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arXiv:2609.30991

Solving Graph Coloring Problems Using Feedback-Based Algorithm for Quantum Optimization

Yuika Kawatomi, Yukina Tatsuta, Kazue Kudo

quant-phSubmitted Sep 25, 2026• Announced Sep 28, 2026• 0 citations
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Abstract

The feedback-based algorithm for quantum optimization (FALQON), which is an extension of the quantum approximate optimization algorithm (QAOA), is a promising approach for finding approximate solutions to combinatorial optimization problems. The FALQON avoids the optimization of variational parameters by using information derived from iterative measurements. To solve a constrained combinatorial optimization problem, it is sometimes more efficient to use an XY mixer rather than the conventional X mixer. We applied the FALQON with the XY mixer to graph coloring and investigated the expectation value of the problem Hamiltonian, success probability of graph coloring, and fair sampling for several small-scale problems. The FALQON with the XY mixer exhibited better performance than that with the X mixer.

Comments
13 pages, 5 figures

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The feedback-based algorithm for quantum optimization (FALQON), which is an extension of the quantum approximate optimization algorithm (QAOA), is a promising approach for finding approximate solutions to combinatorial optimization problems. The FALQON avoids the optimization of variational parameters by using information derived from iterative measurements. To solve a constrained combinatorial optimization problem, it is sometimes more efficient to use an XY mixer rather than the conventional X mixer. We applied the FALQON with the XY mixer to graph coloring and investigated the expectation value of the problem Hamiltonian, success probability of graph coloring, and fair sampling for several small-scale problems. The FALQON with the XY mixer exhibited better performance than that with the X mixer.
In the era of Noisy Intermediate Scale Quantum (NISQ) computing, available quantum resources are limited. Many NP-hard problems can be efficiently addressed using hybrid classical and quantum computational methods. This paper proposes a hybrid variational quantum algorithm designed to solve the kkk-coloring problem of graph vertices. The hybrid classical and quantum algorithms primarily partition the graph into multiple subgraphs through hierarchical techniques. The Quantum Approximate Optimization Algorithm (QAOA) is employed to determine the coloring within the subgraphs, while a classical greedy algorithm is utilized to find the coloring of the interaction graph. Fixed coloring is applied to the interaction graph, and feedback is provided to correct any conflicting colorings within the subgraphs. The merging process into the original graph is iteratively optimized to resolve any arising conflicts. We employ a hierarchical framework that integrates feedback correction and conflict resolution to achieve kkk-coloring of arbitrary graph vertices. Through experimental analysis, we demonstrate the effectiveness of the algorithm, highlighting the rapid convergence of conflict evolution and the fact that iterative optimization allows the classical algorithm to approximate the number of colorings. Finally, we apply the proposed algorithm to optimize the scheduling of a subway transportation network, demonstrating a high degree of fairness.
Feedback-based adaptive quantum optimization (FALQON) is a promising approach for solving combinatorial problems on noisy intermediate-scale quantum (NISQ) devices, requiring only single circuit evaluations per layer. However, standard FALQON relies on fixed hyperparameters that severely limit convergence speed, requiring hundreds to thousands of layers for acceptable solutions. This paper proposes Optimal FALQON, an optimization-based formulation that treats the per-layer time step (δkδ_kδk​) and scaling factor (MkM_kMk​) as decision variables optimized via classical methods. We present a comprehensive empirical study on all 94 non-isomorphic 3-regular graphs with 12 vertices, comparing Optimal FALQON with standard FALQON and multiple QAOA variants. Results demonstrate statistically significant improvements in success probability, evaluation efficiency, and depth-normalized cost across the evaluated benchmarks. Furthermore, initializing QAOA with parameters from Optimal FALQON yields superior warm-start performance compared to fixed initialization.
The design of a good algorithm to solve NP-hard combinatorial approximation problems requires specific domain knowledge about the problems and often needs a trial-and-error problem solving approach. Graph coloring is one of the essential fields to provide an efficient solution for combinatorial applications such as flight scheduling, frequency allocation in networking, and register allocation. In particular, some optimization algorithms have been proposed to solve the multi-coloring graph problems but most of the cases a simple searching method would be the best approach to find an optimal solution for graph coloring problems. However, this naive approach can increase the computation cost exponentially as the graph size and the number of colors increase. To mitigate such intolerable overhead, we investigate the methods to take the advantages of quantum computing properties to find a solution for multi-coloring graph problems in polynomial time. We utilize the variational quantum eigensolver (VQE) technique and quantum approximate optimization algorithm (QAOA) to find solutions for three combinatorial applications by both transferring each problem model to the corresponding Ising model and by using the calculated Hamiltonian matrices. Our results demonstrate that VQE and QAOA algorithms can find one of the best solutions for each application. Therefore, our modeling approach with hybrid quantum algorithms can be applicable for combinatorial problems in various fields to find an optimal solution in polynomial time.
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Graph coloring is a well-known combinatorial optimization problem with broad applications in network security, scheduling, and register allocation. Being an NP-hard problem, exact solutions for large instances are computationally infeasible, calling for other solution strategies. In this study, parameterized and approximation methods to tackle the graph coloring problem are investigated. While approximation algorithms yield near-optimal solutions with theoretical guarantees of performance that are well-suited for large-scale applications, parameterized algorithms utilize graph-specific structural properties to manage constrained instances efficiently. We examine the trade-offs in these approaches, highlighting their strengths and weaknesses. In addition, we introduce a hybrid approach that combines both parameterized and approximation methods to best optimize computational resources and balance accuracy versus feasibility. Furthermore, we discuss the newly realized potential of quantum computing—specifically quantum annealing and the Quantum Approximate Optimization Algorithm (QAOA)—to speed up graph coloring by leveraging quantum parallelism and state superposition. Such quantum-based techniques represent a promising line of attack for breaking up large and hard instances that overwhelm classical approaches. Future research will aim to tighten approximation guarantees, create quantum-augmented hybrid models, and test theoretical results against real-world datasets.