  We present a technique for enhancing the estimation of quantum state
properties by incorporating approximate prior knowledge about the quantum state
of interest. This method involves performing randomized measurements on a
quantum processor and comparing the results with those obtained from a
classical computer that stores an approximation of the quantum state. We
provide unbiased estimators for expectation values of multi-copy observables
and present performance guarantees in terms of variance bounds which depend on
the prior knowledge accuracy. We demonstrate the effectiveness of our approach
through numerical experiments estimating polynomial approximations of the von
Neumann entropy and quantum state fidelities.
