5.4.1 Circuits with multiple gates

Single-qubit gates and two-qubit gates can be combined in a structured sequence to give a quantum circuit containing multiple quantum gates. Consider as an example the circuit shown in Figure 11, which depicts a sequence of gates acting on two qubits, labelled and . Remember that each qubit enters its circuit from the left; first, a Hadamard gate is applied to qubit . Then a CNOT is applied to and , where the control C is and the target T is . Finally, a NOT gate is applied to .

Figure 11 An example of a quantum circuit with multiple quantum gates

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The figure shows two parallel horiontal lines. The upper one is labelled q1 and the lower one is labelled q2. In the left part of the upper line is a square box with a capital letter H in it. Further along the upper line is a large dot. A vertical line leads from this dot to the lower line where it joins a large circle with a large plus sign in it. to the right of this on the lower line is another large circle with a large plus sign in it.

Figure 11 An example of a quantum circuit with multiple quantum gates

The circuit in Figure 11 is a sequence of operations applied to qubits and can be analysed using the methods already introduced, with extra subscripts labelling the single-qubit gates and operations so that the qubit each operation is acting on is clear. Thus acts only on the qubit , leaving unchanged. Therefore, using as a sample input the calculation becomes:

Now acts only on , so

This means that

Next comes the effect of the CNOT gate, courtesy of and gives

Finally, observe that acts on only, so flip of each ket in the superposition: