Complexity of reversible circuits and their quantum implementations

We provide an extensive overview of upper bounds on the number of gates needed in reversible and quantum circuits. As reversible gate libraries we consider single-target gates, mixed-polarity multiple-controlled Toffoli gates, and the set consisting of the NOT, the CNOT, and the two-controlled Toffoli gate. As quantum gate libraries we consider the semi-classical NCV library (consisting of NOT, CNOT, and the square-root of NOT called V  ) as well as the universal and commonly used Clifford+TClifford+T gate library. Besides a summary of known bounds, the paper provides several new and tighter bounds. Several synthesis approaches and mapping schemes were used to calculate the bounds.