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With the recent claim of a quantum advantage demonstration in photonics byZhong et al, the question of the computation of lower-order approximations ofboson sampling with arbitrary quantum states at arbitrary distinguishabilityhas come to the fore. In this work, we present results in this direction,building on the results of Clifford and Clifford. In particular, we show:  1) How to compute marginal detection probabilities (i.e. probabilities of thedetection of some but not all photons) for arbitrary quantum states.  2) Using the first result, how to generalize the sampling algorithm ofClifford and Clifford to arbitrary photon distinguishabilities and arbitraryinput quantum states.  3) How to incorporate truncations of the quantum interference into a samplingalgorithm.  4) A remark considering maximum likelihood verification of the recentphotonic quantum advantage experiment.

Marginal probabilities in boson samplers with arbitrary input states