Native Gate Decomposition

Question

TL;DR: I've got a very small set of gates to use and need to find efficient decompositions for $R_y$ and controlled $R_y$ gates. Does anyone have any better ideas than what I have?

I'm looking to implement something on an ion trap device. My circuit uses $R_y$ and controlled-$R_y$ gates when purely theoretical. I'm trying to find the most efficient representation of the $R_y$ and controlled-$R_y$ gates that I can. The native gates for the device are

\begin{equation} GPI(\phi) = \begin{pmatrix} 0 & e^{-i\phi} \\ e^{i\phi} & 0 \\ \end{pmatrix} \end{equation} \begin{equation} GPI2(\phi) = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & -ie^{-i\phi} \\ -ie^{i\phi} & 1 \\ \end{pmatrix} \end{equation} \begin{equation} GZ(\phi) = \begin{pmatrix} e^{-i\phi/2} & 0 \\ 0 & e^{i\phi/2} \\ \end{pmatrix} \end{equation} \begin{equation} MS=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 0 & 0 & -i \\ 0 & 1 & -i & 0 \\ 0 & -i & 1 & 0 \\ -i & 0 & 0 & 1 \\ \end{pmatrix} \end{equation}

So far I've come up with \begin{equation} i R_y(\phi) = i \begin{pmatrix} \cos(\phi/2) & - \sin(\phi/2) \\ \sin(\phi/2) & \cos(\phi/2) \end{pmatrix} = GPI2(\pi)\cdot GPI(\phi/2)\cdot GPI(\pi) \end{equation}

I don't know how to decompose a CNOT yet but I imagine that's fairly easy. Once I have that I use two CNOTs and two $R_y$ gates to implement a controlled-$R_y$.

Does anyone have any clever ideas on how to implement an $R_y$ gate with less than $3$ native gates, or an controlled-$R_y$ gate with less than two CNOTs and two $R_y$ gates?

Thanks!

Edit 1: Removed the erroneous factor of $1/\sqrt{2}$ from $GZ(\phi)$ and added it to the Molmer Sorenson gate.

Answer 1

You cannot do any better than your stated decompositions.

For example, think about cNOT. It is maximally entangling in the sense that a separable input (e.g. $|+\rangle|0\rangle$) can give a maximally entangled output. Any unitary that is a single controlled-NOT dressed by arbitrary single-qubit operations must also have this property, as single-qubit unitaries cannot change the amount of entanglement present. However, the controlled-$R_y$ does not have this property. It cannot create a maximally entangled state from a separable state (for generic $\phi$). Thus, it must require at least two controlled-nots.

That said, for the controlled-$R_y$, you would prefer a direct minimisation of the number of applications of MS gates and single qubit gates. You are not guaranteed that going via the cNOT construction achieves this for you. I suspect that in this case, you can get there from the cNOT construction and a few circuit identities for reducing the single-qubit gates.

For completeness, I calculated a decomposition of cNOT in terms of MS and standard gates. You should be able to easily convert these into your single-qubit gate set.