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We give a presentation by generators and relations of the group$U_4(\mathbb{Z}[1/\sqrt{2},i])$ of unitary $4\times 4$ matrices with entries inthe ring $\mathbb{Z}[1/\sqrt{2},i]$. This is motivated by the problem of exactsynthesis for the Clifford+T gate set in quantum computation.

Generators and relations for the group $U_4(\mathbb{Z}[1/\sqrt{2},i])$