Unitaries over the rings Z[1/2], Z[1/√2], Z[1/i√2], and Z[1/2,i] are exactly the circuits over four Clifford+T-derived gate sets.
Generators and relations for the group $U_4(\mathbb{Z}[1/\sqrt{2},i])$
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abstract
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 in the ring $\mathbb{Z}[1/\sqrt{2},i]$. This is motivated by the problem of exact synthesis for the Clifford+T gate set in quantum computation.
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quant-ph 1years
2019 1verdicts
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Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits
Unitaries over the rings Z[1/2], Z[1/√2], Z[1/i√2], and Z[1/2,i] are exactly the circuits over four Clifford+T-derived gate sets.