RXX gate

The RXX gate rotates a pair of qubits together about the $X \otimes X$ axis by an angle $\theta$. It acts on two qubits and takes one parameter, $\theta$ in radians. The two qubits are interchangeable. The Circuit panel draws an Xx(θ) box on each wire, with the angle shown as a multiple of π. At $\theta = \pi/2$ on two fresh qubits, the pair reads 00 or 11 with equal probability and never 01 or 10.

Kets such as $\vert 0 \rangle$ name the basis states and rotations are described on the Bloch sphere; the qubits lesson introduces both.

Call

Matrix

The matrix uses the basis order $\vert 00 \rangle, \vert 01 \rangle, \vert 10 \rangle, \vert 11 \rangle$, with $c = \cos\frac{\theta}{2}$ and $s = \sin\frac{\theta}{2}$.

That is $e^{-i\frac{\theta}{2} X \otimes X}$. It is an RZZ gate of the same angle with an H gate on both qubits before and after.

Effect on basis states

At $\theta = \pi/2$ the gate is fully entangling: from $\vert 00 \rangle$ it produces an equal superposition of $\vert 00 \rangle$ and $\vert 11 \rangle$, a Bell state up to a phase.

Inverse

The inverse of rxx(θ) is rxx(-θ).

Usage

from math import pi
from qiskit import QuantumCircuit

qc = QuantumCircuit(2, 2)
qc.rxx(pi / 2, 0, 1)
qc.measure([0, 1], [0, 1])

import { QuantumCircuit } from 'qiskit';

const circuit = QuantumCircuit(2, 2);
circuit.rxx(Math.PI / 2, 0, 1);
circuit.measure([0, 1], [0, 1]);

On a noiseless simulator every shot reads 00 or 11, in close to equal numbers.

On IonQ hardware

RXX is the abstract form of Aria's native entangler: for θ from 0 to π/2, rxx(θ) is exactly MS(0, 0, θ/2π), one native gate with no single-qubit pulses, partially entangling between 0 and π/2. IonQ's MS angle runs from 0 to a quarter turn; above π/2 the compiler rewrites the gate and does not publish the result. On Forte, where the entangler is ZZ, it is one ZZ gate with single-qubit pulses around it. See How your circuit is compiled.

Related

• RYY gate
• RZZ gate
• MS gate
• RX gate
• Gate reference