Variational quantum simulation of the imaginary-time Lyapunov control for accelerating the ground-state preparation

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.