Uniform finite generation of compact Lie groups

Consider a compact connected Lie group G and the corresponding Lie algebra L. Let {X-1,...,X-m} be a set of generators for the Lie algebra L. We prove that G is uniformly finitely generated by {X-1,...,X-m}. This means that every element K is an element of G can be expressed as K = e(Xt1)e(Xt2...)e(Xtl), where the indeterminates X are in the set {X-1,...,X-m}, t(i) is an element of 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. (C) 2002 Elsevier Science B.V. All rights reserved.