Learning Volterra Kernels for Non-Markovian Open Quantum Systems

TL;DR

A data-driven framework for identifying non-Markovian dynamical equations of motion for open quantum systems by vectorizing the reduced density matrix into a four-dimensional state vector and cast the dynamics as a Volterra integro-differential equation with an operator-valued memory kernel.

Abstract

We develop a data-driven framework for identifying non-Markovian dynamical equations of motion for open quantum systems. Starting from the Nakajima--Zwanzig formalism, we vectorize the reduced density matrix into a four-dimensional state vector and cast the dynamics as a Volterra integro-differential equation with an operator-valued memory kernel. The learning task is then formulated as a constrained optimization problem over the admissible operator space, where correlation functions are approximated by rational functions using Padé approximants. We establish well-posedness of the learnin

Figures (4)

• Figure 1: A numerical solution of Problem \ref{['eq:RegProb']}, with $\alpha=0$ and constrained by Equation \ref{['eq:Test1State']}, using the [4/4] Pade approximant expressed by Equation \ref{['eq:PadeHyp']}. We use a subohmic parameter of $p=1/2$ and a superohmic parameter of $p=2.$ The construction of the synthetic data used in this study is discussed in the main text.
• Figure 2: We display the effect of noise on the regression by performing the same study as in the Ohmic case shown in Figure \ref{['fig:Test1']}, yet introduce corruption in the synthetic data in the way described in the main text. We observe that even a small Tikhonov regularization does well to smooth out singular features of the correlation function, while maintaining a fairly reasonable fit to the synthetic data. Decreasing the regularization leads to a montonically decreasing singular feature in the bottom panel near $t=0.5$.
• Figure 3: In the top panel, we display the dynamics, governed by eqs \ref{['eq:rho-clean']}, evolved from an out of training sample in the space of initial density matrices $\varrho$ and generated by the learned correlation functions in bottom panel. In the off-diagonal components, real parts correspond to open circles while imaginary parts correspond to solid circles. We observe that despite a reasonable generalization and that Tikhonov regularization ensures smoothness, the learned correlation functions are not close by any metric to the numerically evaluated correlation functions embedded in eqs \ref{['eq:rho-clean']}.
• Figure 4: Just as in Figure \ref{['fig:Test2']}, we display the dynamics, evolved from an out of training sample in the space of initial density matrices $\varrho$ and generated by learned correlation functions. This shows that a rational hypothesis of the correlation kernel is able to reproduce the dynamics of the density matrix on moderate time scales.