)|0> are swapped (owing to the 1 off-diagonal coefficients). The circuit gate thus corresponds to a CROSSOVER or SWAP function, as Fig. 16.3 indicates with the outputs lb), |a). The equivalent gate representation is also shown at right in the figure. Controlled-U gates In the category of controlled-U gates, U stands for any quantum gate with a 2 x 2 unitary (but not necessarily Hermitian) matrix. For instance, U= X, Y, Z, H (Pauli and Hadamard gates) and S (/2 phase gate), as illustrated in Fig. 16.4 with their X 2 matrices represented in the |0), |1) state basis. The gate controlled-X is the same as CNOT and its matrix (previously called ACNOT) is shown in Eq. (16.29). We notice again from this equation that this 4 X 4 matrix can also be represented in the reduced form controlled-X = (16.33) where the top-left side corresponds to the REPEAT or invariant function (control qubit set to |0), corresponding to 2 x 2 identity matrix I) and the bottom-right side corresponds to the NOT function (control qubit set to |1), corresponding to2x2matrix X). Thus, we can most generally define the matrix of any controlled-U gate (e.g., U= X, Y, Z, H, S. ) according to controlled-U [ ] (16.34) Most generally, any controlled-U gate corresponds to a target qubit rotation characterized by U= R(), where R() is the rotation operator associated with the transformation U on the Bloch sphere, as characterized by a rotation angle 0 about the axis parallel to the unitary vector n (see Appendix N for related definition and properties). Quantum bits and quantum gates Controlled-SWAP |c>a>b>/c>a>b> 00000 001001 010010 011011 100100 101110 10101 111111 Figure 16.5 Controlled-SWAP gate with truth table. Controlled-SWAP gate The schematic representation of a controlled-SWAP gate and its qubit truth table are shown in Fig. 16.5. This gate represents a particular case of the Fredkin gate previously described in Chapter 15, and represented in Fig. 15.7 (with the control qubit C at the bottom). Since this is a 3 x 3 gate, the state basis has eight t