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

06/02/2021

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by   Sarah Meng Li, et al.

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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/√(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.