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arXiv:2605.18393

Quantum Model for CVRPTW

Imran Meghazi, Éric Bourreau

math.OCquant-phSubmitted May 18, 2026• Announced May 19, 2026• 0 citations
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Abstract

This paper proposes a quantum algorithm for the capacitated vehicle routing problem with time windows (CVRPTW) based on Grover Search framework. This problem is often faced by Postal services in the context of package delivery or other time-sensitive operations. We provide an implementation on gate based quantum computer of a model inspired by classical route first, cluster second technique. The quantum paradigm allows to overcome suboptimality inherent property of this decomposition. In the current NISQ (Noisy Intermediate-Scale Quantum) era, the most important limitation is the number of available qubits which makes time windows and capacity constraints hard to tackle. We introduce a qubit-efficient split-inspired modeling which adds only a linear number of decision qubits to standard quantum formulations for Traveling Salesman Problem (TSP).

Journal reference
QUEST-IS 2025, Dec 2025, Paris, France. pp.65-76

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This paper proposes a quantum algorithm for the capacitated vehicle routing problem with time windows (CVRPTW) based on Grover Search framework. This problem is often faced by Postal services in the context of package delivery or other time-sensitive operations. We provide an implementation on gate based quantum computer of a model inspired by classical route first, cluster second technique. The quantum paradigm allows to overcome suboptimality inherent property of this decomposition. In the current NISQ (Noisy Intermediate-Scale Quantum) era, the most important limitation is the number of available qubits which makes time windows and capacity constraints hard to tackle. We introduce a qubit-efficient split-inspired modeling which adds only a linear number of decision qubits to standard quantum formulations for Traveling Salesman Problem (TSP).
Many relevant problems in industrial settings result in NP-hard optimization problems, such as the Capacitated Vehicle Routing Problem (CVRP) or its reduced variant, the Travelling Salesperson Problem (TSP). Even with today's most powerful classical algorithms, the CVRP is challenging to solve classically. Quantum computing may offer a way to improve the time to solution, although the question remains open as to whether Noisy Intermediate-Scale Quantum (NISQ) devices can achieve a practical advantage compared to classical heuristics. The most prominent algorithms proposed to solve combinatorial optimization problems in the NISQ era are the Quantum Approximate Optimization Algorithm (QAOA) and the more general Variational Quantum Eigensolver (VQE). However, implementing them in a way that reliably provides high-quality solutions is challenging, even for toy examples. In this work, we discuss decomposition and formulation aspects of the CVRP and propose an application-driven way to measure solution quality. Considering current hardware constraints, we reduce the CVRP to a clustering phase and a set of TSPs. For the TSP, we extensively test both QAOA and VQE and investigate the influence of various hyperparameters, such as the classical optimizer choice and strength of constraint penalization. Results of QAOA are generally of limited quality because the algorithm does not reach the energy threshold for feasible TSP solutions, even when considering various extensions such as recursive and constraint-preserving mixer QAOA. On the other hand, the VQE reaches the energy threshold and shows a better performance. Our work outlines the obstacles to quantum-assisted solutions for real-world optimization problems and proposes perspectives on how to overcome them.
This study explores Quantum Annealing (QA) versus Classical Computing (CC) approaches in solving the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW), using real-world logistics data from Siemens Advanta. By modeling the problem as a Service Network Design (SND) with binary variables and framing CVRPTW within a quantum mechanical paradigm suitable for QA, this approach leverages quantum annealing in a novel manner to tackle the complex optimization challenges inherent in the VRP. Contrasting it with the performance of the traditional branch and cut solver---a specific CC technique---this research unveils the prospective benefits and existing challenges of QA in complex optimization scenarios, thereby emphasizing the need for advancing quantum computing.
Quantum optimization has recently drawn considerable attention as one of the possible applications of noisy intermediate-scale quantum computation, yet the problem of qubit requirement remains a major bottleneck when combinatorial optimization problems are converted into quantum circuits. This issue becomes especially critical in solving the capacitated vehicle routing problem (CVRP) with the quantum approximate optimization algorithm (QAOA), since the number of required qubits increases polynomially with respect to the number of nodes. This study investigates whether a heuristic divide-and-conquer strategy can be adapted to the quantum setting so as to improve qubit efficiency while preserving the optimization capability to a reasonable extent. The proposed method decomposes a single CVRP into multiple traveling salesman problems (TSPs) by the sweeping-based clustering method, searches for the sector configuration with the smallest angle sum by Grover’s search algorithm, and then solves each sector-wise TSP with the QAOA aided by the gravitational search algorithm. Experiments on five benchmark datasets show that the proposed approach attains feasible solutions within 3.4 to 12.7% of the reinforcement-learning baseline on the main test set. These results suggest that the proposed approach serves as a plausible quantum heuristic framework for constrained routing optimization, with the advantage of reducing the qubit burden by decomposing the original problem into smaller subproblems.
The capacitated vehicle routing problem (CVRP) is an important problem for logistics optimization in supply chain. We propose a novel hybrid classical quantum algorithm that works by splitting the problem into a quantum travelling salesman problem (TSP) phase and a partitioning phase. In order to validate our methods, we run tests using D-Wave's Leap framework on random test cases, as well as a real-world data set from the retail domain. We run a comparison study against existing state of the art hybrid quantum-classical approaches for the CVRP, in addition to classical heuristic methods. Our method is shown to have the additional flexibility to handle fleet extension, and the solution quality is at least as good as the considered state of the art quantum hybrid algorithms. Further, we show that the algorithm can easily be modified for tackling other realistic vehicle routing variants.
Quantum computing (QC) is expected to solve incredibly difficult problems, including finding optimal solutions to combinatorial optimization problems. However, to date, QC alone is still far to demonstrate this capability except on small-sized problems. Hybrid approaches where QC and classical computing work together have shown the most potential for solving real-world scale problems. This work aims to show that we can enhance a classical optimization algorithm with QC so that it can overcome this limitation. We present a new hybrid quantum-classical tabu search (HQTS) algorithm to solve the capacitated vehicle routing problem (CVRP). Based on our prior work, HQTS leverages QC for routing within a classical tabu search framework. The quantum component formulates the traveling salesman problem (TSP) for each route as a QUBO, solved using D-Wave's Advantage system. Experiments investigate the impact of quantum routing frequency and starting solution methods. While different starting solution methods, including quantum-based and classical heuristics methods, it shows minimal overall impact. HQTS achieved optimal or near-optimal solutions for several CVRP problems, outperforming other hybrid CVRP algorithms and significantly reducing the optimality gap compared to preliminary research. The experimental results demonstrate that more frequent quantum routing improves solution quality and runtime. The findings highlight the potential of integrating QC within meta-heuristic frameworks for complex optimization in vehicle routing problems.
In this paper, we explore the potential for quantum annealing to solve realistic routing problems. We focus on two NP-Hard problems, including the Traveling Salesman Problem with Time Windows and the Capacitated Vehicle Routing Problem with Time Windows. We utilize D-Wave's Quantum Annealer and Constrained Quadratic Model (CQM) solver within a hybrid framework to solve these problems. We demonstrate that while the CQM solver effectively minimizes route costs, it struggles to maintain time window feasibility as the problem size increases. To address this limitation, we implement a heuristic method that fixes infeasible solutions through a series of swapping operations. Testing on benchmark instances shows our method achieves promising results with an average optimality gap of 3.86%.