  Variational quantum eigensolver (VQE) is regarded as a promising candidate of
hybrid quantum-classical algorithm for the near-term quantum computers.
Meanwhile, VQE is confronted with a challenge that statistical error associated
with the measurement as well as systematic error could significantly hamper the
optimization. To circumvent this issue, we propose classically-optimized VQE
(co-VQE), where the whole process of the optimization is efficiently conducted
on a classical computer. The efficacy of the method is guaranteed by the
observation that quantum circuits with a constant (or logarithmic) depth are
classically tractable via simulations of local subsystems. In co-VQE, we only
use quantum computers to measure nonlocal quantities after the parameters are
optimized. As proof-of-concepts, we present numerical experiments on quantum
spin models with topological phases. After the optimization, we identify the
topological phases by nonlocal order parameters as well as unsupervised machine
learning on inner products between quantum states. The proposed method
maximizes the advantage of using quantum computers while avoiding strenuous
optimization on noisy quantum devices. Furthermore, in terms of quantum machine
learning, our study shows an intriguing approach that employs quantum computers
to generate data of quantum systems while using classical computers for the
learning process.
