  Obtaining precise estimates of quantum observables is a crucial step of
variational quantum algorithms. We consider the problem of estimating
expectation values of molecular Hamiltonians, obtained on states prepared on a
quantum computer. We propose a novel estimator for this task, which is locally
optimised with knowledge of the Hamiltonian and a classical approximation to
the underlying quantum state. Our estimator is based on the concept of
classical shadows of a quantum state, and has the important property of not
adding to the circuit depth for the state preparation. We test its performance
numerically for molecular Hamiltonians of increasing size, finding a sizable
reduction in variance with respect to current measurement protocols that do not
increase circuit depths.
