dorsal/arxiv
View SchemaUniform Finite Generation of Compact Lie Groups and universal quantum gates
| Authors | D. D'Alessandro |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0111133 |
| URL | https://arxiv.org/abs/quant-ph/0111133 |
Abstract
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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"abstract": "Consider a compact connected Lie group $G$ and the corresponding Lie algebra\n$\\cal L$. Let $\\{X_1,...,X_m\\}$ be a set of generators for the Lie algebra\n$\\cal L$. We prove that $G$ is uniformly finitely generated by\n$\\{X_1,...,X_m\\}$. This means that every element $K \\in G$ can be expressed as\n$K=e^{Xt_1}e^{Xt_2} \\cdot \\cdot \\cdot e^{Xt_l}$, where the indeterminates $X$\nare in the set $\\{X_1,...,X_m \\}$, $t_i \\in \\RR$, $i=1,...,l$, and the number\n$l$ is uniformly bounded. This extends a previous result by F. Lowenthal in\nthat we do not require the connected one dimensional Lie subgroups\ncorresponding to the $X_i$, $i=1,...,m$, to be compact.\n We discuss the consequence of this result to the question of universality of\nquantum gates in quantum computing.",
"arxiv_id": "quant-ph/0111133",
"authors": [
"D. D\u0027Alessandro"
],
"categories": [
"quant-ph"
],
"title": "Uniform Finite Generation of Compact Lie Groups and universal quantum gates",
"url": "https://arxiv.org/abs/quant-ph/0111133"
},
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