Is there relation between IonQ native gates and the rotation gates $R_x$, $R_y$, $R_z$?

Question

I know $GZ = R_z$ and $XX = MS$ but what about $GPI$ and $GPI2$?

Answer 1

I believe that $GPI(\theta)=XR_z(-2\theta)$ and $GPI2(\theta)=R_z(\theta)R_x(\pi/2)R_z(-\theta)$.

Answer 2

According to IonQ website^[1], $$GPI(\phi)=\begin{pmatrix}0&e^{-i\phi}\\e^{i\phi}&0\end{pmatrix}$$ $$GPI2(\phi)=\frac{1}{\sqrt{2}}\begin{pmatrix}1&-ie^{-i\phi}\\-ie^{i\phi}&1\end{pmatrix}$$ Since $GPI$ and $GPI2$ are single-qubit gates, they can be decomposed into rotations about orthogonal axes ($R_x$, and $R_z$ for example). And as @Ken Robbins stated in his answer,

$GPI=R_z(2\phi)X$

$GPI2=R_z(\phi)R_x(\pi/2)R_z(-\phi)$