Studying the dynamics of open quantum systems holds the potential to enable breakthroughs both in fundamental physics and applications to quantum engineering and quantum computation. Due to the high-dimensional nature of the problem, customized deep generative neural networks have been instrumental in modeling the high-dimensional density matrix $\rho$, which is the key description for the dynamics of such systems. 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.

翻译：研究开放量子系统的动力学有望在基础物理以及量子工程和量子计算应用中实现突破。由于该问题的高维特性，定制的深度生成神经网络在建模高维密度矩阵$\rho$方面发挥了关键作用——该矩阵是描述此类系统动力学的核心量。然而，$\rho$的复数值特性、归一化约束及其复杂的动力学过程，阻碍了开放量子系统与深度生成建模最新进展的无缝衔接。本文通过将开放量子系统动力学重新表述为对应的概率分布$Q$（即Husimi Q函数）的偏微分方程(PDE)，突破了这一限制。由此，我们利用归一化流等现成深度生成模型对Q函数进行无缝建模。此外，我们基于欧拉方法和含时变分原理，开发了学习高维PDE控制下归一化流演化的新方法。我们将该方案命名为Q-Flow，并通过耗散谐振子模型和耗散玻色子模型等开放量子系统模拟，验证了Q-Flow的可扩展性与计算效率。Q-Flow优于传统PDE求解器和最先进的物理信息神经网络求解器，尤其在高维系统中表现更佳。