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The three-input TOFFOLI gate is the workhorse of circuit synthesis forclassical logic operations on quantum data, e.g., reversible arithmeticcircuits. In physical implementations, however, TOFFOLI gates are decomposedinto six CNOT gates and several one-qubit gates. Though this decomposition hasbeen known for at least 10 years, we provide here the first demonstration ofits CNOT-optimality.  We study three-qubit circuits which contain less than six CNOT gates andimplement a block-diagonal operator, then show that they implicitly describethe cosine-sine decomposition of a related operator. Leveraging the canonicityof such decompositions to limit one-qubit gates appearing in respectivecircuits, we prove that the n-qubit analogue of the TOFFOLI requires at least2n CNOT gates. Additionally, our results offer a complete classification ofthree-qubit diagonal operators by their CNOT-cost, which holds even if ancillaqubits are available.

On the CNOT-cost of TOFFOLI gates