Ciro Santilli 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)
The first example to look at is the CNOT gate.
The black dot means "control qubit", and "U" means an arbitrary Unitary operation.
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:
- filled black circle: conventional controlled quantum gate, i.e. operate if control qubit is active
- empty (White) circle: operarate if control qubit is inactive
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.
This gate is the first example of a controlled quantum gate that you should study.
To understand why the gate is called a CNOT gate, you should think as follows.
First let's produce a generic quantum state vector where the control qubit is certain to be 0.
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.
If however we take only states where the control qubit is for sure 1:
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 concretelly 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.