Uniform Finite Generation of Compact Lie Groups and universal quantum gates

Abstract

Consider a compact connected Lie group $G$ and the corresponding Lie algebra $\mathcal{L}$. Let $\{X_1,...,X_m\}$ be a set of generators for the Lie algebra $\mathcal{L}$. We prove that $G$ is uniformly finitely generated by $\{X_1,...,X_m\}$. This means that every element $K\in G$ can be expressed as $K=e^{X{t}_1}{e}^{X{t}_2}\cdot \cdot \cdot e^{X{t}_l}$, where the indeterminates $X$ are in the set $\{X_1,...,X_m\}$, $t_i \in \RR$, $i=1,...,l$, and the number $l$ is uniformly bounded. This extends a previous result by F. Lowenthal in that we do not require the connected one dimensional Lie subgroups corresponding to the $X_i$, $i=1,...,m$, to be compact. We discuss the consequence of this result to the question of universality of quantum gates in quantum computing.