The paper studies the main aspects of the realization of 2 x 2 ternary reversible circuits based on cycles, considering the results of the realization of all 362,880 2 x 2 ternary reversible functions. It has been shown that in most cases, realizations obtained with the MMD+ algorithm have a lower complexity (in terms of cost) than realizations based on cycles. In the paper it is shown under which conditions realizations based on transpositions may have a higher or a lower cost than realizations using larger cycles. Finally it is shown that there are a few special cases where realizations based on transpositions have the same cost or possibly lower cost than the MMD+ based realizations. Aspects of scaleability are considered in terms of 2 x 2-based n x n reversible circuits.

中文翻译：

---

三元可逆电路基于循环的综合复杂性

本文研究了基于循环的2 x 2 三元可逆电路实现的主要方面，考虑了所有362,880 个2 x 2 三元可逆函数的实现结果。已经表明，在大多数情况下，使用 MMD+ 算法获得的实现比基于循环的实现具有更低的复杂性（在成本方面）。在本文中，它显示了在哪些条件下基于换位的实现可能比使用更大周期的实现具有更高或更低的成本。最后表明，在一些特殊情况下，基于换位的实现与基于 MMD+ 的实现具有相同的成本或可能更低的成本。根据基于 2 x 2 的 n xn 可逆电路考虑可扩展性方面。