This paper extends the KP time-optimal quantum control solution using the global Cartan $KAK$ decomposition for geodesic-based solutions. Extending recent results on time-optimal constant-θ control, we integrate the Cartan method into an homovariant quantum neural network (EQNN) for quantum control tasks. We show that the finite-depth-constrained EQNN ansatz with Cartan layers can replicate the constant-θ Ahrimani geodesics for the KP problem. We show how gradient-based training with an appropriate cost function can converge to certain global time-optimal solutions for certain types of control problems in Riemannian-symmetric spaces when simple regularity conditions are satisfied. This generalizes previous geometric control theory methods and clarifies how to perform optimal geodesic estimation in the context of quantum machine learning.