Quantum algorithms for solving linear partial differential equations promise substantial speedups over classical methods, but poor conditioning, operator encoding, and measurement readout remain key challenges. In this talk, I'll describe a multiscale wavelet-based encoding of differential operators that, by using a linear combination of unitaries with preconditioning, achieves algorithmic complexity essentially independent of the condition number. Using an SLAC formulation of the derivative based on the Shannon wavelet transform preserves continuum momentum dependence on a lattice while accommodating nonlocal terms with efficient state-preparation circuits.