X-gate (Pauli-X, NOT gate)
$X = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}$
Y-gate (Pauli-Y gate)
$Y = \begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}$
Z-gate (Pauli-Z gate)
$Z = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}$
H-gate (Hadamard gate)
$H = \frac{1}{\sqrt{2}} \begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix}$
T-gate
$T = \begin{pmatrix}
1 & 0 \\
0 & e^{i\pi/4}
\end{pmatrix}$
CNOT-gate (Controlled NOT, CX gate)
$\text{CNOT} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0
\end{pmatrix}$
SWAP-gate
$\text{SWAP} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}$