Variational Methods for Quantum Systems

Hiroshi Takizawa, Akihide Oguchi

Daming Li

Variational methods are highly valuable computational tools for solving high-dimensional quantum systems. In this paper, we explore the effectiveness of three variational methods: density matrix renormalization group (DMRG), Boltzmann machine learning, and variational quantum eigensolver (VQE). We apply these methods to solve 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 three variational methods. Our aim is to demonstrate the power and effectiveness of these variational approaches in tackling complex quantum systems.

Béla Gazdy

R. Giachetti, V. Tognetti, Ruggero Vaia

Felan Amal, O Arundathi, Tavarekere Shivalingaswamy

The variational method in quantum mechanics plays a crucial role in estimating upper bound values of eigenenergies, particularly in the efficient determination of ground state energies. This paper introduces a formalism for the calculation of ground state energies in relativistic quantum systems employing the variational method. The proposed approach is applied to specific physical systems, allowing us to evaluate its effectiveness. Furthermore, we compare the results obtained through this method with existing literature, shedding light on its accuracy and applicability in the context of relativistic quantum systems.

Christopher M. Dawson, Jens Eisert, Tobias J. Osborne

We introduce a unified formulation of variational methods for simulating ground state properties of quantum many-body systems. The key feature is a novel variational method over quantum circuits via infinitesimal unitary transformations, inspired by flow equation methods. Variational classes are represented as efficiently contractible unitary networks, including the matrix-product states of density matrix renormalization, multiscale entanglement renormalization (MERA) states, weighted graph states, and quantum cellular automata. In particular, this provides a tool for varying over classes of states, such as MERA, for which so far no efficient way of variation has been known. The scheme is flexible when it comes to hybridizing methods or formulating new ones. We demonstrate the functioning by numerical implementations of MERA, matrix-product states, and a new variational set on benchmarks.