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I read about the photonic QC Jiŭzhāng that showed quantum advantage by Gaussian boson sampling. I read that boson sampling itself is a sub-universal technique of QC (where they use single-photon states as input states). In the paper, the scientists describe how they use squeezed states for their computation, which can be produced deterministically (making it better realisable then producing single-photon states).

I know the term "squeezed states" from continuous variables approaches (which e.g. Xanadu uses), where a squeezed state is an ellipse in phase space. So I am wondering, whether boson sampling is a special algorithm implemented within the continuous variable approach in QC? Or is it really a totally independent approach of QC?

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    $\begingroup$ Related, though not entirely the same: mattermodeling.stackexchange.com/q/3919/5. Also +1 and welcome to the community! $\endgroup$ Dec 14 '20 at 17:02
  • $\begingroup$ its a big deal yet scratching my head on all this, dont see indication that boson sampling is trying to do "computation". what exactly does "sub universal" mean? not (known to be?) capable of universal computation? a big rationale of Aaronson paper seems to be, look at a physics operation that is hard for a classical system to calculate & then try to prove it. for QC there is decades of research showing the operations (entanglement etc) can be harnessed for logic gates etc, but there seems to be no such indication/ analysis ("yet?") for boson sampling...? $\endgroup$
    – vzn
    Dec 14 '20 at 19:04
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    $\begingroup$ squeezed states arise in what is usually dubbed a "continuous variable formalism", yes. This is just one way to deal with/describe coherences between different Fock states (which you can describe in specific ways in phase-space etc). Gaussian boson sampling is described with this formalism, so in this sense, I guess the answer to the question is yes. But I'm not sure what you mean with it being "a totally independent approach of QC". $\endgroup$
    – glS
    Dec 15 '20 at 9:41
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@gIS's comment effectively answers the question, but to provide a bit more detail Aaronson at shtetl-optimized has a nice blog post on the Gaussian Boson Sampling approach of USTC, contrasting it with Fock state Boson Sampling, wherein the presence/absence of photons correspond to something closer to a conventional digital ($\vert 0\rangle$/$\vert 1\rangle$) perspective of quantum computation.

Section III of Aaronson's blog post summarizes these differences nicely, although I'll admit I don't understand a good chunk of it. I believe conventional single-photon generation has been challenging; spontaneous parametric downconversion (SPDC) requires a lot of post-selection; and in Gaussian Boson Sampling the key point may be something like a Gaussian state is "a state that’s exponential in some quadratic polynomial in the creation operators", which is (apparently) significantly easier to tune.

It's not clear to me yet whether in the end this corresponds to a "continuous variable approach" mentioned in the question. Nonetheless, as the OP suggests it is very likely that neither (Fock state) Boson Sampling nor Gaussian Boson Sampling are universal in the sense of being able to perform arbitrary functions. Indeed, quantum machines that perform Boson Sampling may not even be classically universal, much less capable of performing arbitrary quantum computations.

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