Generators and Relations for the Group On(Z[1/2])

• URL: http://arxiv.org/abs/2106.01175v2
• Date: Mon, 13 Sep 2021 00:52:33 GMT
• Title: Generators and Relations for the Group On(Z[1/2])
• Authors: Sarah Meng Li (Dalhousie University), Neil J. Ross (Dalhousie University), Peter Selinger (Dalhousie University)
• Abstract summary: 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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