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

TL;DR

Q-Flow introduces a novel deep generative modeling approach using normalizing flows to efficiently simulate high-dimensional open quantum system dynamics via a PDE reformulation of the Husimi Q function.

Contribution

The paper presents Q-Flow, a new method that leverages normalizing flows and PDE reformulation to model open quantum system dynamics more effectively than existing techniques.

Findings

Q-Flow outperforms traditional PDE solvers in high-dimensional systems.

Q-Flow demonstrates scalability on complex quantum models.

The approach improves efficiency and accuracy in quantum dynamics simulations.

Abstract

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.