Ising Coupling Gate RZZ

A diagonal two-qubit phase gate generated by the ZZ interaction, the entangling core of QAOA and phase-sum circuits.

Definition

RZZ(θ) = exp(-i θ (Z⊗Z) / 2) is diagonal in the computational basis: it phases each basis state by the product of its two bit-signs. Because Z⊗Z assigns +1 to |00⟩ and |11⟩ and -1 to |01⟩ and |10⟩, RZZ imprints a parity-dependent phase.

Matrix

The like-parity states {00, 11} get phase e^{-iθ/2}; the odd-parity states {01, 10} get e^{+iθ/2}. RZZ is therefore a parity phase gate and is closely tied to the parity-check circuits of quantum error correction.

Decomposition

RZZ(θ) = CNOT·(I⊗RZ(θ))·CNOT. The first CNOT computes the parity into the target, RZ applies the phase conditioned on that parity, and the second CNOT restores the state. This two-CNOT template is one of the most used gadgets in compiled circuits.

import numpy as np
def rzz(theta):
    p, m = np.exp(-1j*theta/2), np.exp(1j*theta/2)
    return np.diag([p, m, m, p])

Uses

RZZ is the cost-layer gate of the Quantum Approximate Optimization Algorithm: each edge of a problem graph contributes an RZZ whose angle encodes the coupling weight. It also builds the Ising-model timestep in digital simulation and appears in phase estimation of diagonal Hamiltonians.

In Kronos digital-twin surrogate experiments, diagonal RZZ layers are a compact way to encode pairwise couplings in a plasma-parameter energy landscape without adding non-diagonal entanglers.