Tracking Non-Markovian Quantum Trajectories of a Superconducting Qubit from a Finite-Memory Bath*

High-fidelity quantum control of superconducting qubits requires determining the time-dependent Hamiltonian of the applied control and environmental coupling. While continuous state tracking has proven effective for determining Markovian qubit time-evolution, coupling to non-computational subspaces or a finite-memory bath can result in non-Markovian dynamics. We use quantum state tracking with continuous weak measurement to experimentally investigate non-Markovianity in a transmon qubit strongly coupled to an unmonitored bath qubit via a cross-resonance interaction. We measure state decay and revival, a signature of non-Markovianity, and train a recurrent neural network to reconstruct quantum trajectories for dynamics that are difficult to describe with single-qubit trajectory theory.