Asymptotically optimal synthesis of reversible circuits

Reversible circuits have been studied extensively and intensively, and have plenty of applications in various areas, such as digital signal processing, cryptography, and especially quantum computing. In 2003, the lower bound $\Omega (2^{n}n/\log n)$ for the synthesis of n-wire reversible circuits was proved. Whether this lower bound has a matching upper bound was listed as one of the future challenging open problems in the survey (Saeedi and Markov (2013) [1]). In this paper we propose an algorithm to implement an arbitrary n-wire reversible circuit with no more than $O(2^{n}n/\log n)$ elementary gates, and thus close the open problem.