Boson sampling can simulate physical problems for which classical simulations are inefficient. However, not all problems simulated by boson sampling are classically intractable. We show explicit classical methods of finding boson sampling distributions when they are known to be highly sparse. In the methods, we first determine a few distributions from restricted number of detectors and then recover the full one using compressive sensing techniques. In general, the latter step could be of high complexity. However, we show that this problem can be reduced to solving an Ising model which under certain conditions can be done in polynomial time. Various extensions are discussed including a version involving quantum annealing. Hence, our results impact the understanding of the class of classically calculable problems. We indicate that boson samplers may be advantageous in dealing with problems which are not highly sparse. Finally, we suggest a hybrid method for problems of intermediate sparsity.
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