We present an extension of K–P time-optimal quantum control solutions using global Cartan KAK decompositions for geodesic-based solutions. Extending recent time-optimal constant– \(\theta \) control results, we integrate Cartan methods into equivariant quantum neural network (EQNN) for quantum control tasks. We show that a finite-depth limited EQNN ansatz equipped with Cartan layers can replicate the constant– \(\theta \) sub-Riemannian geodesics for K–P problems. We demonstrate how, for certain classes of control problem on Riemannian symmetric spaces, gradient-based training using an appropriate cost function converges to global time-optimal solutions when satisfying simple regularity conditions. This generalises prior geometric control theory methods and clarifies how optimal geodesic estimation can be performed in quantum machine learning contexts.