Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE International Conference on Quantum Computing and Engineering (QCE)

Hybrid Quantum-Classical Algorithm for Solving Capacitated Vehicle Routing Problems

Wei‐Hao Huang, Hiromichi Matsuyama, Yu Yamashiro

Published Sep 15, 2024• 0 citations
Publisher

Abstract

This study introduces a hybrid quantum-classical approach to solving the Capacitated Vehicle Routing Problems (CVRP), combining the Column Generation method and the Quantum Alternating Operator Ansatz (QAOAnsatz). The column generation method divides the problem into a master problem and subproblems to effectively solve the problem. We solved the subproblem by using QAOAnsatz to enhance the optimal solution search. Moreover, we generated the problem Hamiltonian using an Augmented Lagrangian-inspired Method to reduce the number of required qubits to handle the inequality. As a result, our experiments obtained an optimal solution for a CVRP problem with 6 customers. This study demonstrates a practical quantum application in logistics and sets a new technique for hybrid quantum-classical optimization frameworks.

DOI
10.1109/qce60285.2024.10369

Related papers

Solving Capacitated Vehicle Routing Problem with Quantum Alternating Operator Ansatz and Column Generation

Wei‐Hao Huang, Hiromichi Matsuyama, Yu Yamashiro

This study proposes a hybrid quantum-classical approach to solving the Capacitated Vehicle Routing Problem (CVRP) by integrating the Column Generation (CG) method with the Quantum Alternating Operator Ansatz (QAOAnsatz). The CG method divides the CVRP into the reduced master problem, which finds the best combination of the routes under the route set, and one or more subproblems, which generate the routes that would be beneficial to add to the route set. This method is iteratively refined by adding new routes identified via subproblems and continues until no improving route can be found. We leverage the QAOAnsatz to solve the subproblems. Our algorithm restricts the search space by designing the QAOAnsatz mixer Hamiltonian to enforce one-hot constraints. Moreover, to handle capacity constraints in QAOAnsatz, we employ an Augmented Lagrangian-inspired method that obviates the need for additional slack variables, reducing the required number of qubits. Experimental results on small-scale CVRP instances (up to 6 customers) show that QAOAnsatz converges more quickly to optimal routes than the standard QAOA approach, demonstrating the potential of this hybrid framework in tackling realworld logistical optimization problems on near-term quantum hardware.

arXiv:2503.17051•International Conference on Quantum Communications, Networking, and Computing (QCNC)•Apr 6, 2026•2 citations

A feasibility-preserved quantum approximate solver for the Capacitated Vehicle Routing Problem

Ningyi Xie, Xinwei Lee, Dongsheng Cai +3 more

Abstract The Capacitated Vehicle Routing Problem (CVRP) is an NP-optimization problem (NPO) that arises in various fields including transportation and logistics. The CVRP extends from the Vehicle Routing Problem (VRP), aiming to determine the most efficient plan for a fleet of vehicles to deliver goods to a set of customers, subject to the limited carrying capacity of each vehicle. As the number of possible solutions skyrockets when the number of customers increases, finding the optimal solution remains a significant challenge. Recently, the Quantum Approximate Optimization Algorithm (QAOA), a quantum-classical hybrid algorithm, has exhibited enhanced performance in certain combinatorial optimization problems compared to classical heuristics. However, its ability diminishes notably in solving constrained optimization problems including the CVRP. This limitation primarily arises from the typical approach of encoding the given problems as penalty-inclusive binary optimization problems. In this case, the QAOA faces challenges in sampling solutions satisfying all constraints. Addressing this, our work presents a new binary encoding for the CVRP, with an alternative objective function of minimizing the shortest path that bypasses the vehicle capacity constraint of the CVRP. The search space is further restricted by the constraint-preserving mixing operation. We examine and discuss the effectiveness of the proposed encoding under the framework of the variant of the QAOA, Quantum Alternating Operator Ansatz (AOA), through its application to several illustrative examples. Compared to the typical QAOA approach, the proposed method not only preserves the feasibility but also achieves a significant enhancement in the probability of measuring optimal solutions.

arXiv:2308.08785•Quantum Information Processing•Jul 29, 2024•24 citations

Hybrid Learning and Optimization methods for solving Capacitated Vehicle Routing Problem

