## Quantum Z gate
**Quantum Z gate** (or **Pauli-Z gate**, **phase flip gate**) is a quantum gate. It rotates the quantum state around the computational axis (the Z-axis) by $\pi$ degrees. In matrix form, it is equivalent to the Pauli-Z matrix $\sigma_z$. This is why it's commonly called Pauli-Z gate.

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

For computational basis states (states on the Z-axis) it leaves the state $\lvert 0\rangle$ unchanged, but flips the phase of $\lvert 1\rangle$ by $\pi$ degrees. This is why it's called a phase flip gate.

$$Z\lvert 0\rangle = \lvert 0\rangle\qquad Z\lvert 1\rangle = -\lvert 1\rangle$$