K-P Quantum Neural Networks
This work addresses quantum control tasks for researchers in quantum machine learning, presenting an incremental extension of geometric methods.
The paper tackled the problem of time-optimal quantum control by integrating Cartan decompositions into equivariant quantum neural networks, showing that this approach can replicate geodesic solutions and converge to global time-optimal solutions under certain conditions.
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-$θ$ 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-$θ$ 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.