Single-qubit gate

The first two that you should study are:

The Hadamard gate takes $|0⟩$ or $|1⟩$ (quantum states with probability 1.0 of measuring either 0 or 1), and produces states that have equal probability of 0 or 1.

The quantum NOT gate swaps the state of $|0⟩$ and $|1⟩$, i.e. it maps: As a result, this gate also inverts the probability of measuring 0 or 1, e.g.

The most common way to construct multi-qubit gates is to use single-qubit gates as part of a controlled quantum gate.

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.

The first example to look at is the CNOT gate.

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.

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 $|00⟩$ and $|01⟩$ should be possible. Therefore, the state must be of the form: where $x$ and $y$ are two complex numbers such that $|x|+|y|=1.0$

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 $|10⟩$ and $|11⟩$ were swapped from $x$ and $y$ to $y$ and $x$ 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.