Scalable Community Detection Using Quantum Hamiltonian Descent and QUBO Formulation

TL;DR

This paper tackles scalable community detection on large graphs by reframing the problem as a QUBO and solving it with Quantum Hamiltonian Descent (QHD), augmented by a multilevel refinement strategy and GPU acceleration. The approach combines modularity-based objectives with penalty terms to enforce valid and balanced partitions, enabling efficient hardware-accelerated optimization. Empirical results show that QHD can outperform traditional solvers on larger instances (up to 5.49% higher modularity in some networks) and match optimal solutions on many smaller cases, highlighting the practical potential of quantum-inspired methods for network analysis. Overall, the work underscores the viability of hybrid quantum-inspired optimization for real-world, large-scale graph problems and points to further gains from advanced sparsity techniques and broader hardware support.

Abstract

We present a quantum-inspired algorithm that utilizes Quantum Hamiltonian Descent (QHD) for efficient community detection. Our approach reformulates the community detection task as a Quadratic Unconstrained Binary Optimization (QUBO) problem, and QHD is deployed to identify optimal community structures. We implement a multi-level algorithm that iteratively refines community assignments by alternating between QUBO problem setup and QHD-based optimization. Benchmarking shows our method achieves up to 5.49\% better modularity scores while requiring less computational time compared to classical optimization approaches. This work demonstrates the potential of hybrid quantum-inspired solutions for advancing community detection in large-scale graph data.

Figures (6)

• Figure 1: Visualization of community structure in a complex network. The network consists of nodes grouped into communities, each represented by a different color.
• Figure 2: Overview of our quantum-inspired community detection approach. The input network is reformulated as a QUBO optimization problem, which is then solved using QHD accelerated by parallel GPU computation. The QHD optimizer uses quantum-inspired dynamics to efficiently identify lowest cost function values in the complex landscape, and delivers superior scalability for instances with thousands of nodes.
• Figure 3: Solution Quality Comparison When GUROBI Hits Time Limit: For 739 instances where GUROBI exceeded time limits (predominantly in larger problems), QHD found better solutions in 71.4% of cases. For larger-scale problems, QHD demonstrated superior performance by finding better solutions than GUROBI within the same time constraints. This advantage stems from QHD's inherent parallel processing capabilities, which can be further enhanced through additional GPU resources and optimized sparse matrix operations, suggesting even greater potential for scaling to larger problem instances.
• Figure 4: Solution Quality Comparison When GUROBI Reaches Optimality: Among 199 instances where GUROBI found optimal solutions (predominantly in smaller problems), QHD achieved identical solutions in 75.4% of cases. In cases where QHD did not match GUROBI's optimal solutions (24.6% of optimally solved instances), the objective value differences remained within a 1.6% relative gap, primarily occurring in smaller instances where the absolute differences in objective values were minimal.
• Figure 5: Performance comparison between QHD and GUROBI on network instances ranging from 52 to 1,034 nodes. QHD achieves superior modularity scores in 80% of test cases with an average improvement of 0.0029, while requiring only 20% of GUROBI's computational time using four GPUs. Results demonstrate QHD's effectiveness across different network densities (3.4%-15.2%).