Variational Neural Network Methods For Open Quantum Systems

Abstract

This study aims to investigate the application of neural network ansätze to open quantum systems with both local and non-local dissipation. Simulating quantum systems has always been a challenging task due to the rapidly growing dimensions of the Hilbert space with the number of constituents. This is especially severe in the case of open systems, which need to be described by density matrices instead of wavefunctions. Recent developments in machine learning techniques offer a promising alternative to efficiently represent quantum many body systems. In this work, we focus on a specific example of neural networks: Restricted Boltzmann Machines with an additional ancillary layer representing the degrees of freedom of the external bath and ensuring a positive semi-definite density matrix. We implement this ansatz for models of spin- 1 2 chains in the presence of an external magnetic field, with local and nonlocal dissipation. For a particular choice of the Hamiltonian parameters, this also corresponds to a realistic model describing a dense chain of two-level atoms that are dipole-dipole interacting and driven by an external field, where collective effects such as superradiance and subradiance emerge. We find that the neural network not only accurately predicts the observables of interest (mean values and correlations in the spin operators), but it also reproduces the exact steady state with high fidelity (>0.98). Moreover, the ansatz performs well and yields sensible results even for system sizes too large to be simulated by direct integration of the master equation. Although some error is present, ongoing research suggests that it will be possible to further minimize it