RZ rotation gate

Definition

$$R_Z(\theta) = e^{-i\theta Z/2} = \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2} \end{pmatrix}$$

$R_Z$ is diagonal in the computational basis, so it commutes with any other $R_Z(\phi)$ and with measurement in the $Z$ basis.

Action on basis states

$$R_Z(\theta)|0\rangle = e^{-i\theta/2}|0\rangle, \qquad R_Z(\theta)|1\rangle = e^{i\theta/2}|1\rangle$$

Each basis state only picks up a phase, so $|\langle 0|R_Z(\theta)|0\rangle|^2 = |\langle 1|R_Z(\theta)|1\rangle|^2 = 1$ for every $\theta$ — a $Z$ rotation is invisible to a $Z$-basis measurement on its own. Its effect only shows up on a qubit already in superposition, e.g. $R_Z(\theta)\tfrac{1}{\sqrt2}(|0\rangle+|1\rangle) = \tfrac{1}{\sqrt2}(e^{-i\theta/2}|0\rangle + e^{i\theta/2}|1\rangle)$, which differs from the input by a relative phase of $e^{i\theta}$.

Key identities

• $P(\lambda) = e^{i\lambda/2}R_Z(\lambda)$: the OpenQASM phase gate $P$ is $R_Z$ up to an overall phase — $P$ fixes the $|0\rangle$ amplitude at exactly $1$ instead of splitting the phase symmetrically.
• $S = R_Z(\pi/2)$ and $T = R_Z(\pi/4)$ up to global phase, so $R_Z$ generalizes the whole Clifford/non-Clifford phase-gate ladder to a continuous parameter.
• $R_X(\theta) = H R_Z(\theta) H$ and $R_Y(\theta) = R_Z(-\pi/2) R_X(\theta) R_Z(\pi/2)$: composing conjugations by $H$ and by $R_Z(\pm\pi/2)$ lets a compiler reach any rotation axis from $R_Z$ alone.
• On superconducting hardware, $R_Z(\theta)$ is frequently realized as a virtual zero-duration frame update rather than a physical microwave pulse, making it effectively free and noiseless compared to $R_X$/$R_Y$.