We give a finite presentation by generators and relations for the group O_n(Z[1/2]) of n-dimensional orthogonal matrices with entries in 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:我们为O_n(Z[1/2])组的正维正方位矩阵和Z[1/2]条目的O_n(Z[1/2])组的生成者和关系作了有限的介绍,然后我们为M/sqrt(2) ⁇ k表的正维正方位矩阵组作了类似的介绍,K是非负整数,M是整数矩阵。两个组都是量子电路研究中产生的。特别是,当尺寸为2的功率时,后一组的元素恰恰是能够用量子电路代表由托夫利门、哈达马德门和计算Ancilla组成的通用门组组成的量子路组成的单一矩阵组。