PDE inference, and design under uncertainty.