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

TL;DR

This paper demonstrates that calculating certain matrix elements in bosonic and fermionic systems with finite-rank operators can be done with polynomial computational complexity, extending known results about permanents.

Contribution

It generalizes the polynomial complexity result from matrix permanents to expectation values in bosonic states, applicable to both bosonic and fermionic systems.

Findings

Polynomial-time computation of matrix elements in bosonic states.

Extension of polynomial complexity results to fermionic systems.

Applicability of Gaussian averaging technique to both bosonic and fermionic cases.

Abstract

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.