Quantum optimization algorithms, particularly the Quantum Approximate Optimization Algorithm (QAOA), have significantly addressed combinatorial optimization problems. While QAOA has been applied to various NP-hard problems, such as Max-Cut and the Traveling Salesman Problem (TSP), its application to the Degree-Constrained Minimum Spanning Tree (DCMST) problem remains unexplored. Inspired by Fowler’s formulation, this work presents the first implementation of the DCMST Hamiltonian within QAOA and insights from its benefits to graph-based machine learning algorithms. We investigated two approaches: (i) the standard QAOA ansatz with an X mixer and (ii) a warm-started QAOA utilizing classical preprocessing to enhance convergence. We used these strategies to provide numerical results for instances with 3 and 4 nodes, employing the COBYLA optimizer and metaheuristic optimization techniques. Our findings serve as a proof of concept, demonstrating the feasibility of applying QAOA to this problem; however, the number of qubits scales as O(N2) with the number of nodes, limiting scalability. Current research focuses on developing more efficient implementations that encode the same number of binary variables using fewer qubits. This study contributes to the expanding field of quantum optimization, highlighting the potential of hybrid quantum-classical algorithms, especially with resource-efficient mixers and warm-starting techniques, to solve complex combinatorial problems and advance the development of scalable quantum algorithms.
Quantum optimization algorithms, particularly the Quantum Approximate Optimization Algorithm (QAOA), have significantly addressed combinatorial optimization problems. While QAOA has been applied to various NP-hard problems, such as Max-Cut and the Traveling Salesman Problem (TSP), its application to the Degree-Constrained Minimum Spanning Tree (DCMST) problem remains unexplored. Inspired by Fowler’s formulation, this work presents the first implementation of the DCMST Hamiltonian within QAOA and insights from its benefits to graph-based machine learning algorithms. We investigated two approaches: (i) the standard QAOA ansatz with an X mixer and (ii) a warm-started QAOA utilizing classical preprocessing to enhance convergence. We used these strategies to provide numerical results for instances with 3 and 4 nodes, employing the COBYLA optimizer and metaheuristic optimization techniques. Our findings serve as a proof of concept, demonstrating the feasibility of applying QAOA to this problem; however, the number of qubits scales as O(N2) with the number of nodes, limiting scalability. Current research focuses on developing more efficient implementations that encode the same number of binary variables using fewer qubits. This study contributes to the expanding field of quantum optimization, highlighting the potential of hybrid quantum-classical algorithms, especially with resource-efficient mixers and warm-starting techniques, to solve complex combinatorial problems and advance the development of scalable quantum algorithms.
In recent years, quantum computing has emerged as a transformative force in the field of combinatorial optimization, offering novel approaches to tackling complex problems that have long challenged classical computational methods. Among these, the Quantum Approximate Optimization Algorithm (QAOA) stands out for its potential to efficiently solve the Max-Cut problem, a quintessential example of combinatorial optimization. However, practical application faces challenges due to current limitations on quantum computational resource. Our work optimizes QAOA initialization, using Graph Neural Networks (GNN) as a warm-start technique. This sacrifices affordable computational resource on classical computer to reduce quantum computational resource overhead, enhancing QAOA's effectiveness. Experiments with various GNN architectures demonstrate the adaptability and stability of our framework, highlighting the synergy between quantum algorithms and machine learning. Our findings show GNN's potential in improving QAOA performance, opening new avenues for hybrid quantum-classical approaches in quantum computing and contributing to practical applications.
In recent years, quantum computing has emerged as a transformative force in the field of combinatorial optimization, offering novel approaches to tackling complex problems that have long challenged classical computational methods. Among these, the Quantum Approximate Optimization Algorithm (QAOA) stands out for its potential to efficiently solve the Max-Cut problem, a quintessential example of combinatorial optimization. However, practical application faces challenges due to current limitations on quantum computational resource. Our work optimizes QAOA initialization, using Graph Neural Networks (GNN) as a warm-start technique. This sacrifices affordable computational resource on classical computer to reduce quantum computational resource overhead, enhancing QAOA's effectiveness. Experiments with various GNN architectures demonstrate the adaptability and stability of our framework, highlighting the synergy between quantum algorithms and machine learning. Our findings show GNN's potential in improving QAOA performance, opening new avenues for hybrid quantum-classical approaches in quantum computing and contributing to practical applications.
