  Unitary Coupled Cluster (UCC) approaches are an appealing route to utilising
quantum hardware to perform quantum chemistry calculations, as quantum
computers can in principle perform UCC calculations in a polynomially scaling
fashion, as compared to the exponential scaling required on classical
computers. Current noisy intermediate scale quantum (NISQ) computers are
limited by both hardware capacity in number of logical qubits and the noise
introduced by the deep circuits required for UCC calculations using the
Variational Quantum Eigensolver (VQE) approach. We present a combined
classical--quantum approach where a stochastic classical UCC pre-processing
step is used to determine the important excitations in the UCC ansatz. The
reduced number of selected excitations are then used in a UCC-based VQE
calculation. This approach gives a systematically improvable approximation, and
we show that significant reductions in quantum resources can be achieved, with
simulations on the CH$_2$, N$_2$ and N$_2$H$_2$ molecules giving
sub-milliHartree errors.
