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

It is known that computing the permanent $\mathop{\rm Per}(1+A)$, where $A$ is a finite-rank matrix requires a number of operations polynomial in the matrix size. I generalize this result to the expectation values $\left\langle\Psi| P(1+A) |\Psi\right\rangle$, where $P()$ is the multiplicative extension of a single-particle operator and $\left|\Psi\right\rangle$ is a product of a large number of identical finite bosonic states (i.e. bosonic states with a bounded number of bosons). I also improve an earlier polynomial estimate for the fermionic version of the same problem.