Uniform finite generation of compact Lie groups

Abstract Consider a compact connected Lie group G and the corresponding Lie algebra L . Let {X1,…,Xm} be a set of generators for the Lie algebra L . We prove that G is uniformly finitely generated by {X1,…,Xm}. This means that every element K∈G can be expressed as K=eXt1eXt2···eXtl, where the indeterminates X are in the set {X1,…,Xm}, t i ∈ R , 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 link the results to the existence of universal logic gates in quantum computing and discuss the impact on bang bang control algorithms for quantum mechanical systems.