A quantum-inspired algorithm for estimating the permanent of positive semidefinite matrices

TL;DR

This paper introduces a classical algorithm inspired by quantum optics to efficiently approximate the permanent of positive semidefinite matrices, leveraging a connection to boson sampling.

Contribution

It presents a novel quantum-inspired classical method for estimating matrix permanents, improving precision and polynomial runtime for specific matrix classes.

Findings

Algorithm achieves polynomial-time approximation for certain matrices.

Provides better precision than existing classical techniques.

Connects quantum optics concepts with classical algorithms.

Abstract

We construct a quantum-inspired classical algorithm for computing the permanent of Hermitian positive semidefinite matrices, by exploiting a connection between these mathematical structures and the boson sampling model. Specifically, the permanent of a Hermitian positive semidefinite matrix can be expressed in terms of the expected value of a random variable, which stands for a specific photon-counting probability when measuring a linear-optically evolved random multimode coherent state. Our algorithm then approximates the matrix permanent from the corresponding sample mean and is shown to run in polynomial time for various sets of Hermitian positive semidefinite matrices, achieving a precision that improves over known techniques. This work illustrates how quantum optics may benefit algorithms development.