quantum gate directory

The $R_{xx}$ gate rotates in the $XX$ interaction basis by angle $\theta$.

$$ R_{xx}(\theta) = \exp\left(-i\frac{\theta}{2}X\otimes X\right) $$

In the computational basis:

$$ \begin{bmatrix} \cos\tfrac{\theta}{2} & 0 & 0 & -i\sin\tfrac{\theta}{2} \\ 0 & \cos\tfrac{\theta}{2} & -i\sin\tfrac{\theta}{2} & 0 \\ 0 & -i\sin\tfrac{\theta}{2} & \cos\tfrac{\theta}{2} & 0 \\ -i\sin\tfrac{\theta}{2} & 0 & 0 & \cos\tfrac{\theta}{2} \end{bmatrix} $$

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.