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We study the computational complexity of quantum-mechanical expectationvalues of single-particle operators in bosonic and fermionic multi-particleproduct states. Such expectation values appear, in particular, infull-counting-statistics problems. Depending on the initial multi-particleproduct state, the expectation values may be either easy to compute (therequired number of operations scales polynomially with the particle number) orhard to compute (at least as hard as a permanent of a matrix). However, if weonly consider full counting statistics in a finite number of finalsingle-particle states, then the full-counting-statistics generating functionbecomes easy to compute in all the analyzed cases. We prove the latterstatement for the general case of the fermionic product state and for thesingle-boson product state (the same as used in the boson-sampling proposal).This result may be relevant for using multi-particle product states as aresource for quantum computing.

Complexity of full counting statistics of free quantum particles in product states