RZZ

RZZ is a two-qubit quantum gate that implements a rotation around the ZZ axis of the two-qubit state space. It is defined by the unitary U = exp(-i θ/2 Z ⊗ Z), where θ is a real parameter and Z is the Pauli-Z operator. In the computational basis, U acts as follows: |00> and |11> acquire a phase of e^{-i θ/2}, while |01> and |10> acquire a phase of e^{+i θ/2}. Thus, the gate applies parity-dependent phases and can generate entanglement for intermediate values of θ.

Physically, RZZ arises as a ZZ-type interaction between two qubits. On superconducting quantum processors, it is implemented by turning on a tunable coupling that produces an effective Z⊗Z interaction for a controlled period, with θ related to the interaction strength and duration by θ = 2 J t / ħ. As a native two-qubit entangling gate, RZZ is widely used in variational quantum algorithms and can serve as a modular building block in quantum circuits. It is often available as a native gate on devices that support tunable couplers or fixed ZZ couplings.

Relation to other gates and usage notes: RZZ(θ) is closely related to controlled-phase gates; with appropriate single-qubit Z-rotations, it can be used to implement a CZ or other phase gates. It is a common choice for entangling qubits in ansatz circuits for VQE and QAOA, where θ is treated as a variational parameter. The gate is compact and expressive, with θ = 0 yielding the identity and θ ≈ π producing strong entanglement.