gates

NOT-Gate (X-Gate): $$\sigma_x = \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}$$ $| 0 ⟩ σ_{x} = | 1 ⟩$ and $| 1 ⟩ σ_{x} = | 0 ⟩$
Phase Gate (NOT-Gate in the diagonal basis):$$\sigma_z = \begin{pmatrix}1 & 0 \ 0 & -1\end{pmatrix}$$ $| 0 ⟩ σ_{z} = | 0 ⟩$ and $| 1 ⟩ σ_{z} = - | 1 ⟩$ . This could also be generalised to: $$\begin{pmatrix}1 & 0 \ 0 & e^{i\phi}\end{pmatrix}$$ which sends $| 0 ⟩ \mapsto | 0 ⟩$ and $| 1 ⟩ \mapsto e^{i ϕ} | 1 ⟩$
**T-Gate( $θ = ϕ / 4$ ): $$\begin{pmatrix}1 & 0 \ 0 & e^{i\pi/4}\end{pmatrix}$$
Hadamard Gate: $$\frac{1}{\sqrt2}\begin{pmatrix} 1 & 1 \ 1 & -1\end{pmatrix}$$ $| 0 ⟩ \mapsto \frac{| 0 ⟩ + | 1 ⟩}{\sqrt{2}}$ and $| 1 ⟩ \mapsto \frac{| 0 ⟩ - | 1 ⟩}{\sqrt{2}}$
Controlled-NOT Gate(CNOT Gate)