Explicit Quantum Search Algorithm for the Densest k-Subgraph Problem

TL;DR

This paper introduces two quantum algorithms for the NP-hard densest k-subgraph problem, demonstrating a quadratic speedup over classical brute-force methods through explicit quantum circuit designs.

Contribution

The paper presents novel quantum approaches, including a gate-based oracle circuit with Dicke states and Quantum Fourier Transform, for solving the densest k-subgraph problem.

Findings

Numerical simulations show quadratic speedup over classical brute-force search.

Proposed quantum algorithms are based on QUBO reduction and Grover's search.

Explicit quantum circuits utilize Dicke states and Quantum Fourier Transform.

Abstract

This paper addresses the problem of finding the densest $k$-vertex subgraph in an arbitrary graph. This problem is NP-hard and has important applications in social network analysis, fraud detection, recommendation systems, and bioinformatics. We propose two quantum approaches to solve this problem: reduction to Quadratic Unconstrained Binary Optimization (QUBO) and using Grover's quantum search algorithm. For the latter approach, we present an explicit gate-based oracle circuit utilizing Dicke states and Quantum Fourier Transform for edge counting. Numerical simulations demonstrate a quadratic speedup over classical Brute-force search.