Generators and relations for $U_n(\mathbb Z [\frac{1}{2}, i])$

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 $\mathbb Z [\frac{1}{2}, i]$, the smallest subring of the complex numbers containing $\frac{1}{2}$ and $i$. Amy, Glaudell, and Ross proved the converse, i.e., any unitary matrix with entries in $\mathbb Z [\frac{1}{2}, i]$ 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 $U_n(\mathbb Z [\frac{1}{2}, i])$, the group of unitary $n\times n$-matrices with entries in $\mathbb Z [\frac{1}{2}, i]$.