  Variational quantum eigensolver (VQE), aiming at determining the ground state
energy of a quantum system described by a Hamiltonian on noisy intermediate
scale quantum (NISQ) devices, is among the most significant applications of
variational quantum algorithms (VQAs). However, the accuracy and trainability
of the current VQE algorithm are significantly influenced due to the barren
plateau (BP), the non-negligible gate error and limited coherence time in NISQ
devices. To tackle these issues, a gradient-sensitive alternate framework with
variable ansatz is proposed in this paper to enhance the performance of the
VQE. We first propose a theoretical framework for VA-VQE via alternately
solving a multi-objective optimization problem and the original VQE, where the
multi-objective optimization problem is defined with respect to cost function
values and gradient magnitudes. Then, we propose a novel implementation method
based on the double $\epsilon$-greedy strategy with the candidate tree and
modified multi-objective genetic algorithm. As a result, the local optima are
avoided both in ansatz and parameter perspectives, and the stability of output
ansatz is enhanced. The experimental results indicate that our framework shows
considerably improvement of the error of the found solution by up to 87.9%
compared with the hardware-efficient ansatz. Furthermore, compared with the
full-randomized VA-VQE implementation, our framework is able to obtain the
improvement of the error and the stability by up to 36.0% and 58.7%,
respectively, with similar quantum costs.
