Exact Synthesis of Single-qubit Unitaries over Clifford-cyclotomic Gate Sets

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. 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 π n. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of π n 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[ei π n ,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%.