Optimal, Qubit-Efficient Quantum Vehicle Routing via Colored-Permutations

2026-04-06

AI Summary

This work addresses the high logical qubit overhead in quantum approaches to the Capacitated Vehicle Routing Problem (CVRP), which arises from explicit encoding of vehicle loads or labels. To overcome this bottleneck, the authors propose a color-based permutation scheme that models the routes of K vehicles as K disjoint partial permutations within a unified permutation matrix. Capacity constraints are implicitly enforced through weighted sums across color layers, eliminating the need for additional logical qubits. By integrating a constraint-augmented Quantum Approximate Optimization Algorithm (QAOA) with manifold-based encoding analysis, the method constructs a multilayer formulation using n²K binary variables. Evaluated on standard benchmark instances, the approach efficiently recovers verified optimal solutions, substantially improving qubit efficiency and algorithmic scalability.

Abstract

We formulate a global-position colored-permutation encoding for the capacitated vehicle routing problem. Each of the $K$ vehicles selects a disjoint partial permutation, and the sum of these $K$ color layers forms a full $n\times n$ permutation matrix that assigns every customer to exactly one visit position. This representation uses $n^2K$ binary decision variables arranged as $K$ color layers over a common permutation structure, while vehicle capacities are enforced by weighted sums over the entries of each color class, requiring no explicit load register and hence no extra logical qubits beyond the routing variables. In contrast, many prior quantum encodings introduce an explicit capacity or load representation with additional qubits. Our construction is designed to exploit the Constraint-Enhanced QAOA framework together with its encoded-manifold analyses. Building on a requirements-based view of quantum utility in CVRP, we develop a routing optimization formulation that directly targets one of the main near-term bottlenecks, namely the additional logical-qubit cost of vehicle labels and explicit capacity constraints. Our proposal shows strong algorithmic performance in addition to qubit efficiency. On a standard benchmark suite, our end-to-end pipeline recovers the independently verified optima. The feasibility oracle may also be of independent interest as a reusable polynomial-time decoding and certification primitive for quantum and quantum-inspired routing pipelines.

Problem

• Capacitated Vehicle Routing Problem
• Quantum Optimization
• Qubit Efficiency
• Permutation Encoding
• Logical Qubits

Innovation

• colored-permutation encoding
• qubit-efficient
• capacitated vehicle routing problem
• Constraint-Enhanced QAOA
• implicit capacity constraints