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arXiv · 2105.14047

Generators and Relations for Un(Z[1/2,i])

Xiaoning Bian·Peter Selinger
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Abstract

Consider the universal gate set for quantum computing consisting of the gates X, CX, CCX, omega^dagger 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 nxn-matrices with entries in Z[1/2,i].

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Xiaoning Bian, Peter Selinger. 2021-05-28. Generators and Relations for Un(Z[1/2,i]). https://doi.org/10.4204/eptcs.343.8

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