We present a general variational approach to determine the steady state of open quantum lattice systems via a neural network approach. The steady-state density matrix of the lattice system is constructed via a purified neural network ansatz in an ext
ended Hilbert space with ancillary degrees of freedom. The variational minimization of cost functions associated to the master equation can be performed using a Markov chain Monte Carlo sampling. As a first application and proof-of-principle, we apply the method to the dissipative quantum transverse Ising model.
We propose an efficient numerical method to compute configuration averages of observables in disordered open quantum systems whose dynamics can be unraveled via stochastic trajectories. We prove that the optimal sampling of trajectories and disorder
configurations is simply achieved by considering one random disorder configuration for each individual trajectory. As a first application, we exploit the present method to the study the role of disorder on the physics of the driven-dissipative Bose-Hubbard model in two different regimes: (i) for strong interactions, we explore the dissipative physics of fermionized bosons in disordered one-dimensional chains; (ii) for weak interactions, we investigate the role of on-site inhomogeneities on a first-order dissipative phase transition in a two-dimensional square lattice.
