Computing the Number of the Equivalence Classes for Reversible Logic Functions

Abstract

Reversible logic function classification plays an important role in reversible logic synthesis. This paper studies the calculation of the number of the equivalence classes for reversible logic functions. In order to do this work, an n-qubit reversible function is expressed as a permutation in the symmetric group $S_{2^{n}}$ , so that a universal formula for calculating the number of equivalence classes of reversible logic functions is derived based on group theory. Based on the calculation method of the number of conjugacy classes of permutation groups, an improved method for calculating the number of equivalence classes of reversible logic functions is obtained. In the experiments with GAP software on a laptop, we can compute the NPNP-equivalence classes for up to 13 qubits, LL-equivalence classes for up to 10 qubits and AA-equivalence classes for up to 10 qubits. Experiment results indicate that our scheme for calculating these equivalence classes of more than 6 qubits over previous published methods is a significant advancement.