Using Gaussian Boson Sampling to Find Dense Subgraphs

TL;DR

This paper demonstrates that Gaussian boson sampling can be effectively used to enhance algorithms for identifying dense subgraphs, leveraging quantum sampling to improve classical search methods.

Contribution

It introduces a novel application of GBS for solving the NP-hard densest k-subgraph problem, linking graph density to Hafnian-based sampling probabilities.

Findings

GBS-enhanced algorithms outperform classical methods in dense subgraph detection

Numerical simulations successfully identify densest subgraphs in 30-vertex graphs

A theoretical link between Hafnian enumeration and graph density is established.

Abstract

Boson sampling devices are a prime candidate for exhibiting quantum supremacy, yet their application for solving problems of practical interest is less well understood. Here we show that Gaussian boson sampling (GBS) can be used for dense subgraph identification. Focusing on the NP-hard densest k-subgraph problem, we find that stochastic algorithms are enhanced through GBS, which selects dense subgraphs with high probability. These findings rely on a link between graph density and the number of perfect matchings -- enumerated by the Hafnian -- which is the relevant quantity determining sampling probabilities in GBS. We test our findings by constructing GBS-enhanced versions of the random search and simulated annealing algorithms and apply them through numerical simulations of GBS to identify the densest subgraph of a 30 vertex graph.