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arXiv:2404.05551

Improving Quantum and Classical Decomposition Methods for Vehicle Routing

Laura S. Herzog, Friedrich Wagner, Christian Ufrecht, Lilly Palackal, Axel Plinge, Christopher Mutschler, Daniel D. Scherer

Published Apr 8, 2024• 2 citations
arXivHTMLPDF

Abstract

Quantum computing is a promising technology to address combinatorial optimization problems, for example via the quantum approximate optimization algorithm (QAOA). Its potential, however, hinges on scaling toy problems to sizes relevant for industry. In this study, we address this challenge by an elaborate combination of two decomposition methods, namely graph shrinking and circuit cutting. Graph shrinking reduces the problem size before encoding into QAOA circuits, while circuit cutting decomposes quantum circuits into fragments for execution on medium-scale quantum computers. Our shrinking method adaptively reduces the problem such that the resulting QAOA circuits are particularly well-suited for circuit cutting. Moreover, we integrate two cutting techniques which allows us to run the resulting circuit fragments sequentially on the same device. We demonstrate the utility of our method by successfully applying it to the archetypical traveling salesperson problem (TSP) which often occurs as a sub-problem in practically relevant vehicle routing applications. For a TSP with seven cities, we are able to retrieve an optimum solution by consecutively running two 7-qubit QAOA circuits. Without decomposition methods, we would require five times as many qubits. Our results offer insights into the performance of algorithms for combinatorial optimization problems within the constraints of current quantum technology.

DOI
10.48550/arxiv.2404.05551

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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.
We present a two-level decomposition strategy for solving the Vehicle Routing Problem (VRP) using the Quantum Approximate Optimization Algorithm. A Problem-Level Decomposition partitions a 13-node (156-qubit) VRP into smaller Traveling Salesman Problem (TSP) instances. Each TSP is then further cut via Circuit-Level Decomposition, enabling execution on near-term quantum devices. Our approach achieves up to 95% reductions in the circuit depth, 96% reduction in the number of qubits and a 99.5% reduction in the number of 2-qubit gates. We demonstrate this hybrid algorithm on the standard edge encoding of the VRP as well as a novel amplitude encoding. These results demonstrate the feasibility of solving VRPs previously too complex for quantum simulators and provide early evidence of potential quantum utility.
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 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.
Abstract—We present a two-level decomposition strategy for solving the Vehicle Routing Problem (VRP) using the Quantum Approximate Optimization Algorithm (QAOA). A Problem-Level Decomposition (PLD) partitions a 9-node (72-qubit) VRP into smaller Traveling Salesman Problem (TSP) instances. Each TSP is then further simplified via Circuit-Level Decomposition (CLD), enabling execution on near-term quantum devices. Our approach achieves up to 90% reductions in circuit depth and qubit count. These results demonstrate the feasibility of solving VRPs previously too complex for quantum simulators and provide early evidence of potential quantum utility.
Transportation systems such as urban logistics, vehicle routing, and infrastructure planning require solving large-scale combinatorial optimization problems under complex constraints. Problems such as the vehicle routing problem (VRP), traveling salesman problem (TSP), and facility location problem (FLP) involve large discrete search spaces and the need to generate multiple feasible solutions in real time. In this work, we develop a hardware-grounded hybrid quantum optimization framework that uses Approximate Quantum Compilation (AQC) to compress early segments of digitized adiabatic evolution into shallow circuits. The compressed prefix is combined with variational layers, enabling a systematic study of how initialization, circuit depth, and expressivity interact on near-term quantum hardware. All experiments are performed on an IBM gate-based quantum computer, and circuits are evaluated as stochastic generators of candidate transportation plans. Results show that moderate prefix compression reduces two-qubit gate depth while maintaining or improving feasible solution discovery, particularly for routing problems. These benefits depend on compatibility between the compressed prefix and the variational ansatz: while standard QAOA effectively leverages AQC initialization, linear-chain QAOA shows limited improvement. Overall, this work demonstrates that hybrid AQC-QAOA methods provide a practical pathway for hardware-efficient quantum optimization, positioning quantum algorithms as candidate generators within transportation decision-making workflows.