Asymptotically optimal synthesis of reversible circuits

TL;DR

This paper presents an algorithm for synthesizing n-wire reversible circuits with gate complexity matching the known lower bound, effectively solving a long-standing open problem in the field.

Contribution

It introduces an algorithm that achieves asymptotically optimal synthesis of reversible circuits, matching the previously established lower bound.

Findings

Algorithm achieves $O(2^n n/ ext{log} n)$ gate complexity

Closes the open problem of matching upper bound to the lower bound

Provides a practical method for optimal reversible circuit synthesis

Abstract

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 (M. Saeedi and I. L Markov, ACM Computing Surveys, 45(2):1-34, 2013). 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.