  Quantum computers have been widely speculated to offer significant advantages
in obtaining the ground state of difficult Hamiltonian in chemistry and
physics. In this work, we first propose a Lyapunov control-inspired strategy to
accelerate the well-established imaginary-time method for ground-state
preparation. We also dig for the source of acceleration of the imaginary-time
process under Lyapunov control with theoretical understanding and dynamic
process visualization. To make the method accessible in the noisy
intermediate-scale quantum era, we further propose a variational form of the
algorithm that could work with shallow quantum circuits. Through numerical
experiments on a broad spectrum of realistic models, including molecular
systems, 2D Heisenberg models, and Sherrington-Kirkpatrick models, we show that
imaginary-time control may substantially accelerate the imaginary time
evolution for all systems and even generate orders of magnitude acceleration
(suggesting exponential-like acceleration) for challenging molecular
Hamiltonians involving small energy gaps as impressive special cases. Finally,
with a proper selection of the control Hamiltonian, the new variational quantum
algorithm does not incur additional measurement costs compared to the original
variational quantum imaginary-time algorithm.
