Synthesis of Quantum Circuits vs. Synthesis of Classical Reversible Circuits

Synthesis of Quantum Circuits vs. Synthesis of Classical Reversible Circuits

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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 group theory of finite groups (in particular the symmetric group ????); the unitary matrices are discussed in group theory of continuous groups (a.k.a. Lie groups, in particular the unitary group U(??))...