Learning thermodynamic master equations for open quantum systems

Abstract

The characterization of Hamiltonians and other components of open quantum dynamical systems plays a crucial role in quantum computing and other applications. Scientific machine learning techniques have been applied to this problem in a variety of ways, including by modeling with deep neural networks. However, the majority of mathematical models describing open quantum systems are linear, and the natural nonlinearities in learnable models have not been incorporated using physical principles. We present a data-driven model for open quantum systems that includes learnable, thermodynamically consistent terms. The trained model is interpretable, as it directly estimates the system Hamiltonian and linear components of coupling to the environment. We validate the model on synthetic two and three-level data, as well as experimental two-level data collected from a quantum device at Lawrence Livermore National Laboratory.

Figures (9)

• Figure 1: Expectation of trace distance over test trajectories. The shaded region indicates the maximum and minimum values of the trace distance for each time step.
• Figure 2: Predicted Bloch vector trajectories for a two-level quantum system governed by the nonlinear thermodynamic master equation. The trajectory yielding the largest average trace distance over time is chosen. The corresponding initial condition is $v_1(0) = 0.2216$, $v_2(0) = -0.1476$, and $v_3(0) = 0.9596$
• Figure 3: Predicted Bloch vector trajectories for a two-level quantum system governed by the linear Lindblad equation. The initial condition is identical to the trajectory in Figure \ref{['fig:two_level_one_trajectory']}, i.e., $v_1(0) = 0.2216$, $v_2(0) = -0.1476$, and $v_3(0) = 0.9596$.
• Figure 4: The third Bloch component of the exact trajectories for the Lindblad equation and nonlinear thermodynamic master equation. The initial condition for the Bloch vector is $v_1(0) = 0.2216$, $v_2(0) = -0.1476$, and $v_3(0) = 0.9596$.
• Figure 5: Expectation (solid line) and maximum and minimum (shaded region), over all controls, of trace distance between the density matrices from learned and true dynamics of a qutrit system trained over different training intervals (black vertical lines).