Variational methods for solving high dimensional quantum systems

Variational methods are highly valuable computational tools for solving high-dimensional quantum systems. In this paper, we explore the effectiveness of three variational methods: the density matrix renormalization group (DMRG), Boltzmann machine learning, and the variational quantum eigensolver (VQE). We apply these methods to two different quantum systems: the fermi-Hubbard model in condensed matter physics and the Schwinger model in high energy physics. To facilitate the computations on quantum computers, we map each model to a spin 1/2 system using the Jordan-Wigner transformation. This transformation allows us to take advantage of the capabilities of quantum computing. We calculate the ground state of both quantum systems and compare the results obtained using the three variational methods. By doing so, we aim to demonstrate the power and effectiveness of these variational approaches in tackling complex quantum systems.