Boson Sampling represents a promising approach to obtain an evidence
of the supremacy of quantum systems as a resource for the solution of computational
problems. The classical hardness of Boson Sampling has been related to the so called
Permanent-of-Gaussians Conjecture and has been extended to some generalizations
such as Scattershot Boson Sampling, approximate and lossy sampling under some
reasonable constraints. However, it is still unclear how demanding these techniques
are for a quantum experimental sampler. Starting from a state of the art analysis
and taking account of the foreseeable practical limitations, we evaluate and discuss the
bound for quantum supremacy for different recently proposed approaches, accordingly
to today’s best known classical simulators.