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

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 ntimes 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.