Synthesis of Quantum Circuits vs. Synthesis of Classical Reversible Circuits

At first sight, quantum computing is completely different from classical computing. Nevertheless, a link is provided by reversible computation.

Whereas an arbitrary quantum circuit, acting on ?? qubits, is described by an ?? × ?? unitary matrix with ??=2??, a reversible classical circuit, acting on ?? bits, is described by a 2?? × 2?? permutation matrix. The permutation matrices are studied in grou...