In the era of noisy intermediate-scale quantum computing, this paper proposes a hierarchical adaptive quantum approximate optimization algorithm (QAOA) to address quantum resource constraints in graph-coloring problems (GCPs). The algorithm employs a divide-and-conquer strategy by partitioning a large graph into multiple subgraphs, solving subgraph coloring using various QAOA variants, and incorporating a greedy search as a fallback mechanism. During the merging stage, conflicts are resolved through iterative optimization. Experimental results demonstrate that the conflict rate rapidly converges to zero and iterative optimization reduces the number of colors required. Tests on the COLOR benchmark dataset and 50-node $d$-regular graphs show that the adaptive QAOA outperforms the standard QAOA and adaptive noise QAOA in efficiency. For some complex graphs, this algorithm surpasses classical algorithms in terms of the number of colors used. The GCP solution can be applied to subway transportation network instances, achieving effective optimization of scheduling operations. This study presents a hierarchical and precise quantum-classical hybrid computing scheme for NP-hard problems, offering significant value for graph coloring and related practical applications.

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• Quantum algorithms & computation
• Quantum circuits
• Quantum communication, protocols & technology
• Quantum computation