Controlled quantum gates are gates that have two types of input qubits:These gates can be understood as doing a certain unitary operation only if the control qubits are enabled or disabled.
- control qubits
- operand qubits (terminology made up by Ciro Santilli just now)
Generic controlled quantum gate symbol
. Source. When the operand has a conventional symbol, e.g. the Figure "Quantum NOT gate symbol" for the quantum NOT gate to form the CNOT gate, that symbol is used in the operand instead.
Some authors use the convention of:
The CNOT gate is a controlled quantum gate that operates on two qubits, flipping the second (operand) qubit if the first (control) qubit is set.
On the standard basis:we see that this means that only and should be possible. Therefore, the state must be of the form:where and are two complex numbers such that
If we operate the CNOT gate on that state, we obtain:and so the input is unchanged as desired, because the control qubit is 0.
Therefore, in that case, what happened is that the probabilities of and were swapped from and to and respectively, which is exactly what the quantum NOT gate does.
So from this we understand more concretely what "the gate only operates if the first qubit is set to one" means.
Now go and study the Bell state and understand intuitively how this gate is used to produce it.
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