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, a quantum-classical hybrid algorithm known as Quantum Approximate Optimization Algorithm (QAOA) can provide better solutions in some cases of combinatorial optimization problems, compared to classical heuristics. However, the QAOA exhibits a diminished ability to produce high-quality solutions for some constrained optimization problems including the CVRP. One potential approach for improvement involves a variation of the QAOA known as the Grover-Mixer Quantum Alternating Operator Ansatz (GM-QAOA). In this work, we attempt to use GM-QAOA to solve the CVRP. We present 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 Grover-Mixer. We examine and discuss the effectiveness of the proposed solver through its application to several illustrative examples.

翻译：容量受限车辆路径问题（CVRP）是一个NP优化问题（NPO），广泛出现在交通运输与物流等多个领域。CVRP是车辆路径问题（VRP）的扩展，旨在确定一支车队在每辆车有限载重约束下将货物送达一组客户的最优配送方案。随着客户数量增加，可能的解空间呈指数级增长，因此寻找最优解仍是一个重大挑战。近年来，一种名为量子近似优化算法（QAOA）的量子-经典混合算法在某些组合优化问题上能够比经典启发式算法提供更优的解。然而，对于包括CVRP在内的某些约束优化问题，QAOA在生成高质量解方面的能力有所减弱。一种潜在的改进途径是采用QAOA的变体——Grover-Mixer量子交替算子拟设（GM-QAOA）。本文尝试使用GM-QAOA求解CVRP问题。我们提出了一种新的CVRP二进制编码方案，并设计了一个替代目标函数，该函数通过最小化最短路径来规避CVRP的车辆容量约束。搜索空间进一步通过Grover-Mixer进行限制。我们通过将该求解器应用于若干示例案例，检验并讨论了其有效性。