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.

翻译：我们针对带容量约束的车辆路径问题提出了一种全局位置着色排列编码方法。每辆$K$车辆选取一个不相交的部分排列，这些$K$个颜色层的总和构成一个完整的$n\times n$置换矩阵，该矩阵将每位顾客精确分配至一个访问位置。该表示采用$n^2K$个二元决策变量，在公共排列结构上排列为$K$个颜色层，同时通过各颜色类条目上的加权和强制实施车辆容量约束，既不需要显式负载寄存器，也无需在路径变量之外额外增加逻辑量子比特。相比之下，许多现有量子编码会引入显式容量或负载表示并需要额外量子比特。我们的构造旨在利用"约束增强型量子近似优化算法"框架及其编码流形分析。基于对CVRP中量子效用的需求视角，我们开发了一种路由优化公式，直接针对近期主要瓶颈之一——即车辆标签和显式容量约束带来的附加逻辑量子比特成本。除量子比特效率外，我们的方案还展现出强劲的算法性能。在标准基准测试集上，我们的端到端流水线能够恢复经独立验证的最优解。此外，可行性预查功能作为一种可复用的多项式时间解码与认证基元，可为量子及量子启发式路由流水线提供独立参考价值。