Large products of double cosets for symmetric subgroups

We consider the problem of classifying pairs $x,y\in G$ such that $KxKyK=G$ 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 $G=\mathrm{SU}(n)$ and any symmetric $K\subseteq G$ except the type AIII case $K\simeq \mathrm{S}(\mathrm{U}(p)\times \mathrm{U}(n-p))$ with $p\mathrm{\ne }n\mathrm{/}2$. We also present some applications of these results to gate decompositions in quantum computing.