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It is known that computing the permanent of the matrix $1+A$, where $A$ is afinite-rank matrix, requires a number of operations polynomial in the matrixsize. Motivated by the boson-sampling proposal of restricted quantumcomputation, I extend this result to a generalization of the matrix permanent:an expectation value in a product of a large number of identical bosonic stateswith a bounded number of bosons. This result complements earlier studies on thecomputational complexity in boson sampling and related setups. The proposedtechnique based on the Gaussian averaging is equally applicable to bosonic andfermionic systems. This also allows us to improve an earlier polynomialcomplexity estimate for the fermionic version of the same problem.

Polynomial computational complexity of matrix elements of finite-rank-generated single-particle operators in products of finite bosonic states