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。每辆车辆 $K$ 选择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，该矩阵将每个客户精确分配到一个访问位置。这种表示使用 $n^2K$ 个二进制决策变量，排列为公共置换结构上的 $K$ 个颜色层，同时车辆容量通过对每个颜色类条目的加权求和来强制执行，无需显式负载寄存器，因此除了路径变量外不需要额外的逻辑量子比特。相比之下，许多先前的量子编码引入了带有额外量子比特的显式容量或负载表示。我们的构造旨在利用约束增强型QAOA框架及其编码流形分析。基于CVRP中量子效用的需求视角，我们开发了一种路径优化公式，直接针对近期的瓶颈之一，即车辆标签和显式容量约束带来的额外逻辑量子比特成本。我们的方案在量子比特效率之外还展示了强大的算法性能。在标准基准测试集上，我们的端到端流程恢复了独立验证的最优解。可行性判定器作为一种可复用的多项式时间解码与认证原语，对于量子及量子启发的路径规划流程也可能具有独立的研究价值。