  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 $\rho$, which is the fundamental
description for the dynamics of such systems, is high-dimensional, customized
deep generative neural networks have been instrumental in modeling $\rho$.
However, the complex-valued nature and normalization constraints of $\rho$, 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
probability 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 the
dissipative harmonic oscillator and the dissipative bosonic model. Q-Flow is
superior to conventional PDE solvers and state-of-the-art physics-informed
neural network solvers, especially in high-dimensional systems.
