On the CNOT-cost of the TOFFOLI gate

Three-input TOFFOLI gates are heavily used when performing classical logic operations on quantum data, e.g., in reversible arithmetic circuits. However, in physical implementations TOFFOLI gates are decomposed into six CNOT gates and several one-qubit gates. Though this decomposition has been known for at least 10 years, we provide here the first demonstration of its CNOT-optimality. We first prove that any circuit (i) containing less than six CNOT gates and (ii) implementing a diagonal operator, can be rearranged to form the cosine-sine decomposition of a related operator. Leveraging the canonicity of such decompositions to limit one-qubit gates appearing in respective circuits, we completely classify three-qubit diagonal operators by their CNOT-cost. As a consequence, we deduce that the TOFFOLI cannot be implemented using fewer than six CNOTs. Circuit-analysis techniques developed in our work also produce new lower bounds for related gates and the n-qubit analogue of the TOFFOLI.