quantum gate directory
$xx$ Rotation
Symbol
$R_{xx}$
Description:
Two-qubit rotation generated by $X \otimes X$.
Alternate notations:
- $R_{xx}(\theta)$
- $\mathrm{RXX}(\theta)$
- $XX(\theta)$
SDK Support
| SDK | Name |
|---|---|
| Qiskit |
qiskit.circuit.library.RXXGate
|
| PennyLane |
pennylane.IsingXX
|
| Cirq |
cirq.XXPowGate
ⓘ
|
| Q# | — |
| PyQuil | — |
| Braket | — |
| BQSKit | — |
The $R_{xx}$ gate rotates in the $XX$ interaction basis by angle $\theta$.
$$ R_{xx}(\theta) = \exp\left(-i\frac{\theta}{2}X\otimes X\right) = \begin{bmatrix} \cos\frac{\theta}{2} & 0 & 0 & -i\sin\frac{\theta}{2} \\ 0 & \cos\frac{\theta}{2} & -i\sin{\frac{\theta}{2}} & 0 \\ 0 & -i\sin{\frac{\theta}{2}} & \cos{\frac{\theta}{2}} & 0 \\ -i\sin{\frac{\theta}{2}} & 0 & 0 & \cos{\frac{\theta}{2}} \end{bmatrix} $$
Properties
- Entangling for $\theta$ not a multiple of $\pi$.
- Related to $R_{zz}$ by basis change: $R_{xx}(\theta) = (H \otimes H),R_{zz}(\theta),(H \otimes H)$.
- Generated by an $XX$ interaction Hamiltonian.
Usage
- Native entangling gate in trapped-ion and some superconducting platforms.
- Variational circuits and Hamiltonian simulation of $XX$ couplings.
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