A Quantum Photonic Approach to Graph Coloring

Gaussian Boson Sampling (GBS) is a quantum computational model that leverages linear optics to solve sampling problems believed to be classically intractable. Recent experimental breakthroughs have demonstrated quantum advantage using GBS, motivating its application to real-world combinatorial optimization problems. In this work, we reformulate the graph coloring problem as an integer programming problem using the independent set formulation. This enables the use of GBS to identify cliques in the complement graph, which correspond to independent sets in the original graph. Our method is benchmarked against classical heuristics and exact algorithms on two sets of instances: Erdős-Rényi random graphs and graphs derived from a smart-charging use case. The results demonstrate that GBS can provide competitive solutions, highlighting its potential as a quantum-enhanced heuristic for graph-based optimization.

💡 Research Summary

The paper investigates the use of Gaussian Boson Sampling (GBS), a photonic quantum‑computing model, as a heuristic accelerator for the graph coloring problem, which is NP‑hard in general. The authors first reformulate graph coloring as an integer‑programming problem based on independent sets: a proper k‑coloring corresponds to a partition of the vertex set into k independent sets. By exploiting the well‑known complementarity between independent sets and cliques, the problem is transformed into finding large cliques in the complement graph (\bar G).

GBS can be programmed to encode the adjacency matrix of any undirected graph into a squeezed‑state interferometer. When the device is measured, each photon‑click pattern (s) corresponds to a subgraph (G