Graph Partitioning using Quantum Annealing on the D-Wave System (1705.03082v1)

Abstract: In this work, we explore graph partitioning (GP) using quantum annealing on the D-Wave 2X machine. Motivated by a recently proposed graph-based electronic structure theory applied to quantum molecular dynamics (QMD) simulations, graph partitioning is used for reducing the calculation of the density matrix into smaller subsystems rendering the calculation more computationally efficient. Unconstrained graph partitioning as community clustering based on the modularity metric can be naturally mapped into the Hamiltonian of the quantum annealer. On the other hand, when constraints are imposed for partitioning into equal parts and minimizing the number of cut edges between parts, a quadratic unconstrained binary optimization (QUBO) reformulation is required. This reformulation may employ the graph complement to fit the problem in the Chimera graph of the quantum annealer. Partitioning into 2 parts, 2^N parts recursively, and k parts concurrently are demonstrated with benchmark graphs, random graphs, and small material system density matrix based graphs. Results for graph partitioning using quantum and hybrid classical-quantum approaches are shown to equal or out-perform current "state of the art" methods.

• The paper explores implementing graph partitioning using quantum annealing on the D-Wave system, focusing on its application to graphs from quantum molecular dynamics simulations.
• Numerical results show that quantum solutions often achieve comparable or superior partitioning quality, particularly regarding the number of cut edges, compared to classical tools like METIS on benchmark and scientific graphs.
• The research highlights the potential of quantum annealing for graph partitioning to improve efficiency, leveraging hybrid classical-quantum methods to address current hardware constraints for larger graph problems.

Graph Partitioning using Quantum Annealing on the D-Wave System: An Expert Overview

This paper presents a comprehensive paper on implementing graph partitioning (GP) using quantum annealing with the D-Wave 2X system. Aimed at enhancing computational efficiency for quantum molecular dynamics (QMD) simulations, the authors introduce graph-based electronic structure theory as a backbone for reducing the density matrix calculations into more manageable subsystems.

Quantum Annealing and its Application

Quantum annealing (QA) leverages quantum mechanical effects, such as tunneling and entanglement, to optimize combinatorial problems effectively. The D-Wave system executes these annealing processes by minimizing an Ising objective function. It employs qubits interconnected within a sparse graph, known as the Chimera graph, allowing programmable weights and coupling strengths. The advantage of QA over classical methods is most evident in addressing NP-hard problems across domains, including optimization and machine learning.

Graph Partitioning Methodologies

Graph partitioning is articulated through unconstrained and constrained approaches. Unconstrained GP emphasizes community clustering via a modularity metric, which can be mapped directly into the Hamiltonian of a quantum annealer. Conversely, constrained GP demands a reformulation into quadratic unconstrained binary optimization (QUBO) to partition graphs into equal parts, minimize cut edges, and successfully integrate into the Chimera graph architecture.

Numerical Results and Benchmarking

The paper demonstrates GP into two parts, recursively into $2^n$ parts, and concurrently into $k$ parts using quantum and hybrid classical-quantum methodologies. These approaches are applied on benchmark graphs, random graphs, and electronic structure graphs from QMD simulations. The D-Wave system's quantum solutions were compared against existing tools such as METIS and KaHIP and often showed comparable or superior partitioning quality, particularly when evaluating the number of cut edges as a metric.

Methodological Innovations

The paper introduces innovative techniques such as the graph complement method for dense graphs to optimize embedding on the D-Wave hardware. Thresholding the modularity matrix effectively reduces the coupler count, minimizing qubit requirements. Moreover, k-concurrent approaches enable partitioning without recursive steps, further demonstrating the flexibility of quantum methods over conventional recursive bisection.

Implications and Future Directions

The results indicate that quantum annealing for GP can enhance partitioning quality and efficiency, particularly for complex scientific graphs. The practical limitations posed by the physical constraints of the D-Wave system were mitigated using hybrid classical-quantum approaches like qbsolv, enabling larger graph processing. This research proposes further exploration of quantum approaches in new domains and extensions into clustering and community detection.

Conclusion

This paper establishes the efficacy of quantum annealing for graph partitioning, presenting substantial evidence for its application in reducing computational overhead in QMD simulations. While current systems are confined by hardware constraints, the hybrid methods offer plausible solutions to expansive and intricate graph problems. Future endeavours may focus on scaling k-concurrent approaches and exploring quantum solutions for other computational fields.