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We describe boson sampling of interacting atoms from the noncondensedfraction of Bose-Einstein-condensed (BEC) gas confined in a box trap as a newplatform for studying computational #P-hardness and quantum supremacy ofmany-body systems. We calculate the characteristic function and statistics ofatom numbers via newly found hafnian master theorem. Using Bloch-Messiahreduction, we find that interatomic interactions give rise to two equallyimportant entities - eigen-squeeze modes and eigen-energy quasiparticles -whose interplay with sampling atom states determines behavior of the BEC gas.We infer that two necessary ingredients of #P-hardness, squeezing andinterference, are self-generated in the gas and, contrary to Gaussian bosonsampling in linear interferometers, external sources of squeezed bosons are notrequired.

Towards the simplest model of quantum supremacy: Atomic boson sampling in a box trap