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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 representedby square matrices that are unitary and Hermitian. However, with a simpleminded argument, it is established that the most of the allowed reversiblegates are non-self-inverse (unitary but non-Hermitian) in nature. It is alsoshown that the % of non-Hermitian gates increases with the dimension. Forexample, 58.33% of the 2-bit gates, 98.10% of the 3-bit gates and 99.99% of the4-bit gates are non-Hermitian. As classical reversible gates are essentiallypermutation gates, above statistics is strictly valid for classical reversiblegates, but the argument can be easily extended to include quantum gates and toestablish 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 thatcannot be decomposed as a product of two single bit gates) among all possiblegates has been computed as 83.3%, and the applicability of this analysis in theoptimization of circuit cost is discussed.

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