We give a finite presentation by generators and relations for the group $\mathrm{O}_n(\mathbb{Z}[1/2])$ of $n$-dimensional orthogonal matrices with entries in $\mathbb{Z}[1/2]$. We then obtain a similar presentation for the group of $n$-dimensional orthogonal matrices of the form $M/\sqrt{2}{}^k$, where $k$ is a nonnegative integer and $M$ is an integer matrix. Both groups arise in the study of quantum circuits. In particular, when the dimension is a power of $2$, the elements of the latter group are precisely the unitary matrices that can be represented by a quantum circuit over the universal gate set consisting of the Toffoli gate, the Hadamard gate, and the computational ancilla.

中文翻译：

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群的生成器和关系 $\mathrm{O}_n(\mathbb{Z}[1/2])$

我们通过生成器和关系给出了一组 $\mathrm{O}_n(\mathbb{Z}[1/2])$ 的有限表示，其中包含 $\mathbb{Z}[ 1/2]$。然后我们获得了形式为 $M/\sqrt{2}{}^k$ 的 $n$ 维正交矩阵组的类似表示，其中 $k$ 是一个非负整数，$M$ 是一个整数矩阵. 这两个群体都出现在量子电路的研究中。特别地，当维度是 $2$ 的幂时，后一组的元素恰好是酉矩阵，可以用量子电路在由 Toffoli 门、Hadamard 门和计算门组成的通用门集上表示。附属物。