The mean photon number per mode = Trace(Covariance matrix)/N - 0.5
where N is the number of modes
Hi @MUSHKAN_SUREKA,
I can’t seem to find the formula you mention. Where in the paper is it located?
Hello Sebastian,
Even I cannot find a relevant paper but I asked my professor and he told me about this. Th most relevant link I could find is continuous variable - How to write the covariance matrix of a quantum gaussian state as a function of photon numbers? - Quantum Computing Stack Exchange.
Kindly look into it and I would update you if I find a paper which mentions this.
a small proof of that can be:-
n = a⁺a = (q ‑ ip)(q + ip) = q² + i (qp – pq) + p² = q² + p² – ℏ/2
Hence, for a single mode of 2×2 covariance matrix V and displacement 0, ⟨n⟩= ⟨q²⟩ + ⟨p²⟩ – ℏ/2 = Tr V – ℏ/2
I uploaded a reference on the drive. Eq 5.9 on page 50 of the thesis should help you.
Hi @MUSHKAN_SUREKA, we’re taking a look at this. It is possible that the result is correct and indeed a large number of photons is needed, but since the paper was just showing a proof of concept this was not relevant for them.
We’ll look into this and come back with a more definite answer.
Hi @MUSHKAN_SUREKA, yes you are correct. The mean photon number (in the case of zero displacement) is given by the trace of the covariance matrix minus ℏ/2. Note that by default in SF we use ℏ=2.
If you have multiple modes, the formula still works, except you have to subtract out the vacuum energy of all modes: \bar{n} = \textit{Tr}(V) - N ℏ/2, where N is the number of modes.
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Hello Nathan, yeah however even after that the mean photon number is too high
Hello Nathan,
https://arxiv.org/pdf/2301.01232.pdf Appendix A section 2 might be useful. Could you please confirm if equation A7 is correct? Since it is nowhere mentioned in Xanadu/s graph similarity paper.
Thanks!
Hi @MUSHKAN_SUREKA,
If all of the formulas are correct and you still get that you need a lot of photons then maybe it means that you really need a lot of photons.
I think the latest paper that you mentioned was written by @Amanuel. @Amanuel, do you have any thoughts you could share with @MUSHKAN_SUREKA?
Thanks for tagging me @CatalinaAlbornoz, and thank you @MUSHKAN_SUREKA for the interest in our paper. Formula A7 is actually taken from a paper written by some Xanadu folks. It is Eq. 14. That paper along with this one were the most helpful when learning about the encoding of a graph into a GBS device. The latter I believe is where the detailed theory was first developed. This paper may also be of interest.
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Thank you Amanuel! Unfortunately, I never came across the detailed paper you mentioned. It seems like that direct formula is correct. @CatalinaAlbornoz yeah I guess practically, one does need a lot of photons for the actual embedding. I was just wondering if that is possible or not on hardware.
Thank you for your help and apologies for constantly pinging you haha.
Hi @Amanuel, thank you for your response!
@MUSHKAN_SUREKA, the paper on Gaussian Boson Sampling for perfect matchings of arbitrary graphs is the one where Amanuel mentions he found a lot of details. Given the large number of photons needed it would seem that it’s not possible to run this on current hardware. Thank you for your questions! There’s no need to apologize for them. I’m glad we can help.
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