# Calculating nullifier and modfied tdm photonic circuit

I want to run an experiment on Borealis which is similar to the given example in time domain photonic circuits documentation (Generation of EPR pair)

https://strawberryfields.ai/photonics/demos/run_time_domain.html

Q1) it requires a calculation of nullifier since I can’t perform the homodyne measurement, is there any workaround with Borealis

Q2) can we create our own modified time domain circuit in Borealis where might only want 2D cluster state formation

Welcome to the Forum!

Since Borealis was built specifically for GBS it can be limited in what you can do with it. In our advanced Borealis tutorial, right before the section on calculating the transfer matrix, we have a comparison between the TDM circuit and how Borealis works with Fock measurements. I don’t know if this answers your first question, so please let me know if it doesn’t.

Regarding your second question, yes, you can modify the parameters in Borealis to create specific circuits given the constraints in Borealis. In the beginner Borealis tutorial you will notice the three loops. The first one allows you to entangle one qumode with the next. This is the first dimension. The second loop allows you to entangle one qumode with the one that comes 6 clock cycles later. This is the second dimension. The third loop allows you to entangle one qumode with the one that comes 36 clock cycles later. This is the third dimension. The key here is that you can define the gates manually, so you can control the entanglement of the third loop, and if you want there to be no entanglement you can simply set the transmission of that beamsplitter (T_2) to T=1 for all modes. You can learn more about this on the section of the tutorial called Define your gates manually.

I hope this helps you keep exploring and running experiments on Borealis!

Hi @CatalinaAlbornoz, Thank you for your response.
just another query related to the Q1
Is it possible to confirm whether cluster state formation occurred? because we can’t calculate the nullifier.

Hey @Souvik_Sarkar! I spoke to someone on our hardware team and this is what they said:

Usually, you would derive the nullifier operator of your cluster and compute the variance of its measurement outcomes. However, this nullifier operator is defined in terms of quadrature operators and Borealis users cannot measure those since that would require a different kind of detector (homodyne instead of photon-number resolving). So this user is asking if there is a “photon-number equivalent” to the quadrature nullifiers that they can use to verify their cluster. And I think there is: the noise-reduction factor (NRF). Problem is, we only know the NRF equation for two-mode squeezed states, i.e. a special and very simple type of cluster. Not what the user is looking for.

I’ll look into this a bit more!

Hey @Souvik_Sarkar, just a quick update here that I’m still waiting to hear back from some folks but haven’t forgotten about you! Will get back to you ASAP

Okay! Here’s an expert’s thoughts:

Something that can be done is to verify entanglement in the system using Schwinger spins that can be measured with PNR (https://arxiv.org/pdf/1310.5119.pdf). They can be related with a cluster state in some situations but in general I don’t know if that is possible. Also we can try to define observables that can be measure with PNR and derive an entanglement witness with them (https://arxiv.org/pdf/1303.6403.pdf) or use tricks like the ones used to derive van loock-furusawa criteria with such observables (https://arxiv.org/pdf/quant-ph/0210155.pdf) or use non gaussian criteria like here (https://www.pnas.org/doi/epdf/10.1073/pnas.0908329106). Finding the correct observables could be a difficult task and proving entanglement doesn’t prove, in general, that we have a cluster state, except for 2 and 3 mode gaussian states. This is because a cluster state is a very specific entangled state and there are many other states (even gaussian states) that are not cluster states (except, again, for two and three modes, all gaussian states with 3 or less modes are cluster states).

Hey! @isaacdevlugt Thank you very much for your response.
That’s a lot of paper to go through.

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Let us know if you come up with a solution!

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