Читать книгу Principles of Superconducting Quantum Computers - Daniel D. Stancil - Страница 25

The gates that we have considered so far involve operations that are independently applied to separate qubits—there is no qubit–qubit interaction. If we are to entangle two qubits, then we need classes of gates where the operation on one qubit depends on the state of another. One of the most important such gates is the controlled-NOT, or CNOT gate. For this gate, one of the input qubits is the “control,” and the other is the “target.” If the control qubit is zero, then nothing is done to the target qubit, but if the control qubit is one, then the target qubit is flipped. For example, if the right qubit in our notation is the control and the left qubit is the target, then the CNOT gate transforms the basis states as follows:

(1.49)

The effect of a CNOT can be compactly represented by UCN|t⟩|c⟩=|c⊕t⟩|c⟩, where ⊕ represents exclusive-OR or modulo-2 addition (e.g., 0+1=1, but 1+1=0). The matrix representation of the CNOT gate is

(1.50)

and the circuit symbol is shown in Figure 1.4.

Figure 1.4 Symbol for a CNOT gate, and its effect on basis states.

It is worth noting at this point that we are putting the least-significant qubit at the top of a circuit diagram, and on the right on the state labels used with kets. This labels states with the natural binary order. However, there are different conventions in use, and this can be a point of confusion. Some authors put the top-most qubit in a circuit diagram on the left when they label kets. In this alternate notation, UCN′|c⟩|t⟩=|c⟩|c⊕t⟩ so that

(1.51)

The matrix representation of the CNOT gate in this alternate convention is

(1.52)

We will consistently use the first convention, with the least-significant qubit (top-most on a circuit diagram) on the right when writing state labels.