J. Math. Phys. 56, 082201 (2015); http://dx.doi.org/10.1063/1.4927100

**Abstract:**We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov, and Mosca [Quantum Inf. Comput. 13(7,8), 607–630 (2013)]. Their algorithm takes as input an exactly synthesizable single-qubit unitary—one which can be expressed without error as a product of Clifford and T gates—and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer n, we consider the “Clifford-cyclotomic” gate set consisting of the Clifford group plus a z-rotation by πn . We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of πn z-rotations. For the Clifford+T case n = 4, the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring Z[eiπn,1/2] . We prove that this characterization holds for a handful of other small values of n but the fraction of positive even integers for which it fails to hold is 100%.

*Source:* Email from journals_AT_aip-info.org, 16 Sep. 2015, http://scitation.aip.org/content/aip/journal/jmp/56/8/10.1063/1.4927100