Learning Volterra Memory Kernels for Non-Markovian Qubit Dynamics

We develop a data-driven framework for identifying non-Markovian equations of motion for open quantum systems, demonstrated here for qubit-environment dynamics.

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 Pade approximants.

We establish well-posedness of the learning problem, ensuring existence of minimizers.

To assess performance, we construct synthetic data sets from representative test problems of increasing complexity: (i) exactly solvable pure dephasing, with correlation functions expressed in terms of special functions, (ii) a damped Jaynes-Cummings model with an analytic coherence kernel, (iii) a transverse Born model with frequency-resolved bath integrals and population-coherence coupling, and (iv) a non-rotating-wave quantum Rabi model whose memory kernel has no closed form.

Numerical experiments demonstrate that Pade captures nontrivial temporal structures such as oscillatory memory, algebraic tails, and phase-sensitive coherence transfer, and that the learned models generalize across ensembles of physically admissible initial states.

We perform a parametrization-invariant sensitivity analysis and show that the trajectories are insensitive to the unrecoverable parts of the kernel, so the learned models stay predictive despite severe ill-conditioning in kernel recovery.

These results together illustrate that data-driven rational approximation provides an effective route to identifying non-Markovian kernels of practical relevance in quantum technologies.