Q-Flow: Generative Modeling for Differential Equations of Open Quantum Dynamics with Normalizing Flows
This work addresses a bottleneck in quantum physics and engineering by enabling more efficient simulations of open quantum systems, which is incremental as it builds on existing generative modeling techniques.
The paper tackles the challenge of modeling open quantum system dynamics by reformulating the problem into a partial differential equation for the Husimi Q function, enabling the use of off-the-shelf deep generative models like normalizing flows, and demonstrates superior scalability and efficiency compared to conventional PDE solvers and state-of-the-art physics-informed neural network solvers in high-dimensional systems.
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 $ρ$, which is the fundamental description for the dynamics of such systems, 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 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.