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

翻译：已知计算矩阵 $1+A$ 的积和式（其中 $A$ 为有限秩矩阵）所需的运算次数随矩阵规模呈多项式增长。受限制量子计算中玻色子采样方案的启发，本文将这一结果推广至广义矩阵积和式：即具有有界玻色子数的大量全同玻色子态乘积中的期望值。该结果补充了早期关于玻色子采样及相关场景中计算复杂度的研究。所提出的基于高斯平均的技术同样适用于玻色子和费米子系统。这一方法还使我们能够改进同一问题的费米子版本中先前多项式复杂度估计的结果。