2024/04/02 by Perrier, Elija, Jackson, Christopher S.
#Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2404.02358
Geometric methods have useful application for solving problems in a range of quantum information disciplines, including the synthesis of time-optimal unitaries in quantum control. In particular, the use of Cartan decompositions to solve problems in optimal control, especially lambda systems, has given rise to a range of techniques for solving the so-called KP-problem, where target unitaries belong to a semi-simple Lie group manifold G whose Lie algebra admits a \mathfrakg=\mathfrakk ⊕ \mathfrakp decomposition and time-optimal solutions are represented by subRiemannian geodesics synthesised via a distribution of generators in \mathfrakp. In this paper, we propose a new method utilising global Cartan decompositions G=KAK of symmetric spaces G/K for generating time-optimal unitaries for targets -iX ∈ [\frakp,\frakp] ⊂ \frakk with controls -iH(t) ∈ \frakp. Target unitaries are parametrised as U=kac where k,c ∈ K and a = eiΘ with Θ∈ \fraka. We show that the assumption of dΘ=0 equates to the corresponding time-optimal unitary control problem being able to be solved analytically using variational techniques. We identify how such control problems correspond to the holonomies of a compact globally Riemannian symmetric space, where local translations are generated by \mathfrakp and local rotations are generated by [\mathfrakp,\mathfrakp].