RXX

RXX is a two-qubit "Ising coupling" gate - it entangles two qubits by an amount that depends continuously on an angle \( \theta \), based on the \( X \otimes X \) interaction.

Qubits	2
Parameters	\( \theta \) - coupling angle, in radians
Inverse	RXX(−θ)
Also called	XX-rotation, Ising XX coupling gate

What it does

RXX(θ) is defined as \( e^{-i\theta X \otimes X / 2} \). It mixes pairs of basis states that differ in both qubits: \( |00\rangle \) with \( |11\rangle \), and \( |01\rangle \) with \( |10\rangle \).

Matrix

\[ RXX(\theta) = \begin{pmatrix} \cos(\theta/2) & 0 & 0 & -i\sin(\theta/2) \\ 0 & \cos(\theta/2) & -i\sin(\theta/2) & 0 \\ 0 & -i\sin(\theta/2) & \cos(\theta/2) & 0 \\ -i\sin(\theta/2) & 0 & 0 & \cos(\theta/2) \end{pmatrix} \]

Action on basis states

• \( |00\rangle \rightarrow \cos(\theta/2)|00\rangle - i\sin(\theta/2)|11\rangle \)
• \( |01\rangle \rightarrow \cos(\theta/2)|01\rangle - i\sin(\theta/2)|10\rangle \)
• \( |10\rangle \rightarrow \cos(\theta/2)|10\rangle - i\sin(\theta/2)|01\rangle \)
• \( |11\rangle \rightarrow \cos(\theta/2)|11\rangle - i\sin(\theta/2)|00\rangle \)

Example

RXX with \( \theta = \pi/2 \) applied to q[0], q[1] (both starting at \( |0\rangle \)) produces \( \dfrac{1}{\sqrt{2}}(|00\rangle - i|11\rangle) \) - an entangled state built in a single gate, without a separate H + CNOT pair.

Other notations

Naming conventions across ecosystems; exact syntax can vary by library version, and this does not claim to be Qompile's generated output unless otherwise noted.

Language	Typical form
OpenQASM 2.0	rxx(theta) q[0], q[1];
Qiskit	circuit.rxx[theta, q[0], q[1]]
Cirq	cirq.XXPowGate(exponent=theta / math.pi)(q[0], q[1])
Q#	-

Related

• RZZ - the equivalent coupling gate built from Z⊗Z