Uniform Finite Generation of Compact Lie Groups and universal quantum gates

2001/11/24 by Domenico D’Alessandro, D. D'Alessandro, D'Alessandro, D. · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Physical sciences #Quantum Physics (quant-ph) #quant-ph

paper · pdf · doi:10.48550/arxiv.quant-ph/0111133

arxiv created 2001/11/24 · openalex publication_date 2001/11/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider a compact connected Lie group G and the corresponding Lie algebra \cal L. Let \X₁,...,X_m\ be a set of generators for the Lie algebra \cal L. We prove that G is uniformly finitely generated by \X₁,...,X_m\. This means that every element K ∈ G can be expressed as K=e^Xt₁e^Xt₂ ⋅ ⋅ ⋅ e^Xt_l, where the indeterminates X are in the set \X₁,...,X_m \, t_i ∈ \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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• Geometric Control Methods for Quantum Computations

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