  The variational quantum eigensolver (VQE) is a hybrid quantum-classical
algorithm for finding the minimum eigenvalue of a Hamiltonian that involves the
optimization of a parameterized quantum circuit. Since the resulting
optimization problem is in general nonconvex, the method can converge to
suboptimal parameter values which do not yield the minimum eigenvalue. In this
work, we address this shortcoming by adopting the concept of variational
adiabatic quantum computing (VAQC) as a procedure to improve VQE. In VAQC, the
ground state of a continuously parameterized Hamiltonian is approximated via a
parameterized quantum circuit. We discuss some basic theory of VAQC to motivate
the development of a hybrid quantum-classical homotopy continuation method. The
proposed method has parallels with a predictor-corrector method for numerical
integration of differential equations. While there are theoretical limitations
to the procedure, we see in practice that VAQC can successfully find good
initial circuit parameters to initialize VQE. We demonstrate this with two
examples from quantum chemistry. Through these examples, we provide empirical
evidence that VAQC, combined with other techniques (an adaptive termination
criteria for the classical optimizer and a variance-based resampling method for
the expectation evaluation), can provide more accurate solutions than "plain"
VQE, for the same amount of effort.
