Constrained-optimization Approach Delivers Superior Classical Performance for Graph Partitioning via Quantum-ready Method

Abstract

Graph partitioning is one of an important set of well-known compute-intense (NP-hard) graph problems that devolve to discrete constrained optimization. We sampled solutions to the problem via two different quantum-ready methods to understand the strengths and weaknesses of each method. First we formulated and sampled the problem as a quadratic unconstrained binary optimization (QUBO) problem, via the best known QUBO formulation, using a best-in-class QUBO sampler running purely classically. Second, we formulated the problem at a higher level, as a set of constraints and an objective function, and sampled it with a recently developed constrained-optimization sampler (which generates QUBOs and also samples them classically). We find that both approaches often deliver better partitions than the purpose-built classical graph partitioners. Further, we find that the constrained-optimization approach is often able to deliver better partitions in less time than the bespoke-QUBO approach, without specific knowledge of the graph-partitioning problem. Stepping back from graph partitioning itself, one key question is whether bespoke algorithms for high-value problems or general tools for classes of problems are more likely to deliver the power of QCs to a broad market of real-world users. We believe this early evidence, though anecdotal, supports the proposition that general tools may contribute significant benefit to a range of problems, expanding the impact of QCs on industry and society. The fact that this benefit is independent of the low-level sampler employed, whether classical software or one of a variety of QC architectures, reinforces the need for further work on high-level optimization. The commercial availability in the cloud of such software today, delivering superior classical performance for some problems, enables quantum-forward organizations to migrate to quantum-ready methods now, accelerating progress toward quantum advantage and ensuring sampler software evolves to address the problems such organizations value.