Gradient Flows for Optimisation and Quantum Control: Foundations and Applications

Abstract

For addressing optimisation tasks on finite dimensional quantum systems, we give a comprehensive account of the foundations of gradient flows on Riemannian manifolds including new developments: we extend former results from Lie groups such as the full unitary group to closed subgroups like partitionings by factorisation into tensor products, where the finest partitioning consists of purely local unitary operations. Moreover, the common framework is kept sufficiently general and allows for setting up gradient flows on (sub-)manifolds, Lie (sub-)groups, quotient groups, and reductive homogeneous spaces. Relevant convergence conditions are discussed meant to serve as justification for recent and new achievements, and as foundation for further research. Exploiting the differential geometry of quantum dynamics under different scenarios helps to provide highly useful algorithms: (a) On an abstract level, gradient flows may establish the exact upper bounds of pertinent quality functions, i.e. upper bounds reachable within the underlying manifold of the state space dynamics; (b) in a second stage referring to a concrete experimental setting, gradient flows on the manifold of piecewise constant control amplitudes $\R^m$ may be set up to provide (approximations to) optimal control of quantum devices under realistic conditions. Illustrative examples and new applications are given relating to distance measures of pure-state entanglement. We establish the correspondence to best rank-1 approximations of higher-order tensors.