Quantum advantage and #P-hard complexity of many-body systems explained via eigen-squeeze modes as opposed to quasiparticles

We show that the eigen-squeeze modes, not quasiparticles, are responsible for the computational #P-hardness of quantum many-body statistics in interacting systems [V.V. Kocharovsky et al., PRA 106, 063312 (2022), Entropy 25, 1584 (2023)]. We consider a generic example of the Bose-Einstein-condensed (BEC) gas. We suggest boson sampling of atom numbers in the noncondensed fraction of BEC gas as a new experimental platform for studying quantum advantage of many-body interacting systems over classical computers. In contrast to Gaussian boson sampling in linear interferometers, it does not require inputting squeezed bosons by sophisticated external sources. Besides, it overcomes the major limitation of optical boson sampling - an exponential growth of losses with an increasing number of modes. We present a new theoretical technique for analysis of quantum statistical properties which are #P-hard for computing. It is based on the newly found hafnian master theorem. We calculate atom sampling statistics and show its #P-hardness for computing. We explain the atom boson sampler as a quantum generator of random strings of excited-atom numbers based on a natural process of persistent equilibrium fluctuations as opposed to a quantum simulator of some input signal or controlled process.