Simulability of partially distinguishable superposition and Gaussian boson sampling

We study the hardness of classically simulating boson sampling with superposition and Gaussian input states at nonzero photon indistinguishability. We find that, similar to regular boson sampling, distinguishability causes exponential attenuation of the many-photon interference terms in both these boson sampling variants. For superposition sampling, we find that it is not simulable with out method at zero indistinguishability, which is evidence for the computational hardness of this problem, and we find that it is simulable at any level of particle distinguishability, similar to regular boson sampling. If an efficient classical algorithm to approximate a given sum over permanents is found, this approach also leads to an efficient classical algorithm to simulate Gaussian boson sampling in the presence of distinguishability.