  The problem of finding the ground state energy of a Hamiltonian using a
quantum computer is currently solved using either the quantum phase estimation
(QPE) or variational quantum eigensolver (VQE) algorithms. For precision
$\epsilon$, QPE requires $O(1)$ repetitions of circuits with depth
$O(1/\epsilon)$, whereas each expectation estimation subroutine within VQE
requires $O(1/\epsilon^{2})$ samples from circuits with depth $O(1)$. We
propose a generalised VQE algorithm that interpolates between these two regimes
via a free parameter $\alpha\in[0,1]$ which can exploit quantum coherence over
a circuit depth of $O(1/\epsilon^{\alpha})$ to reduce the number of samples to
$O(1/\epsilon^{2(1-\alpha)})$. Along the way, we give a new routine for
expectation estimation under limited quantum resources that is of independent
interest.
