  In this paper, we show the equivalence of the set of unitaries computable by
the circuits over the Clifford and T library and the set of unitaries over the
ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$, in the single-qubit case. We report an
efficient synthesis algorithm, with an exact optimality guarantee on the number
of Hadamard and T gates used. We conjecture that the equivalence of the sets of
unitaries implementable by circuits over the Clifford and T library and
unitaries over the ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$ holds in the
$n$-qubit case.
