Fidelity for bosonic states

Hi,

I am wondering if it’s possible to compute fidelity between two bosonic states (linear combinations of Gaussians). I know this can be done with Gaussian states, but not linear combinations of them.

I first thought about combining the vectors of means and covariance matrices from the Gaussians in the linear combination to get a single Gaussian, but now I am not sure I have enough data to do something like that.

Another method I’m thinking of is to compare each Gaussian of a set to its “corresponding” Gaussian of the other set. But here, I don’t know what the correct ordering is to carry on this comparison.

For some more context: I am comparing output GKP states from a CSUM circuit and its teleported version, both of which run using the bosonic backend.
Previously, I had been converting these output states into a Fock basis representation (using density matrix) but this process takes a while (and I want to compare this circuit for several choices of circuit parameters).
Now I am wondering if it’s possible to compare the output states using their linear combination of Gaussians expressions and not having to convert them to the Fock basis.

Thanks, and let me know if you need more information!

Hey @Francisco_J_Estrella :wave:! Thanks for the question :smile:! We’re looking into an answer for you. In the mean time, please share any relevant code to your question if you have it :computer:.

Hi Francisco. That’s a good question and I don’t have a ready answer fo you. I think your first attempt was on the right track though. Maybe the combinations have to be taken with care: I haven’t tried, but you may want to try computing the fidelity between all the pairs of Gaussians from each set and only then combine the results.
Let us know if it works out.

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Hi @Francisco_J_Estrella were you able to figure this out?