Controlled-RZ gate (CRZ)

Definition

$$\mathrm{CRZ}(\theta) = |0\rangle\langle 0|\otimes I + |1\rangle\langle 1|\otimes R_Z(\theta) = \begin{pmatrix} 1&0&0&0\\0&1&0&0\\0&0&e^{-i\theta/2}&0\\0&0&0&e^{i\theta/2} \end{pmatrix}$$

with the control as the first (leftmost) qubit.

Action on basis states

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

The control is untouched. When the control is $|1\rangle$, both target values pick up a phase — $|10\rangle$ gets $e^{-i\theta/2}$, not just $1$ — which is the key difference from $\mathrm{CP}(\theta)$, whose $|10\rangle$ amplitude is always left exactly at $1$.

Decomposition

$$\mathrm{CRZ}(\theta) = (I\otimes R_Z(\theta/2))\;\mathrm{CX}\;(I\otimes R_Z(-\theta/2))\;\mathrm{CX}$$

For control $0$, both CNOTs act trivially and the target sees $R_Z(\theta/2)R_Z(-\theta/2)=I$. For control $1$, the first CNOT flips the target, the surrounding $R_Z(\mp\theta/2)$ gates pick up phases that depend on the (opposite-sign) bit values before and after the flip, and the second CNOT flips the target back — a direct phase-kickback calculation shows the two contributions add to exactly $e^{i\theta\,\mathrm{sign}(t)/2}$, reproducing $R_Z(\theta)$ on the original target value $t$.

Key identities

• $\mathrm{CRZ}(\theta) \ne \mathrm{CP}(\theta)$ in general: they agree on $|00\rangle,|01\rangle,|11\rangle$ only up to how each defines phase on $|10\rangle$, and differ by a target-controlled global-phase factor overall.
• $\mathrm{CRZ}(\theta_1)\,\mathrm{CRZ}(\theta_2) = \mathrm{CRZ}(\theta_1+\theta_2)$: controlled Z-rotations on the same qubit pair compose additively, just like the underlying $R_Z$.
• $\mathrm{CRZ}(2\pi) = I\otimes I$ and $\mathrm{CRZ}(0) = I \otimes I$, since $R_Z(2\pi) = R_Z(0) = I$ up to global phase considerations that cancel in the controlled version.
• $\mathrm{CRZ}$ is the natural entangling primitive whenever an algorithm needs a data-dependent $Z$-rotation, e.g. controlled time-evolution steps in quantum simulation.