ZZ gate (Ising coupling)

ZZ gate is a parametric two-qubit gate implementing correlated Z rotations. Among the most commonly available parametric gates.

Matrix:

$$\text{ZZ}(\theta) = \exp(-i\theta Z_1 Z_2 / 2) = \begin{pmatrix} e^{-i\theta/2} & 0 & 0 & 0 \\ 0 & e^{i\theta/2} & 0 & 0 \\ 0 & 0 & e^{i\theta/2} & 0 \\ 0 & 0 & 0 & e^{-i\theta/2} \end{pmatrix}$$

Action: applies correlated phase rotations; leaves computational basis unchanged, only phases states.

• $\text{ZZ}(0) = I$
• $\text{ZZ}(\pi/2)$ phase entanglement (not strength entanglement)
• $\text{ZZ}(\pi) = -\text{CZ}$ (up to global phase)

• Parametric: tune interaction angle
• Diagonal: only applies phases, doesn't mix basis states
• Symmetric: both qubits treated equally
• Hamiltonian form: $\text{ZZ}(\theta) = \exp(-i\theta Z \otimes Z / 2)$

• Superconducting qubits: inductive or capacitive coupling naturally produces ZZ (static and parametric)
• Trapped ions: driven ZZ interactions via laser detuning
• Spin systems: Ising-type ZZ coupling (very common in condensed matter)

• Variational algorithms: ubiquitous in QAOA, VQE, Ising-problem ansatze
• Quantum simulation: simulate ZZ-coupled systems (Ising model, quantum magnetism)
• Native operations: highly optimized on most platforms; preferred for performance

• Special case: $\text{CZ} = \text{ZZ}(\pi)$ (up to global phase)

• From CNOT and RZ: possible but complex if ZZ not native
• Advantage: native ZZ on most platforms (don't decompose)