put qubits |a) /x), the CNOT gate can generate any of the four EPR or Bell states, as defined in Eq. (16.2). It is easily verified that the matrix ACNOT is unitary and that its inverse matrix is ACNOT A CNOT, or ACNOT = I (I= 4 x 4 identity matrix). This last result is expected, since the repeated action of CNOT (with same control qubit, by inherent circuit con- struction) must leave the target qubit invariant. CROSSOVER or SWAP gate It is also possible to build circuits made of concatenated CNOT gates, where the control and target qubits exchange roles, i.e., an output target qubit from a first gate serving as an input control qubit for the next gate. Consider, for instance, the three-gate circuit shown in Fig. 16.3, which (as we shall see) corresponds to the CROSSOVER or SWAP operator. The circuit is seen to include in the middle a CNOT gate arranged upside down, thus, using the target output qubit of the first CNOT gate as a control for the second CNOT gate. For simplicity, we assume that the control qubits that are input to CNOT gates must be in a pure state. Under this assumption, this whole circuit must be input only with pure states |a), |b) = |0), |1). The matrix representation of the CNOT gate makes it possible to construct the circuit shown in Fig. 16.3, with care to properly define the matrix BCNOT corresponding to the reversed arrangement. As an exercise, one may show that in the computational base (|0)|0), |0)|1), |1)|0>, |1)|1>} the matrix BCNOT takes the form BCNOT (16.31) A second exercise is to compute the circuit matrix ASWAP from the matrix definitions in Eqs. (16.29) and (16.30). The computation yields ASWAP (16.32) 16.3 Quantum gates with multiple inputs and outputs X(ii) Y|(id) Figure 16.4 (a) Controlled-U gate, (b) corresponding matrix gate possibilities U = X, Y, Z, H, and S. It is seen from the coefficients in the matrix ASWAP that the input tensor states |0) |0) and |1)|1) are left invariant (owing to the 1 coefficients in the matrix-diagonal), while the input tensor states |0)|1) and |1