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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.