quantum gate directory

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

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

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 $\theta$ that are not an integer multiple of $\pi$.
• Related to $R_{xx}$ by single-qubit phase rotations: $R_{yy}(\theta) = (S \otimes S)\,R_{xx}(\theta)\,(S^\dagger \otimes S^\dagger)$.
• Generated by a $YY$ interaction Hamiltonian.

Usage

• Entangling layers in variational algorithms.
• Basis-changed implementations of $XX$ or $ZZ$ couplings.