Non-Hermitian quantum gates are more common than Hermitian quantum gates
Anirban Pathak · arXiv (Cornell University) · 2013
Most of the frequently used quantum gates (e.g., NOT, Hadamard, CNOT, SWAP, Toffoli, Fredkin and Pauli gates) are self-inverse (Hermitian). However, with a simple minded argument it is established that most of the allowed quantum gates are non-Hermitian (non-self-inverse). It is also shown that the % of non-Hermitian gates increases with the dimension. For example, 58.33% of the 2-qubit gates, 98.10% of the 3-qubit gates and 99.99% of the 4-qubit gates are non-Hermitian. As classical reversible gates are essentially permutation gates so the above statistics is strictly valid for classical reversible gates. Further, since Hermiticity is not of much interest in context of the classical reversible gate, hence the result implies that most of the allowed classical reversible gates are non-self-inverse.