  In the current noisy intermediate scale quantum era of quantum computation,
available hardware is severely limited by both qubit count and noise levels,
precluding the application of many current hybrid quantum-classical algorithms
to non-trivial quantum chemistry problems. In this paper we propose applying
some of the fundamental ideas of conventional Quantum Monte Carlo algorithms --
stochastic sampling of both the wavefunction and the Hamiltonian -- to quantum
algorithms in order to significantly decrease quantum resource costs. In the
context of an imaginary-time propagation based projective quantum eigensolver,
we present a novel approach to estimating physical observables which leads to a
two order of magnitude reduction in the required sampling of the quantum state
to converge the ground state energy of a system relative to current
state-of-the-art eigensolvers. The method can be equally applied to
excited-state calculations and, combined with stochastic approximations of the
system Hamiltonian, provides a promising near-term approach to Hamiltonian
simulation for general chemistry on quantum devices.
