Learned potentials for bosonic matter wave manipulation

An important capability recently developed in the field of variational quantum algorithms (VQAs) is the learning of a target unitary operation using a parameterized quantum circuit, so-called quantum-assisted quantum compilation (QAQC). In the context of discrete or continuous-variable quantum systems, the QAQC procedure allows to encode a target unitary in a depth-optimal or, more generally, resource-optimal circuit. In this work, we extend the concept of QAQC to bosonic systems, specifically for experiments exploiting the matter-wave nature of Bose-Einstein Condensates (BEC) trapped in reconfigurable optical potentials. In this case, what would be considered a quantum algorithm for a discrete quantum system becomes an experimental protocol for a BEC system. The VQA for QAQC in our BEC setting involves minimization of a cost function, which quantifies the distinguishability of BEC states acted upon by either the target or parameterized unitary.



We show that the cost obtained from the fully quantum mechanical dynamics in the two-mode approximation is equivalent to the cost obtained from simulation of the GPE dynamics of an interacting system. We also test how the evolution time of the system in the GPE simulation, and the choice of training set impact the cost function and its convergence. To quantify the performance of our BEC QAQC scheme on generic matter wave states, we prove covering number-based generalization bounds which depend on the number of atoms and number of matter wave states used in computing the cost function.