We generalize an e\ufb03cient exact synthesis algorithm for single-qubit unitaries over the Cli\ufb00ord+T gate set which was presented by Kliuchnikov, Maslov and Mosca. Their algorithm takes as input an exactly synthesizable single-qubit unitary\u2013one which can be expressed without error as a product of Cli\ufb00ord and T gates\u2013and 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 \u201cCli\ufb00ord-cyclotomic\u201d gate set consisting of the Cli\ufb00ord group plus a z-rotation by \u03c0 n. We present an e\ufb03cient exact synthesis algorithm which outputs a decomposition using the minimum number of \u03c0 n z-rotations. For the Cli\ufb00ord+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 \u03c0 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%.<\/p>\n","protected":false},"excerpt":{"rendered":"
We generalize an e\ufb03cient exact synthesis algorithm for single-qubit unitaries over the Cli\ufb00ord+T gate set which was presented by Kliuchnikov, Maslov and Mosca. Their algorithm takes as input an exactly synthesizable single-qubit unitary\u2013one which can be expressed without error as a product of Cli\ufb00ord and T gates\u2013and outputs a sequence of gates which implements it. 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