  The application of machine learning techniques to solve problems in quantum
control together with established geometric methods for solving optimisation
problems leads naturally to an exploration of how machine learning approaches
can be used to enhance geometric approaches to solving problems in quantum
information processing. In this work, we review and extend the application of
deep learning to quantum geometric control problems. Specifically, we
demonstrate enhancements in time-optimal control in the context of quantum
circuit synthesis problems by applying novel deep learning algorithms in order
to approximate geodesics (and thus minimal circuits) along Lie group manifolds
relevant to low-dimensional multi-qubit systems, such as SU(2), SU(4) and
SU(8). We demonstrate the superior performance of greybox models, which combine
traditional blackbox algorithms with prior domain knowledge of quantum
mechanics, as means of learning underlying quantum circuit distributions of
interest. Our results demonstrate how geometric control techniques can be used
to both (a) verify the extent to which geometrically synthesised quantum
circuits lie along geodesic, and thus time-optimal, routes and (b) synthesise
those circuits. Our results are of interest to researchers in quantum control
and quantum information theory seeking to combine machine learning and
geometric techniques for time-optimal control problems.
