Large products of double cosets for symmetric subgroups

TL;DR

This paper classifies when products of double cosets cover simple compact Lie groups, with applications to quantum gate decompositions, providing a complete classification for most cases in SU(n).

Contribution

It offers a necessary condition for such products to cover the group and fully classifies these cases for SU(n) excluding a specific symmetric subgroup type.

Findings

Complete classification for SU(n) with most symmetric subgroups

Necessary condition for double coset products to cover the group

Applications to quantum gate decomposition techniques

Abstract

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.