Statistical distribution of the reversible gates: what percentage of them are self-inverse?

TL;DR

This paper analyzes the distribution of reversible gates, revealing that most are non-self-inverse, especially as the number of bits increases, and discusses implications for quantum and classical circuit optimization.

Contribution

It provides a statistical analysis of the proportion of self-inverse versus non-self-inverse reversible gates across different dimensions and extends the discussion to quantum gates.

Findings

Most 2-bit gates are self-inverse (58.33%), but this drops sharply for higher bits.

Over 98% of 3-bit gates and 99.99% of 4-bit gates are non-self-inverse.

Approximately 83.3% of 2-bit gates are genuinely non-decomposable.

Abstract

It is well known that most of the frequently used reversible logic gates (e.g., NOT, CNOT, SWAP, Toffoli, Fredkin) are self-inverse and are represented by square matrices that are unitary and Hermitian. However, with a simple minded argument, it is established that the most of the allowed reversible gates are non-self-inverse (unitary but non-Hermitian) in nature. It is also shown that the % of non-Hermitian gates increases with the dimension. For example, 58.33% of the 2-bit gates, 98.10% of the 3-bit gates and 99.99% of the 4-bit gates are non-Hermitian. As classical reversible gates are essentially permutation gates, above statistics is strictly valid for classical reversible gates, but the argument can be easily extended to include quantum gates and to establish that the majority of the quantum gates are also non-self-inverse. Further, the % of genuinely 2-bit reversible gates (i.e., 2-bit gates that cannot be decomposed as a product of two single bit gates) among all possible gates has been computed as 83.3%, and the applicability of this analysis in the optimization of circuit cost is discussed.