What would the operations be needed to run the V terms from the Bose Hubbard CV paper on the X8?

I was reading this paper by Christian Weedbrook, Timjan Kalajdzievski, and Patrick Rebentrost on running Bose Hubbard models on CV devices (https://arxiv.org/pdf/1801.06565.pdf) and wanted to implement it myself on the X8. While there are CV compatible operations given for running the J gate, there are no such operations given for running the U gate or the V gates, and instead they are listed in text (end of section D, page 6).

Could you please spell out what these operations would be on the device?

Also, I am aware @josh had written a guide for this already (https://sfopenboson.readthedocs.io/en/latest/tutorials/bosehubbard.html, thanks Josh this was so cool!) but I just wanted to be clear on the operations needed for the X8, as those would be different from what can be run locally on fock backend.

Hi @Samarth_Sandeep, that’s a great question.

It seems like the U gate (the on-site interaction term) is a Kerr gate, while the V gate (dipole-dipole interaction term) are cross-Kerr interactions.

Both of these are unavailable on X8.
The X8 only supports beamsplitters and rotation gates, as it is only a GBS device.

Hi Catalina, I am aware that the U and V gates as originally designed cannot be run on the X8 or any Gaussian backend.

However, the paper describes ways that they can be run on a Gaussian device using commutators to decompose the quartic and cubic phases down to quadratic ones, which should be able to be seen in section D on P. 6. That being said, because its all in text and not in the form of a decomposed circuit, I just wanted to double check exactly what that decomposition was.

Hi @Samarth_Sandeep!

I consulted with Tim, one of the authors of the paper, and he mentioned the following:

The paper says that cubic gates can be implemented optically with photon counting measurement (for non linearity), and the quartic gate can be decomposed with commutators and more cubic gates. Non Gaussian operations like V and U cannot be decomposed in terms of strictly gaussian operations, as a result they are required for universality!

For the quartic gate, the decomposition into other gates including cubic is given here, specifically you can go to equations (51) and (56) to (58).

I hope this helps!

Hi @CatalinaAlbornoz,

Thanks for the response and to Tim for adding some details! Do you think it would be possible for the gates for that cubic gate to be shared? Still not fully understanding how to write that out on the device.

Happy Sunday,

Hi @Samarth_Sandeep,

You can go to reference 36 in the paper. That’s where they got the optical implementation info for that gate.

Timjan mentioned that he has more detail in his PhD thesis.

In there you can look at equation (23) and the additional references 30 and 31.

Please let me know if this helps!