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.
翻译:我们用发电机和关系有限地展示$\mathrm{O ⁇ n(\mathbb ⁇ [1/2])组的美元-维正方位矩阵,其条目为$\mathbb ⁇ [1/2]$。然后,我们用类似的方式展示了表格$M/sqrt{2 ⁇ k$(美元是非负整数,美元是整数矩阵)中的美元-维正方位矩阵。两个组都出现在量子电路研究中。特别是当尺寸为$2的功率时,后一组的元素正是由托夫利门、哈达马德门和计算轴组成的通用门的量子路代表的单一矩阵。