  We find a universal analytic formula for a characteristic function (Fourier
transform) of a joint probability distribution for the particle occupation
numbers in a BEC gas and the Hafnian Master Theorem generalizing the famous
Permanent Master Theorem of MacMahon. We suggest an appealing model, a
multi-qubit BEC trap formed by a set of qubit potential wells, and discuss
specifics of such an atomic boson-sampling system vs a photonic one. Finally,
the process of many-body fluctuations in a BEC trap is #P-hard for computing.
It could serve as a basis for demonstrating quantum supremacy of the many-body
interacting systems over classical simulators.
