Generators and relations for U_n(ℤ [1/2, i])

05/28/2021

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by   Xiaoning Bian, et al.

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Consider the universal gate set for quantum computing consisting of the gates X, CX, CCX, ω†H and S. All of these gates have matrix entries in the ring ℤ [1/2, i], the smallest subring of the complex numbers containing 1/2 and i. Amy, Glaudell, and Ross proved the converse, i.e., any unitary matrix with entries in ℤ [1/2, i] can be realized by a quantum circuit over the above gate set using at most one ancilla. In this paper, we give a finite presentation by generators and relations of U_n(ℤ [1/2, i]), the group of unitary n× n-matrices with entries in ℤ [1/2, i].