K-P Quantum Neural Networks

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 \emph{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 certain 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.

@article{perrier2025_2504.01673,
  title={ K-P Quantum Neural Networks },
  author={ Elija Perrier },
  journal={arXiv preprint arXiv:2504.01673},
  year={ 2025 }
}