Uniform Finite Generation of Compact Lie Groups and universal quantum gates

Details Uniform Finite Generation of Compact Lie Groups and universal quantum gates Uniform Finite Generation of Compact Lie Groups and universal quantum gates

2001-11-24

arxiv.org/abs/quant-ph/0111133v1

Physics

Quantum Physics

Scientific paper

Consider a compact connected Lie group $G$ and the corresponding Lie algebra $\cal L$. Let $\{X_1,...,X_m\}$ be a set of generators for the Lie algebra $\cal L$. 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 $K=e^{Xt_1}e^{Xt_2} \cdot \cdot \cdot e^{Xt_l}$, 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.

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