Classical simulation of boson sampling with sparse output

Boson sampling can simulate physical problems for which classical simulations are inefficient. However, not all problems simulated by boson sampling are classically intractable. We consider a situation in which it is known that the outcome from boson sampling is sparse. It can be determined from a few marginal distributions which are classically calculable. Still, recovering of the joint distribution can be of high complexity. We show classically efficient methods of the recovery assuming high sparsity of the joint distribution. Various extensions are discussed including a version involving quantum annealing.