Arul Rhik Mazumder

We develop quantum algorithms for estimating properties of general matrix functions using product formulae, with applications to phase estimation, Green’s function evaluation, and sampling from time-evolved states. The resulting methods exhibit low depth, commutator scaling similar to that found for product formulae, and require only a single ancillary qubit. Our central primitive applies Richardson extrapolation to product formulae. By considering a randomized compilation scheme, we also give a protocol to statistically approximate measurement statistics of quantum states, which extends previous settings beyond observable estimation. We give refined analyses of gate complexities for k-local and power law systems; matrix ensembles with long tails; systems where commutator scaling is only understood up to a fixed order; and settings where there is a prior on input states -- those with fixed Fermion number.