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：我们为容量受限的车辆路径问题提出了一种全局定位的彩色排列编码方案。每辆$K$辆车选取一个不相交的部分排列，这些$K$个颜色层之和构成一个完整的$n \times n$置换矩阵，将每个客户分配到恰好一个访问位置。该表示使用$n^2K$个二元决策变量，排列为$K$个颜色层，覆盖在一个公共排列结构之上，而车辆容量则通过每个颜色类条目的加权和来强制执行，无需显式的负载寄存器，因此在路由变量之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显式的容量或负载表示，需要额外的量子比特。我们的构造旨在利用约束增强型QAQA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，直接针对近期的关键瓶颈之一，即车辆标签和显式容量约束的额外逻辑量子比特成本。我们的方案除量子比特效率外，还表现出强大的算法性能。在标准基准测试套件上，我们的端到端流水线恢复了独立验证的最优解。可行性预言机也可能作为可重用的多项式时间解码与认证原语，独立应用于量子和量子启发式路由流水线中。