# Z-gate (Relative phase gate) Definition $Z=\begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix}$, $Z|0\rangle = |0\rangle$ and $Z|1\rangle=-|1\rangle$. if $|\psi\rangle = \alpha|0\rangle+\beta|1\rangle$, then $Z|\psi\rangle = \alpha|0\rangle-\beta|1\rangle$ i.e., it applies relative phase of $\pi$ to $|1\rangle$. note: $Z^2=I$ ## Created 2026-01-14 14:19