Unsupervised Deep Neural Network Approach To Solve Fermionic Systems

Abstract

Solving the Schr\"{o}dinger equation for interacting many-body quantum systems faces computational challenges due to exponential scaling with system size. This complexity limits the study of important phenomena in materials science and physics. We develop an Artificial Neural Network (ANN)-driven algorithm to simulate fermionic systems on lattices. Our method uses Pauli matrices to represent quantum states, incorporates Markov Chain Monte Carlo sampling, and leverages an adaptive momentum optimizer. We demonstrate the algorithm's accuracy by simulating the Heisenberg Hamiltonian on a one-dimensional lattice, achieving results with an error in the order of $10^{-4}$ compared to exact diagonalization. Furthermore, we successfully model a magnetic phase transition in a two-dimensional lattice under an applied magnetic field. Importantly, our approach avoids the sign problem common to traditional Fermionic Monte Carlo methods, enabling the investigation of frustrated systems. This work demonstrates the potential of ANN-based algorithms for efficient simulation of complex quantum systems, opening avenues for discoveries in condensed matter physics and materials science.