The Traveling Salesman Problem (TSP) is a fundamental challenge in combinatorial optimization, widely applied in logistics and transportation. As the size of TSP instances grows, traditional algorithms often struggle to produce high-quality solutions within reasonable timeframes. This study investigates the potential of the Quantum Approximate Optimization Algorithm (QAOA), a hybrid quantum-classical method, to solve TSP under realistic constraints. We adopt a QUBO-based formulation of TSP that integrates real-world logistical constraints reflecting operational conditions, such as vehicle capacity, road accessibility, and time windows, while ensuring compatibility with the limitations of current quantum hardware. Our experiments are conducted in a simulated environment using high-performance computing (HPC) resources to assess QAOA's performance across different problem sizes and quantum circuit depths. In order to improve scalability, we propose clustering QAOA (Cl-QAOA), a hybrid approach combining classical machine learning with QAOA. This method decomposes large TSP instances into smaller sub-problems, making quantum optimization feasible even on devices with a limited number of qubits. The results offer a comprehensive evaluation of QAOA's strengths and limitations in solving constrained TSP scenarios. This study advances quantum optimization and lays groundwork for future large-scale applications.
Quantum optimization algorithms, such as the Quantum Approximate Optimization Algorithm, are emerging as promising heuristics for solving complex combinatorial problems. To improve performance, several extensions to the standard QAOA framework have been proposed in recent years. Two notable directions include: warm-starting techniques, which incorporate classical approximate solutions to guide the quantum evolution, and custom mixer Hamiltonians, such as XY mixers, which constrain the search to feasible subspaces aligned with the structure of the problem. In this work, we propose an approach that integrates these two strategies: a warm-start initialization with an XY mixer ansatz, enabling constraint-preserving quantum evolution biased toward high-quality classical solutions. The method begins by reformulating the combinatorial problem as a MaxCut instance, solved approximately using the Goemans-Williamson algorithm. The resulting binary solution is relaxed and used to construct a biased superposition over valid one-hot quantum states, maintaining compatibility with the XY mixer's constraints. We evaluate the approach on 5-city instances of the Traveling Salesperson Problem, a canonical optimization problem frequently encountered as a subroutine in real-world Vehicle Routing Problems. Our method is benchmarked against both the standard XY-mixer QAOA and a warm-start-only variant based on MaxCut relaxation. Results show that the proposed combination consistently outperforms both baselines in terms of the percentage and rank of optimal solutions, demonstrating the effectiveness of combining structured initializations with constraint-aware quantum evolution for optimization problems.
This paper is a comparison between classical and quantum inspired methods for solving the max-flow problem, which is commonly a foundational challenge in optimization and graph theory. This study will investigate the feasibility and performance of applying the Quantum Approximate Optimization Algorithm (QAOA) and other hybrid quantum-classical approaches to address the max-flow problem. This paper starts with comparing quantum approaches with well-established classical methods such as the Ford-Fulkerson and Edmonds-Karp.The main aim of this work is to asses how close quantum-inspired formulations can get to classical approaches in terms of their flow value as well as their runtime. Implementations are developed using the Qiskit framework on simulated quantum backends and compared against classical baselines across synthetic graph instances of varying sizes and densities. Experimentally, it can be seen that quantum approaches implemented using the QAOA technique managed to attain flow solutions comparable to classical ones, demonstrating similar accuracy, and in certain very dense graph configurations, competitive runtime performance due to parallelization in state-vector computing.These findings suggest that although current quantum-inspired approaches may not yet surpass classical algorithms in efficiency, but they can replicate their outcomes with reasonable fidelity. This provides an important validation of their applicability to combinatorial optimization. This study also lays groundwork for future research to explore hardware-level implementations, scalability enhancements, and also hybrid integration strategies for quantum optimization in network flow problems.
The traveling salesman problem (TSP) is one of the most often-used NP-hard problems in computer science to study the effectiveness of computing models and hardware platforms. In this regard, it is also heavily used as a vehicle to study the feasibility of the quantum computing paradigm for this class of problems. In this paper, we tackle the TSP using the quantum approximate optimization algorithm (QAOA) approach by formulating it as an optimization problem. By adopting an improved qubit encoding strategy and a layer-wise learning optimization protocol, we present numerical results obtained from the gate-based digital quantum simulator, specifically targeting TSP instances with 3, 4, and 5 cities. We focus on the evaluations of three distinctive QAOA mixer designs, considering their performances in terms of numerical accuracy and optimization cost. Notably, we find that a well-balanced QAOA mixer design exhibits more promising potential for gate-based simulators and realistic quantum devices in the long run, an observation further supported by our noise model simulations. Furthermore, we investigate the sensitivity of the simulations to the TSP graph. Overall, our simulation results show that the digital quantum simulation of problem-inspired ansatz is a successful candidate for finding optimal TSP solutions.