Universal quantum circuit for two-qubit transformations with three controlled-NOT gates

We consider quantum circuits made of controlled-NOT (CNOT) gates and single-qubit unitary gates and look for constructions that minimize the use of CNOT gates. We show, by means of an explicit quantum circuit, that three CNOT gates are necessary and sufficient in order to implement an arbitrary unitary transformation of two qubits. We also identify the subset of two-qubit gates that can be performed with only two CNOT gates and provide a simple characterization for them.