Reinforcement Learning–based Quantum Approximate Optimization Algorithm (QAOA-RL) is used to solve the Vehicle Routing Problem (VRP) by transforming it into a Quadratic Unconstrained Binary Optimization (QUBO) form. The RL agent learns to optimize QAOA parameters for better route selection. Results show that QAOA-RL achieves a lower total cost compared to exact ILP and heuristic Greedy Search methods. Although the training time is higher, the solution quality improves significantly for larger problem sizes. This study highlights QAOA-RL as a scalable and promising hybrid quantum–classical approach for real-world logistics optimization.
Reinforcement Learning–based Quantum Approximate Optimization Algorithm (QAOA-RL) is used to solve the Vehicle Routing Problem (VRP) by transforming it into a Quadratic Unconstrained Binary Optimization (QUBO) form. The RL agent learns to optimize QAOA parameters for better route selection. Results show that QAOA-RL achieves a lower total cost compared to exact ILP and heuristic Greedy Search methods. Although the training time is higher, the solution quality improves significantly for larger problem sizes. This study highlights QAOA-RL as a scalable and promising hybrid quantum–classical approach for real-world logistics optimization.
Vehicle Routing Problem (VRP) is one of the most complex NP-hard combinatorial optimization problem in transportation and logistics that requires a dynamic solution approach. In this paper we present a new hybrid approach that combines the Quantum Approximate Optimization Algorithm (QAOA) into the QRL policy network, instead of the usual variational layers, QAOA mixing and cost Hamiltonian layers. This enhancement enables the agent to exploit problem specific particular quantum correlations when learning policies, and so richer exploration of the routing solution space. The QAOA-augmented QRL framework shows quicker convergence in training and can tackle larger VRP instances that are beyond the reach of Grover's Adaptive Search (GAS) and Quantum Reinforcement Learning (QRL) approaches. Experiments on standard VRP instances demonstrate better solutions, fewer episodes to converge and good memory usage on near term quantum hardware simulators. These findings demonstrate QAOA- integrated QRL as a viable approach to scalable, high quality quantum-assisted combinatorial optimization.
Introduction Finding the most efficient routes for a fleet of delivery vehicles remains a critical and costly challenge for nearly any logistics business. This core idea is known as the Vehicle Routing Problem (VRP). The primary goal is to minimize operational costs, such as the total travel distance or the number of trucks deployed, while ensuring all the customers are served. Because VRP is NP-hard, finding a perfect solution is often impossible, forcing industries to rely on classical approximate algorithms. However, quantum variational algorithms offer a new paradigm by iteratively optimizing parameters to navigate the solution space, presenting a new and promising avenue for finding better and faster solutions. Methods In this paper, we introduce Symmetric Arc Reduction in Quantum Approximate Optimization Algorithm (SAR-QAOA), which systematically eliminates redundant symmetric routing variables in the binary integer programming formulation prior to the Quadratic Unconstrained Binary Optimization (QUBO) formulation. The proposed SAR-QAOA encodes edge-based VRP while reducing the qubit requirement from n ( n −1) to n ( n −1)/2, thereby improving the resource efficiency of the QUBO formulation by prioritizing regions of the solution space that are more likely to contain high-quality solutions; SAR enhances both search accuracy and computational efficiency. Results Experimental results on noiseless quantum simulators validate the feasibility of the proposed formulation and its integration with the QAOA framework, providing a promising foundation for future investigations on larger routing problems and evolving quantum hardware.
