Some common quantum gates

Identity: $\ket{x} \mapsto \ket{x}$

$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

NOT (Pauli X): $\ket{x} \mapsto \ket{1 \oplus x}$

$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

Pauli Y:

$Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$

Pauli Z:

$Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

Hadamard:

$H = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$

CNOT: $\ket{x} \ket{y} \mapsto \ket{x} \ket{x \oplus y}$

$C = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0\end{pmatrix}$

Phase shift:

$R_\theta = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\theta} \end{pmatrix}$

(To be continued)