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 w qubits, is described by an n x n unitary matrix with n=2w, a reversible classical circuit, acting on w bits, is described by a 2w x 2w permutation matrix. The permutation matrices are studied in group theory of finite groups (in particular the symmetric group Sn); the unitary matrices are discussed in group theory of continuous groups (a.k.a. Lie groups, in particular the unitary group U(n). Both the synthesis of a reversible logic circuit and the synthesis of a quantum logic circuit take advantage of the decomposition of a matrix: the former of a permutation matrix, the latter of a unitary matrix. In both cases the decomposition is into three matrices. In both cases the decomposition is not unique.