  With the recent claim of a quantum advantage demonstration in photonics by
Zhong et al, the question of the computation of lower-order approximations of
boson sampling with arbitrary quantum states at arbitrary distinguishability
has 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 the
detection of some but not all photons) for arbitrary quantum states.
  2) Using the first result, how to generalize the sampling algorithm of
Clifford and Clifford to arbitrary photon distinguishabilities and arbitrary
input quantum states.
  3) How to incorporate truncations of the quantum interference into a sampling
algorithm.
  4) A remark considering maximum likelihood verification of the recent
photonic quantum advantage experiment.
