Resource-efficient quantum matrix processing with commutator scaling

We develop quantum algorithms for estimating properties of general matrix functions, with applications to phase estimation, Green’s function evaluation, and recovering measurement distributions of time-evolved states. The resulting methods exhibit commutator scaling in matrix parameters similar to that usually found for product formulae, lower circuit depth in all other parameters, and require only a single ancillary qubit. Our central primitive consists of classically postprocessing randomly chosen product formulae circuits, which mathematically corresponds to an approximation of a Richardson extrapolation.