RXX (XX interaction) gate

RXX matters wherever a model's interaction term is naturally expressed in the X basis rather than Z — Heisenberg-type spin chains, or trapped-ion hardware whose native multi-qubit gate is generated by a collective XX-type coupling rather than ZZ.

Takes2 qubitsWhat joins here

Takes a 2-qubit register and states no assumption about what is on it. That is the whole condition: anything returning 2 qubits can feed it.

16 entries meet this end. Named below.

Returns2 qubitsWhat joins here

Returns a 2-qubit register a next stage can take. This is the end that joins.

11 entries meet this end. Named below.

A unitary. The same register goes in and comes out, and this entry states no assumption about what is on it — which is what lets anything of the same width feed it. See all 29 →

Fits before this 16

Fits after this 11

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.

Takes2 qubitsWhat joins here

Takes a 2-qubit register and states no assumption about what is on it. That is the whole condition: anything returning 2 qubits can feed it.

16 entries meet this end. Named below.

Returns2 qubitsWhat joins here

Returns a 2-qubit register a next stage can take. This is the end that joins.

11 entries meet this end. Named below.

A unitary. The same register goes in and comes out, and this entry states no assumption about what is on it — which is what lets anything of the same width feed it. See all 29 →

Fits before this 16

Fits after this 11

Qubits: 2
Depth: 1 native / 5-gate decomposition (H, H, CX, RZ, CX, H, H)
Parameter: θ (radians)
Matrix: cos(θ/2)·I⊗I - i sin(θ/2)·X⊗X
Generator: exp(-iθ XX/2)
Decomposition: H⊗H · RZZ(θ) · H⊗H
Gate family: Non-Clifford (generic θ); Clifford at θ=π/2

RXX(π/2) on q[0], q[1]

P(|00⟩) at θ=π/250%

P(|11⟩) at θ=π/250%

Quantum vs classical

Classical baseline

Use a classical state-vector or matrix simulation at the same width, precision, and measurement objective.

Quantum claim

The quantum record demonstrates a state or operator behavior; it does not make classical simulation or communication costs disappear.

How to compare

Compare fidelity, samples, gate depth, noise, memory, and the cost of preparing and reading the state.

Nobody has reviewed this record for gaps yet.

FrameworkNative

rxx-interaction.qasm

OPENQASM 3.0;
include "stdgates.inc";
qubit[2] q;
rxx(pi/2) q[0], q[1];

Related entries

RZZ (ZZ interaction) gate
Gates
RX rotation gate
Gates
Transverse-field Ising simulation
Algorithms
Heisenberg XXZ spin-chain operator
Operators