# K-P Quantum Neural Networks

arXiv: https://arxiv.org/abs/2504.01673 · HTML: https://mlsanity.com/paper/2504.01673/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 19 references that are in the map.
- First posted on arXiv 2 Apr 2025.

## Builds on (most cited first, 4 of 4)

- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Theory for Equivariant Quantum Neural Networks](https://mlsanity.com/paper/2210.08566/) (2022)
- [Group-Invariant Quantum Machine Learning](https://mlsanity.com/paper/2205.02261/) (2022)
- [Representation Theory for Geometric Quantum Machine Learning](https://mlsanity.com/paper/2210.07980/) (2022)

## Abstract (arXiv)

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.