  The ability of near-term quantum computers to represent
classically-intractable quantum states has brought much interest in using such
devices for estimating the ground and excited state energies of fermionic
Hamiltonians. The usefulness of such near-term techniques, generally based on
the Variational Quantum Eigensolver (VQE), however, is limited by device noise
and the need to perform many circuit repetitions. This paper addresses these
challenges by generalizing VQE to consider wavefunctions in a subspace spanned
by classically tractable states and states that can be prepared on a quantum
computer. The manuscript shows how the ground and excited state energies can be
estimated using such "classical-boosting" and how this approach can be combined
with VQE Hamiltonian decomposition techniques. Unlike existing VQE approaches,
the sensitivity to sampling error and device noise approaches zero in the limit
where the classically tractable states are able to describe an eigenstate. A
detailed analysis of the measurement requirements in the simplest case, where a
single computational basis state is used to boost conventional VQE, shows that
the ground-state energy estimation of several closed-shell homonuclear diatomic
molecules can be accelerated by a factor of approximately 10-1000. The analysis
also shows that the measurement reduction of such single basis state boosting,
relative to conventional VQE, can be estimated using only the overlap between
the ground state and the computational basis state used for boosting.
