Q-Flow: Generative Modeling for Open Quantum Dynamics with Normalizing Flows

Studying the dynamics of open quantum systems can enable breakthroughs both in fundamental physics and applications to quantum engineering and quantum computation. Since the density matrix ρ is high-dimensional, customized deep generative neural networks have been instrumental in modeling ρ. However, the complex-valued nature and normalization constraints of ρ, as well as its complicated dynamics, prohibit a seamless connection between open quantum systems and the recent advances in deep generative modeling. Here, we lift that limitation by utilizing a reformulation of open quantum system dynamics to a partial differential equation (PDE) for a corresponding quasiprobability distribution Q, the Husimi Q function. Thus, we model the Q function seamlessly with off-the-shelf deep generative models such as normalizing flows. Additionally, we develop novel methods for learning normalizing flow evolution governed by high-dimensional PDEs based on the Euler method and the application of the time-dependent variational principle. We name the resulting approach Q-Flow and demonstrate the scalability and efficiency of Q-Flow on open quantum system simulations, including Fokker-Plank and dissipative Bose-Hubbard models. Q-Flow is superior to conventional PDE solvers and state-of-the-art physics-informed neural network solvers, especially in high-dimensional systems.