RXX (XX interaction) gate

Definition

$$\mathrm{RXX}(\theta) = e^{-i\theta (X\otimes X)/2} = \begin{pmatrix} \cos(\theta/2)&0&0&-i\sin(\theta/2)\\0&\cos(\theta/2)&-i\sin(\theta/2)&0\\0&-i\sin(\theta/2)&\cos(\theta/2)&0\\-i\sin(\theta/2)&0&0&\cos(\theta/2) \end{pmatrix}$$

Since $(X\otimes X)^2 = I\otimes I$, the exponential splits as $\cos(\theta/2)\,I\otimes I - i\sin(\theta/2)\,X\otimes X$, and because $X\otimes X$ swaps $|00\rangle\leftrightarrow|11\rangle$ and $|01\rangle\leftrightarrow|10\rangle$, the off-diagonal terms connect exactly those pairs.

Action on basis states

$$\mathrm{RXX}(\theta)|00\rangle = \cos(\theta/2)|00\rangle - i\sin(\theta/2)|11\rangle, \qquad \mathrm{RXX}(\theta)|01\rangle = \cos(\theta/2)|01\rangle - i\sin(\theta/2)|10\rangle$$

At $\theta=\pi/2$, $\mathrm{RXX}(\pi/2)|00\rangle = \tfrac{1}{\sqrt2}(|00\rangle - i|11\rangle)$ — an entangled Bell-like state produced from a product-state input, giving equal $1/2$ probability on $|00\rangle$ and $|11\rangle$.

Decomposition

$$\mathrm{RXX}(\theta) = (H\otimes H)\;\mathrm{RZZ}(\theta)\;(H\otimes H)$$

Since $X = HZH$, we have $X\otimes X = (H\otimes H)(Z\otimes Z)(H\otimes H)$, so conjugating the entire $\mathrm{RZZ}(\theta)$ circuit — itself $\mathrm{CX}\,(I\otimes R_Z(\theta))\,\mathrm{CX}$ — by Hadamards on both qubits rotates the interaction axis from $Z\otimes Z$ to $X\otimes X$, giving the five-gate circuit $H,H,\mathrm{CX},R_Z(\theta),\mathrm{CX},H,H$.

Key identities

• $\mathrm{RXX}(\pi/2)$ is Clifford: it equals $(H\otimes H)\,\mathrm{RZZ}(\pi/2)\,(H\otimes H)$, a conjugation of the Clifford gate $\mathrm{RZZ}(\pi/2)$ by Clifford Hadamards, hence itself Clifford; for generic $\theta$ it is not.
• $\mathrm{RXX}(\theta_1)\,\mathrm{RXX}(\theta_2) = \mathrm{RXX}(\theta_1+\theta_2)$: successive XX pulses on the same pair compose additively.
• $\mathrm{RXX}(\pi) = -i\,X\otimes X$: a full-$\pi$ pulse reproduces a simultaneous bit-flip on both qubits up to global phase.
• $\mathrm{RXX}$ is the generating gate of the trapped-ion Mølmer–Sørensen interaction, which natively entangles pairs (or larger groups) of ions via their shared motional mode.