Hierarchical Quantum Optimization for Large-Scale Vehicle Routing: A Multi-Angle QAOA Approach with Clustered Decomposition

TL;DR

This work tackles large-scale vehicle routing by introducing a hierarchical quantum optimization framework that decomposes a 13-location VRP into three balanced clusters of four nodes each. Intra-cluster routing is solved exactly using standard QAOA on Open Loop TSP subproblems, while inter-cluster routing is addressed with Multi-Angle QAOA (MA-QAOA) using SPSA for parameter optimization, aided by a classical post-processing step to ensure feasibility. Across 10 synthetic datasets, intra-cluster solutions match classical Gurobi optima, and inter-cluster MA-QAOA achieves near-optimal performance with average approximation ratios in the high 90s percentile, demonstrating scalability beyond prior quantum VRP results. The hierarchical decomposition reduces quantum resource requirements and provides a viable path toward applying quantum optimization to realistically sized VRP instances on near-term devices, with strong implications for logistics and supply-chain optimization.

Abstract

We present a quantum optimization methodology for solving large-scale Vehicle Routing Problem (VRP) using a combination of standard and Multi-Angle Quantum Approximate Optimization Algorithms (MA-QAOA). The approach decomposes 13-locations based VRP problems through clustering into three balanced clusters of 4 nodes each, then applies standard QAOA for intra-cluster Open Loop Traveling Salesman Problem (OTSP) and MA-QAOA for inter-cluster VRP routing. Validation across 10 distinct datasets demonstrates that standard QAOA consistently identifies optimal solutions for intra-cluster routing, which is matching classical Gurobi optimizer results exactly. More significantly, MA-QAOA with Simultaneous Perturbation Stochastic Approximation(SPSA) optimizer demonstrates competitive performance against classical optimization methods, ultimately converging towards a solution that closely approximates the classical Gurobi optimizer result.The clustered decomposition enables quantum optimization of problem sizes generally larger than previous quantum VRP implementations, advancing from 4-6 location limits to 13-location problems while maintaining solution quality.

Figures (5)

• Figure 1: flowchart for largescale VRP solution: (a) clustering using K-means algorithm, (b) Intra-cluster routing using OTSP with standard QAOA, (c) Inter-cluster routing using MA-QAOA for enhanced performance.
• Figure 2: K-means clustering of 12 customer locations into three clusters of 4 nodes each. Black crosses indicate cluster centroids used for depot determination and inter-cluster distance calculations.
• Figure 3: Optimal Open Loop TSP routes for the three clusters obtained using standard QAOA ($p=1$). Each cluster shows the intra-cluster routing path connecting the designated initial node to the final node while visiting all intermediate nodes exactly once. The solutions match classical Gurobi optimizer results exactly.
• Figure 4: Complete hierarchical VRP solution showing inter-cluster routing obtained through MA-QAOA (thick lines connecting cluster representatives and depot) and intra-cluster OTSP routes within each cluster (thin lines). The two-vehicle solution demonstrates balanced workload distribution across clusters with optimal connectivity.
• Figure 5: Comparative energy performance of MA-QAOA (SPSA) and Gurobi optimizer across 10 VRP datasets. .