Quantum supremacy of the many-body fluctuations in the occupations of the excited particle states in a Bose-Einstein-condensed gas

TL;DR

This paper introduces a universal formula for particle occupation fluctuations in a Bose-Einstein condensate, proposing a model that could demonstrate quantum supremacy in many-body quantum systems.

Contribution

It presents a new analytic formula for occupation number distributions and a model for atomic boson-sampling systems, extending the Hafnian Master Theorem.

Findings

Derived a universal characteristic function for BEC occupation numbers

Proposed a multi-qubit BEC trap model for boson-sampling

Showed that many-body fluctuation calculations are #P-hard

Abstract

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.