  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, offering an
alternative to gradient-based selection procedures. 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 disentangled
(factorized) 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 in terms of energy accuracy for a given number of variational
parameters.
