The Capacitated Vehicle Routing Problem (CVRP) is an NP-optimization problem (NPO) that arises in various fields including transportation and logistics. The CVRP extends from the Vehicle Routing Problem (VRP), aiming to determine the most efficient plan for a fleet of vehicles to deliver goods to a set of customers, subject to the limited carrying capacity of each vehicle. As the number of possible solutions skyrockets when the number of customers increases, finding the optimal solution remains a significant challenge. Recently, the Quantum Approximate Optimization Algorithm (QAOA), a quantum-classical hybrid algorithm, has exhibited enhanced performance in certain combinatorial optimization problems compared to classical heuristics. However, its ability diminishes notably in solving constrained optimization problems including the CVRP. This limitation primarily arises from the typical approach of encoding the given problems as penalty-inclusive binary optimization problems. In this case, the QAOA faces challenges in sampling solutions satisfying all constraints. Addressing this, our work presents a new binary encoding for the CVRP, with an alternative objective function of minimizing the shortest path that bypasses the vehicle capacity constraint of the CVRP. The search space is further restricted by the constraint-preserving mixing operation. We examine and discuss the effectiveness of the proposed encoding under the framework of the variant of the QAOA, Quantum Alternating Operator Ansatz (AOA), through its application to several illustrative examples. Compared to the typical QAOA approach, the proposed method not only preserves the feasibility but also achieves a significant enhancement in the probability of measuring optimal solutions.

翻译：容量受限车辆路径问题（CVRP）是一个出现在运输和物流等多个领域的NP优化问题（NPO）。CVRP由车辆路径问题（VRP）扩展而来，旨在为车队在每辆车有限载重能力约束下，确定向一组客户配送货物的最有效方案。随着客户数量增加，可能解的数量急剧上升，寻找最优解仍是一项重大挑战。近年来，量子近似优化算法（QAOA）作为一种量子-经典混合算法，在某些组合优化问题中展现出优于经典启发式算法的性能。然而，其在求解包括CVRP在内的约束优化问题时能力显著下降。这一局限主要源于将给定问题编码为包含惩罚项的二进制优化问题的典型方法。在这种情况下，QAOA在采样满足所有约束的解时面临困难。针对这一问题，我们的工作提出了一种新的CVRP二进制编码，并采用替代目标函数——通过绕过CVRP车辆容量约束来最小化最短路径。搜索空间通过保持约束的混合操作进一步限制。我们通过将其应用于几个示例，在QAOA变体——量子交替算子拟设（AOA）框架下检验并讨论了所提编码的有效性。与典型的QAOA方法相比，所提方法不仅保持了可行性，而且在测量最优解的概率方面实现了显著提升。