A controlled-U gate has two qubits as inputs: a control qubit and a target qubit. Nothings happens to the control qubit (x). The target qubit ly) is acted on by the 2 x 2 unitary U if the control is [1), but nothing happens if it is |0). [Iin) [out) lyin) U yout) The CNOT gate is the same as a controlled-X gate. It has two equivalent representa- tions as a circuit diagram: 3 [I out) lyin) -X II out) In classical computing, the logical NOT operation flips a bit 0 1. The quantum operator X similarly interchanges the computational basis states |0) (1). Of course X can also act on linear combinations of |0) and [1), so it really is quite different from the NOT operator. Yet, the similarity is sufficient that the controlled-X gate is referred to as CNOT. Consider a two qubit system that is acted on by one-qubit gates as well as CNOT with the usual convention that 1 is the control and 2 is the target. Denoting the CNOT(s) by C, establish the following identities: (a) CXIC = XIX2 . (b) CYC = YIX2 . (c) CZIC = Z1 . (d) CY2C = Z1Y2