# Native Gate Decomposition

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

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

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

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.

• MS doesn't seem right. Are you missing a factor of $1/\sqrt{2}$? And you've got an extra $1/\sqrt{2}$ in $GZ(\phi)$? Commented Jan 7, 2022 at 7:37
• DaftWullie is exactly correct; I edited it to correct their mistakes. Commented Jan 7, 2022 at 14:18

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.