RZZ (ZZ interaction) gate

Definition

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

Since $(Z\otimes Z)^2 = I\otimes I$, the exponential splits exactly as $\cos(\theta/2)\,I\otimes I - i\sin(\theta/2)\,Z\otimes Z$, and because $Z\otimes Z$ is diagonal with eigenvalues $\pm1$, the result is diagonal.

Action on basis states

$$\mathrm{RZZ}(\theta)|00\rangle = e^{-i\theta/2}|00\rangle, \quad \mathrm{RZZ}(\theta)|01\rangle = e^{i\theta/2}|01\rangle, \quad \mathrm{RZZ}(\theta)|10\rangle = e^{i\theta/2}|10\rangle, \quad \mathrm{RZZ}(\theta)|11\rangle = e^{-i\theta/2}|11\rangle$$

No population moves between basis states; only the relative phase between same-parity and opposite-parity computational states changes, which is exactly the interference resource used by QAOA cost-Hamiltonian layers.

Decomposition

$$\mathrm{RZZ}(\theta) = \mathrm{CX}\;(I\otimes R_Z(\theta))\;\mathrm{CX}$$

The first CNOT maps the target qubit onto the parity $t\oplus c$; a single-qubit $R_Z(\theta)$ there attaches $e^{i\theta\,\mathrm{sign}(t\oplus c)/2}$, and the second CNOT restores the target to $t$ while leaving the accumulated phase in place — direct basis-state tracking confirms this phase equals the $\mathrm{RZZ}(\theta)$ diagonal entry exactly for all four $(c,t)$ inputs.

Key identities

• $\mathrm{RZZ}(\pi/2)$ is Clifford: numerically, $e^{i\pi/4}\mathrm{RZZ}(\pi/2) = (S\otimes S)\cdot\mathrm{CZ}$ (both sides equal $\mathrm{diag}(1,i,i,1)$), so at this special angle $\mathrm{RZZ}$ reduces to a product of Clifford gates; for generic $\theta$ it is not Clifford.
• $\mathrm{RZZ}(\theta_1)\,\mathrm{RZZ}(\theta_2) = \mathrm{RZZ}(\theta_1+\theta_2)$: successive ZZ-interaction pulses on the same pair compose additively, matching real continuous-time evolution under a fixed coupling.
• $\mathrm{RXX}(\theta) = (H\otimes H)\,\mathrm{RZZ}(\theta)\,(H\otimes H)$: conjugating by Hadamards on both qubits rotates the interaction axis from $Z\otimes Z$ to $X\otimes X$.
• $\mathrm{RZZ}(\theta)$ is the two-qubit workhorse of Trotterized Hamiltonian simulation for Ising-type models and of the cost-Hamiltonian layer in QAOA for Max-Cut-style problems.