5
$\begingroup$

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.

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

1 Answer 1

4
$\begingroup$

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. enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.