  Boson sampling, a computational problem conjectured to be hard to simulate on
a classical machine, is a promising candidate for an experimental demonstration
of quantum advantage using bosons. However, inevitable experimental noise and
imperfections, such as loss in the interferometer and random counts at the
detectors, could challenge the sampling task from entering the regime where
quantum advantage is achievable. In this work we introduce higher order
non-linearities as a mean to enhance the computational complexity of the
problem and the protocol's robustness against noise, i.e. increase the noise
threshold that allows to perform an efficient classical simulation of the
problem. Using a phase-space method based on the negativity volume of the
relevant quasi-probability distributions, we establish a necessary
non-classicality condition that any experimental proof of quantum advantage
must satisfy. Our results indicate that the addition of single-mode Kerr
non-linearity at the input state preparation level, while retaining a
linear-optical evolution, makes the Boson sampling protocol more robust against
noise and consequently relaxes the constraints on noise parameters required to
show quantum advantage.
