  Variational quantum eigensolver (VQE) emerged as a first practical algorithm
for near-term quantum computers. Its success largely relies on the chosen
variational ansatz, corresponding to a quantum circuit that prepares an
approximate ground state of a Hamiltonian. Typically, it either aims to achieve
high representation accuracy (at the expense of circuit depth), or uses a
shallow circuit sacrificing the convergence to the exact ground state energy.
Here, we propose the approach which can combine both low depth and improved
precision, capitalizing on a genetically-improved ansatz for hardware-efficient
VQE. Our solution, the multiobjective genetic variational quantum eigensolver
(MoG-VQE), relies on multiobjective Pareto optimization, where topology of the
variational ansatz is optimized using the non-dominated sorting genetic
algorithm (NSGA-II). For each circuit topology, we optimize angles of
single-qubit rotations using covariance matrix adaptation evolution strategy
(CMA-ES) -- a derivative-free approach known to perform well for noisy
black-box optimization. Our protocol allows preparing circuits that
simultaneously offer high performance in terms of obtained energy precision and
the number of two-qubit gates, thus trying to reach Pareto-optimal solutions.
Tested for various molecules (H$_2$, H$_4$, H$_6$, BeH$_2$, LiH), we observe
nearly ten-fold reduction in the two-qubit gate counts as compared to the
standard hardware-efficient ansatz. For 12-qubit LiH Hamiltonian this allows
reaching chemical precision already at 12 CNOTs. Consequently, the algorithm
shall lead to significant growth of the ground state fidelity for near-term
devices.
