Ising Coupling Gate RXX

A symmetric two-qubit entangler generated by the XX interaction, native to trapped-ion hardware.

Definition

The RXX(θ) gate implements exp(-i θ (X⊗X) / 2). It is generated by the Ising XX coupling between two spins and is a natural, hardware-level entangler on trapped-ion machines, where a shared motional mode mediates an effective X⊗X interaction.

Matrix

At θ = π/2 the RXX gate is locally equivalent to CNOT — it becomes a maximally entangling gate that a single pair of single-qubit rotations converts into a CNOT. The Mølmer-Sørensen gate is the physical realization of RXX on ions.

Symmetry and commutation

RXX is symmetric under exchange of the two qubits because X⊗X is symmetric. Two RXX gates on the same pair commute and add their angles: RXX(α)·RXX(β) = RXX(α+β). This makes calibration and error accumulation straightforward on ion hardware.

import numpy as np
def rxx(theta):
    c, s = np.cos(theta/2), -1j*np.sin(theta/2)
    return np.array([[c,0,0,s],[0,c,s,0],[0,s,c,0],[s,0,0,c]])

Uses

RXX directly simulates the transverse-field-Ising and Heisenberg couplings that appear in condensed-matter models and in variational chemistry ansätze. Because it is native to ions, compilers targeting those platforms express entanglement in RXX rather than CNOT, saving the extra single-qubit dressing a CNOT would require.

The trio RXX, RYY, RZZ spans the local-equivalence class of all two-qubit entanglers through the KAK / canonical decomposition. See canonical gate and Mølmer-Sørensen.