  We develop a quantum filter diagonalization method (QFD) that lies somewhere
between the variational quantum eigensolver (VQE) and the phase estimation
algorithm (PEA) in terms of required quantum circuit resources and conceptual
simplicity. QFD uses a set of of time-propagated guess states as a variational
basis for approximate diagonalization of a sparse Pauli Hamiltonian. The
variational coefficients of the basis functions are determined by the
Rayleigh-Ritz procedure by classically solving a generalized eigenvalue problem
in the space of time-propagated guess states. The matrix elements of the
subspace Hamiltonian and subspace metric matrix are each determined in quantum
circuits by a one-ancilla extended swap test, i.e., statistical convergence of
a one-ancilla PEA circuit. These matrix elements can be determined by many
parallel quantum circuit evaluations, and the final Ritz estimates for the
eigenvectors can conceptually be prepared as a linear combination over separate
quantum state preparation circuits. The QFD method naturally provides for the
computation of ground-state, excited-state, and transition expectation values.
We numerically demonstrate the potential of the method by classical simulations
of the QFD algorithm for an N=8 octamer of BChl-a chromophores represented by
an 8-qubit ab initio exciton model (AIEM) Hamiltonian. Using only a handful of
time-displacement points and a coarse, variational Trotter expansion of the
time propagation operators, the QFD method recovers an accurate prediction of
the absorption spectrum.
