SubRiemannian geodesics and cubics for efficient quantum circuits

Previous work argued that efficient quantum circuits can be obtained from special curves called geodesics, and they studied geodesics in Riemannian manifolds equipped with a penalty metric where the penalty was taken to infinity. Taking such limits seems problematic, because it is not clear that all extremals of a limiting optimal control problem can be arrived at as limits of solutions. To rectify this we use the Pontryagin Maximum Principle to construct equations for normal and abnormal subRiemannian geodesics and cubics. We also investigate whether neural networks can be trained to help generate quantum circuits.