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Subspace diagonalisation methods have appeared recently as promising means toaccess the ground state and some excited states of molecular Hamiltonians byclassically diagonalising small matrices, whose elements can be efficientlyobtained by a quantum computer. The recently proposed Variational Quantum PhaseEstimation (VQPE) algorithm uses a basis of real time-evolved states, for whichthe 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 arbitrarymolecular systems and assess its performance and costs for the H2, H3+ and H6molecules. We also propose using Variational Fast Forwarding (VFF) to decreaseto quantum depth of time-evolution circuits for use in VQPE. We show that theapproximation provides a good basis for Hamiltonian diagonalisation even whenits 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, preservingthe linear cost of exact VQPE.

Variational Phase Estimation with Variational Fast Forwarding