Gaussian Boson Sampling for perfect matchings of arbitrary graphs

TL;DR

This paper introduces a quantum computing approach using Gaussian Boson Sampling to efficiently estimate the number of perfect matchings in arbitrary graphs, leveraging the hafnian of the adjacency matrix.

Contribution

It establishes a novel connection between Gaussian Boson Sampling and graph theory, enabling faster and more energy-efficient estimation of perfect matchings.

Findings

The probability of measuring specific photon patterns relates to the hafnian of the adjacency matrix.

Encoding adjacency matrices into Gaussian states allows for efficient sampling strategies.

The method outperforms classical algorithms in speed and energy consumption.

Abstract

A famously hard graph problem with a broad range of applications is computing the number of perfect matchings, that is the number of unique and complete pairings of the vertices of a graph. We propose a method to estimate the number of perfect matchings of undirected graphs based on the relation between Gaussian Boson Sampling and graph theory. The probability of measuring zero or one photons in each output mode is directly related to the hafnian of the adjacency matrix, and thus to the number of perfect matchings of a graph. We present encodings of the adjacency matrix of a graph into a Gaussian state and show strategies to boost the sampling success probability. With our method, a Gaussian Boson Sampling device can be used to estimate the number of perfect matchings significantly faster and with lower energy consumption compared to a classical computer.