Monit Sharma, Hoong Chuin Lau

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.

arXiv:2509.15262•Sep 18, 2025•0 citations

A Hybrid Solution Method for the Capacitated Vehicle Routing Problem Using a Quantum Annealer

Sebastian Feld, Christoph Roch, Thomas Gabor +5 more

The Capacitated Vehicle Routing Problem (CVRP) is an NP-optimization problem (NPO) that has been of great interest for decades for both, science and industry. The CVRP is a variant of the vehicle routing problem characterized by capacity constrained vehicles. The aim is to plan tours for vehicles to supply a given number of customers as efficiently as possible. The problem is the combinatorial explosion of possible solutions, which increases superexponentially with the number of customers. Classical solutions provide good approximations to the globally optimal solution. D-Wave's quantum annealer is a machine designed to solve optimization problems. This machine uses quantum effects to speed up computation time compared to classic computers. The problem on solving the CVRP on the quantum annealer is the particular formulation of the optimization problem. For this, it has to be mapped onto a quadratic unconstrained binary optimization (QUBO) problem. Complex optimization problems such as the CVRP can be translated to smaller subproblems and thus enable a sequential solution of the partitioned problem. This work presents a quantum-classic hybrid solution method for the CVRP. It clarifies whether the implemenation of such a method pays off in comparison to existing classical solution methods regarding computation time and solution quality. Several approaches to solving the CVRP are elaborated, the arising problems are discussed, and the results are evaluated in terms of solution quality and computation time.

arXiv:1811.07403•Frontiers in ICT•Jun 25, 2019•190 citations

An Advanced Hybrid Quantum Tabu Search Approach to Vehicle Routing Problems

James B. Holliday, Eneko Osaba, Khoa Luu

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.

arXiv:2501.12652•Jan 22, 2025•0 citations

Hybrid Quantum Tabu Search for Solving the Vehicle Routing Problem

James B. Holliday, Braeden Morgan, Hugh Churchill +1 more

There has never been a more exciting time for the future of quantum computing than now. Real-world quantum computing usage is now the next XPRIZE. With that challenge in mind we have explored a new approach as a hybrid quantum-classical algorithm for solving NP-Hard optimization problems. We have focused on the classic problem of the Capacitated Vehicle Routing Problem (CVRP) because of its real-world industry applications. Heuristics are often employed to solve this problem because it is difficult. In addition, meta-heuristic algorithms have proven to be capable of finding reasonable solutions to optimization problems like the CVRP. Recent research has shown that quantum-only and hybrid quantum/classical approaches to solving the CVRP are possible. Where quantum approaches are usually limited to minimal optimization problems, hybrid approaches have been able to solve more significant issues. Still, the hybrid approaches often need help finding solutions as good as their classical counterparts. In our proposed approach, we created a hybrid quantum/classical meta-heuristic algorithm capable of finding the best-known solution to a classic CVRP problem. Our experimental results show that our proposed algorithm often outperforms other hybrid approaches.

