  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 $\mathfrak{g}=\mathfrak{k} \oplus \mathfrak{p}$ decomposition and
time-optimal solutions are represented by subRiemannian geodesics synthesised
via a distribution of generators in $\mathfrak{p}$. 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 $-iX \in
[\frak{p},\frak{p}] \subset \frak{k}$ with controls $-iH(t) \in \frak{p}$.
Target unitaries are parametrised as $U=kac$ where $k,c \in K$ and $a =
e^{i\Theta}$ with $\Theta \in \frak{a}$. We show that the assumption of
$d\Theta=0$ 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
$\mathfrak{p}$ and local rotations are generated by
$[\mathfrak{p},\mathfrak{p}]$.
