Here we list some of the more commonly used gates in Quantum Computing. Some are easier to grasp than others, and those familiar with classical computing will recognize a few of the gates such as NOT which turns a 1 into 0 and a 0 into a 1 and does exactly the same on a Qubit.

#### State Vectors

We can represent our quantum state vector as a superposition of two states 0 and 1. We show their representations here. $\left| 0 \right> = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$

and $\left| 1 \right> = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$

Any generic vector can be comprised of a linear combination of these two states $\left|\psi \right> = \alpha \left|0 \right> + \beta \left|1 \right>$

#### Pauli Gates

These are gates that act on a single qubit and can be thought of as some of the most fundamental gates in Quantum Computing and they are also the easiest to learn and remember for most people.

#### NOT Gate (X)

Perhaps the most basic gate and easily interpret able because the behaviour is the same in both the classical and Quantum case. We basically take a statement that is False and make it True, and vice versa. Just like normal programming. The matrix is represented below: $\sigma_{x} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \ \end{bmatrix}$

#### Y Gate $\sigma_{y} = \begin{bmatrix} 0 & i \\ i & 0 \ \end{bmatrix}$

#### Z Gate $\sigma_{z} = \begin{bmatrix} 1 & 0 \\ 0 & -1 \ \end{bmatrix}$

#### Example NOT gate on two different states (0/1)

We can easily see how any of the Pauli gates work on a single Qubit. As follows we take a 0 and put it through a NOT gate and the same with a 1. Lets check we get an output that we expect. $X( \left| 0 \right>) = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \left| 1 \right>$

And $X( \left| 1 \right>) = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \left| 0 \right>$

Perhaps one of the most interesting of Quantum gates Hadamard creates a superposition from each of the states 0 and 1. It performs the following mapping  and its matrix is as follows. $H = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 1 \\ 1 & -1 \ \end{bmatrix}$

Many circuits involve the Hadamard Gate, so its worth understanding well.

#### Phase Shift Gate $Phase = \begin{bmatrix} 1 & 0 \\ 0 & e^{i\phi} \\ \end{bmatrix}$

#### CNOT

The C stands for controlled, which mean it only operates a NOT operation if another condition is met, that is a control qubit is 1, otherwise it does nothing. It operates on 2 qubits and therefore is a 4×4 matrix which is represented below. $CNOT = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}$

#### SWAP

Another 4×4 matrix which operates on 2 qubits. It takes the state of state A and swaps with state B. A very useful process to have happen for a number of process. It is easy to see that this works. $SWAP = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 &1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$