  Subspace diagonalisation methods have appeared recently as promising means to
access the ground state and some excited states of molecular Hamiltonians by
classically diagonalising small matrices, whose elements can be efficiently
obtained by a quantum computer. The recently proposed Variational Quantum Phase
Estimation (VQPE) algorithm uses a basis of real time-evolved states, for which
the energy eigenvalues can be obtained directly from the unitary matrix U =
exp(-iHt), which can be computed with cost linear in the number of states used.
In this paper, we report a circuit-based implementation of VQPE for arbitrary
molecular systems and assess its performance and costs for the H2, H3+ and H6
molecules. We also propose using Variational Fast Forwarding (VFF) to decrease
to quantum depth of time-evolution circuits for use in VQPE. We show that the
approximation provides a good basis for Hamiltonian diagonalisation even when
its fidelity to the true time evolved states is low. In the high fidelity case,
we show that the approximate unitary U can be diagonalised instead, preserving
the linear cost of exact VQPE.
