Quantum Algorithm for Maximum Biclique Problem

TL;DR

This paper introduces a quantum algorithm that significantly speeds up the maximum biclique problem, an NP-hard challenge, using quantum computing techniques and demonstrates its potential through IBM quantum simulations.

Contribution

It proposes the first quantum algorithm with quadratic speed-up for the maximum biclique problem and details encoding methods for bipartite graphs on quantum computers.

Findings

Quantum algorithm qMBS achieves O^*(2^(n/2)) complexity.

Proof-of-principle experiments validate the algorithm's practical potential.

Two variants for different biclique problems are developed.

Abstract

Identifying a biclique with the maximum number of edges bears considerable implications for numerous fields of application, such as detecting anomalies in E-commerce transactions, discerning protein-protein interactions in biology, and refining the efficacy of social network recommendation algorithms. However, the inherent NP-hardness of this problem significantly complicates the matter. The prohibitive time complexity of existing algorithms is the primary bottleneck constraining the application scenarios. Aiming to address this challenge, we present an unprecedented exploration of a quantum computing approach. Efficient quantum algorithms, as a crucial future direction for handling NP-hard problems, are presently under intensive investigation, of which the potential has already been proven in practical arenas such as cybersecurity. However, in the field of quantum algorithms for graph databases, little work has been done due to the challenges presented by the quantum representation of complex graph topologies. In this study, we delve into the intricacies of encoding a bipartite graph on a quantum computer. Given a bipartite graph with n vertices, we propose a ground-breaking algorithm qMBS with time complexity O^*(2^(n/2)), illustrating a quadratic speed-up in terms of complexity compared to the state-of-the-art. Furthermore, we detail two variants tailored for the maximum vertex biclique problem and the maximum balanced biclique problem. To corroborate the practical performance and efficacy of our proposed algorithms, we have conducted proof-of-principle experiments utilizing IBM quantum simulators, of which the results provide a substantial validation of our approach to the extent possible to date.