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 $S_{2^{n}}$ , 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.

中文翻译：

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计算可逆逻辑函数的等价类数

可逆逻辑功能分类在可逆逻辑综合中起着重要作用。本文研究可逆逻辑函数等价类数的计算。为了完成这项工作，将一个 n -qubit 可逆函数表示为对称群 S 2 n $S_{2^{n}}$ 中的一个置换，从而得到计算可逆等价类数的通用公式逻辑函数是基于群论推导出来的。基于置换群共轭类数的计算方法，得到了一种计算可逆逻辑函数等价类数的改进方法。在笔记本电脑上使用 GAP 软件进行的实验中，我们可以计算多达 13 个量子位的 NPNP 等价类，最多 10 个量子位的 LL 等价类和最多 10 个量子位的 AA 等价类。实验结果表明，与之前发表的方法相比，我们计算这些超过 6 个量子比特的等价类的方案是一个重大进步。