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 Z[2β1β,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 Z[2β1β,i] holds in the
n-qubit case.Comment: 23 pages, 3 figures, added the proof of T-optimality of the circuits
synthesized by Algorithm