arXiv:2404.13203•IEEE International Conference on Quantum Computing and Engineering (QCE)•Sep 15, 2024•10 citations
This study introduces a hybrid quantum-classical approach to solving the Capacitated Vehicle Routing Problems (CVRP), combining the Column Generation method and the Quantum Alternating Operator Ansatz (QAOAnsatz). The column generation method divides the problem into a master problem and subproblems to effectively solve the problem. We solved the subproblem by using QAOAnsatz to enhance the optimal solution search. Moreover, we generated the problem Hamiltonian using an Augmented Lagrangian-inspired Method to reduce the number of required qubits to handle the inequality. As a result, our experiments obtained an optimal solution for a CVRP problem with 6 customers. This study demonstrates a practical quantum application in logistics and sets a new technique for hybrid quantum-classical optimization frameworks.
This study proposes a hybrid quantum-classical approach to solving the Capacitated Vehicle Routing Problem (CVRP) by integrating the Column Generation (CG) method with the Quantum Alternating Operator Ansatz (QAOAnsatz). The CG method divides the CVRP into the reduced master problem, which finds the best combination of the routes under the route set, and one or more subproblems, which generate the routes that would be beneficial to add to the route set. This method is iteratively refined by adding new routes identified via subproblems and continues until no improving route can be found. We leverage the QAOAnsatz to solve the subproblems. Our algorithm restricts the search space by designing the QAOAnsatz mixer Hamiltonian to enforce one-hot constraints. Moreover, to handle capacity constraints in QAOAnsatz, we employ an Augmented Lagrangian-inspired method that obviates the need for additional slack variables, reducing the required number of qubits. Experimental results on small-scale CVRP instances (up to 6 customers) show that QAOAnsatz converges more quickly to optimal routes than the standard QAOA approach, demonstrating the potential of this hybrid framework in tackling realworld logistical optimization problems on near-term quantum hardware.
Abstract The Capacitated Vehicle Routing Problem (CVRP) is an NP-optimization problem (NPO) that arises in various fields including transportation and logistics. The CVRP extends from the Vehicle Routing Problem (VRP), aiming to determine the most efficient plan for a fleet of vehicles to deliver goods to a set of customers, subject to the limited carrying capacity of each vehicle. As the number of possible solutions skyrockets when the number of customers increases, finding the optimal solution remains a significant challenge. Recently, the Quantum Approximate Optimization Algorithm (QAOA), a quantum-classical hybrid algorithm, has exhibited enhanced performance in certain combinatorial optimization problems compared to classical heuristics. However, its ability diminishes notably in solving constrained optimization problems including the CVRP. This limitation primarily arises from the typical approach of encoding the given problems as penalty-inclusive binary optimization problems. In this case, the QAOA faces challenges in sampling solutions satisfying all constraints. Addressing this, our work presents a new binary encoding for the CVRP, with an alternative objective function of minimizing the shortest path that bypasses the vehicle capacity constraint of the CVRP. The search space is further restricted by the constraint-preserving mixing operation. We examine and discuss the effectiveness of the proposed encoding under the framework of the variant of the QAOA, Quantum Alternating Operator Ansatz (AOA), through its application to several illustrative examples. Compared to the typical QAOA approach, the proposed method not only preserves the feasibility but also achieves a significant enhancement in the probability of measuring optimal solutions.
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.
The Capacitated Vehicle Routing Problem (CVRP) is an NP-optimization problem (NPO) that has been of great interest for decades for both, science and industry. The CVRP is a variant of the vehicle routing problem characterized by capacity constrained vehicles. The aim is to plan tours for vehicles to supply a given number of customers as efficiently as possible. The problem is the combinatorial explosion of possible solutions, which increases superexponentially with the number of customers. Classical solutions provide good approximations to the globally optimal solution. D-Wave's quantum annealer is a machine designed to solve optimization problems. This machine uses quantum effects to speed up computation time compared to classic computers. The problem on solving the CVRP on the quantum annealer is the particular formulation of the optimization problem. For this, it has to be mapped onto a quadratic unconstrained binary optimization (QUBO) problem. Complex optimization problems such as the CVRP can be translated to smaller subproblems and thus enable a sequential solution of the partitioned problem. This work presents a quantum-classic hybrid solution method for the CVRP. It clarifies whether the implemenation of such a method pays off in comparison to existing classical solution methods regarding computation time and solution quality. Several approaches to solving the CVRP are elaborated, the arising problems are discussed, and the results are evaluated in terms of solution quality and computation time.
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.
There has never been a more exciting time for the future of quantum computing than now. Real-world quantum computing usage is now the next XPRIZE. With that challenge in mind we have explored a new approach as a hybrid quantum-classical algorithm for solving NP-Hard optimization problems. We have focused on the classic problem of the Capacitated Vehicle Routing Problem (CVRP) because of its real-world industry applications. Heuristics are often employed to solve this problem because it is difficult. In addition, meta-heuristic algorithms have proven to be capable of finding reasonable solutions to optimization problems like the CVRP. Recent research has shown that quantum-only and hybrid quantum/classical approaches to solving the CVRP are possible. Where quantum approaches are usually limited to minimal optimization problems, hybrid approaches have been able to solve more significant issues. Still, the hybrid approaches often need help finding solutions as good as their classical counterparts. In our proposed approach, we created a hybrid quantum/classical meta-heuristic algorithm capable of finding the best-known solution to a classic CVRP problem. Our experimental results show that our proposed algorithm often outperforms other hybrid approaches.