We investigate the potential of the Quantum Approximate Optimization Algorithm (QAOA) for reducing energy consumption in route planning, a key challenge in logistics due to the NP-hard nature of the Traveling Salesman and Vehicle Routing Problems. By encoding route optimization as a Quadratic Unconstrained Binary Optimization (QUBO) problem and implementing QAOA circuits at depth p = 3-5 alongside classical baselines of Simulated Annealing (SA) and Genetic Algorithms (GA), we perform systematic benchmarks on Euclidean graphs of sizes N = 5, 10, and 20. Our results demonstrate that QAOA attains higher solution quality with approximation ratios of 0.953 (N = 5), 0.921 (N = 10), and 0.903 (N = 20), outperforming SA and GA by 2.7-4.4%. Wall-clock runtimes for QAOA are 2-3x faster than SA across all tested sizes, and energy consumption measurements reveal a three-order-of-magnitude reduction, remaining in the picojoule range versus nanojoules for classical methods. Translating these gains to real-world logistics suggests an 8.2% improvement in routing efficiency could save approximately 2.62 EJ of fuel annually in the U.S., avoiding nearly 1.94 x 10^8 tonnes of CO2 emissions. These findings highlight QAOA's promise as a fast, energy-efficient optimizer for sustainable logistics applications and underscore its potential role in next-generation fleet-management systems.
The vehicle routing problem (VRP) is a core NP‐hard combinatorial optimization problem in logistics and supply chain management. Quantum computing, particularly the Quantum Approximate Optimization Algorithm (QAOA), is being explored as a promising heuristic tool for tackling such problems. Therefore, this study employs the QAOA approach to solve the airport cargo VRP. Subsequently, through a concrete airport cargo case study, the paper analyzes the process of solving single‐vehicle TSP and multivehicle VRP problems using QAOA. It compares the performance of different classical optimizers and discusses the impact of parameter settings (e.g., penalty factors, QAOA depth, and number of iterations) on the solution quality and feasibility, especially comparing the convergence speed of dynamic and fixed penalty coefficients. The paper also describes the design principles of QAOA quantum circuits for such VRP problems. Furthermore, the study presents a preliminary benchmark on a real quantum computer, demonstrating that its computation time for the problem’s QUBO matrix (14 ms) is approximately comparable to that of classical algorithms. Finally, it summarizes the current scalability challenges and limitations faced by QAOA in VRP applications concerning qubit count, circuit depth, and classical optimizer performance, and offers an outlook on future development directions. The research indicates that although QAOA demonstrates solution capabilities on specific small‐scale VRP instances, the complexity of parameter tuning and sensitivity to the spatial distribution of aircraft stands are critical considerations before practical application.
The Capacitated Vehicle Routing Problem (CVRP) is a fundamental NP-hard problem in logistics. Augmented Lagrangian Methods (ALM) for solving CVRP performance depends heavily on well-tuned penalty parameters. In this paper, we propose a hybrid optimization approach that integrates deep reinforcement learning (RL) to automate the selection of penalty parameter values within both classical (RL-C-ALM) and quantum-enhanced (RL-Q-ALM) ALM solvers. Using Soft Actor-Critic, our approach learns penalty values from CVRP instance features and constraint violations. In RL-Q-ALM, subproblems are encoded as QUBOs and solved using Variational Quantum Eigensolvers (VQE). The agent learns across episodes by maximizing solution feasibility and minimizing cost. Experiments show that RL-C-ALM outperforms manually tuned ALM on synthetic and benchmark CVRP instances, achieving better solutions with fewer iterations. Also, RL-Q-ALM matches classical solution quality on small instances but incurs higher runtimes due to quantum overhead. Our results highlight the potential of combining RL with classical and quantum solvers for scalable, adaptive combinatorial optimization.
Intelligent transportation systems (ITS) are a critical component of Industry 4.0 and 5.0, particularly having applications in logistic management. One of their crucial utilization is in supply-chain management and scheduling for optimally routing transportation of goods by vehicles at a given set of locations. This paper discusses the broader problem of vehicle traffic management, more popularly known as the Vehicle Routing Problem (VRP), and investigates the possible use of near-term quantum devices for solving it. For this purpose, we give the Ising formulation for VRP and some of its constrained variants. Then, we present a detailed procedure to solve VRP by minimizing its corresponding Ising Hamiltonian using a hybrid quantum-classical heuristic called Quantum Approximate Optimization Algorithm (QAOA), implemented on the IBM Qiskit platform. We compare the performance of QAOA with classical solvers such as CPLEX on problem instances of up to 15 qubits. We find that performance of QAOA has a multifaceted dependence on the classical optimization routine used, the depth of the ansatz parameterized byp, initialization of variational parameters, and problem instance itself.