  We present a new hybrid quantum-classical algorithm for optimizing unitary
coupled-cluster (UCC) wave functions deemed the projective quantum eigensolver
(PQE), amenable to near-term noisy quantum hardware. Contrary to variational
quantum algorithms, PQE optimizes a trial state using residuals (projections of
the Schr\"{o}dinger equation) rather than energy gradients. We show that the
residuals may be evaluated by simply measuring two energy expectation values
per element. We also introduce a selected variant of PQE (SPQE) that uses an
adaptive ansatz built from arbitrary-order particle-hole operators and
circumvents the expensive gradient-based selection procedures used in adaptive
variational quantum algorithms. PQE and SPQE are tested on a set of molecular
systems covering both the weak and strong correlation regimes, including
hydrogen clusters with 4-10 atoms and the BeH$_{2}$ molecule. When employing a
fixed ansatz, we find that PQE can converge UCC wave functions to essentially
identical energies as variational optimization while requiring fewer
computational resources. A comparison of SPQE and adaptive variational quantum
algorithms shows that - for ans\"{a}tze containing the same number of
parameters - the two methods yield results of comparable accuracy. Finally, we
show that SPQE performs similar to, and in some cases, better than selected
configuration interaction and the density matrix renormalization group on 1-3
dimensional strongly correlated H$_{10}$ systems.
