Fast and efficient exact synthesis of single qubit unitaries generated by Clifford and T gates

In this paper, we show the equivalence of the set of unitaries computable by the circuits over Clifford and T library and the set of unitaries over the ring \mathbbZ[\frac1\sqrt2,i], in the single qubit case. We report an efficient synthesis algorithm, with exact optimality guarantee on the number of Hadamard gates used. We conjecture that the equivalence of the sets of unitaries implementable by circuits over Clifford and T library and unitaries over the integer ring \mathbbZ[\frac1\sqrt2,i] holds in the n-qubit case.