Large products of double cosets for symmetric subgroups

We consider the problem of classifying pairs $x,y \in G$ such that $K x K y K = G$ where $G$ is a simple compact connected Lie group and $K$ is a symmetric subgroup. We give a necessary condition on $x,y$ for all simply connected $G$, and a complete classification when $G = \operatorname{SU}(n)$ and any symmetric $K \subseteq G$ except the type AIII case $K \simeq \operatorname{S}(\operatorname{U}(p) \times \operatorname{U}(n-p))$ with $p \neq n/2$. We also present some applications of these results to gate decompositions in quantum computing.