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：容量约束车辆路径问题（CVRP）是一个在运输与物流等领域广泛出现的NP优化问题（NPO）。作为车辆路径问题（VRP）的扩展，CVRP旨在确定车队在每辆车辆有限载重约束下，向一组客户配送货物的最高效方案。随着客户数量增加，可行解数量呈爆炸式增长，寻找最优解仍是一项重大挑战。近期，一种名为量子近似优化算法（QAOA）的量子-经典混合算法在部分组合优化问题中展现出优于经典启发式算法的求解能力。然而，对于包括CVRP在内的约束优化问题，QAOA生成高质量解的能力有所减弱。一种潜在的改进方案是采用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。我们提出了一种新的CVRP二进制编码方案，并设计了替代目标函数——通过最小化绕过CVRP车辆容量约束的最短路径。进一步利用Grover-Mixer约束搜索空间。通过多个实例应用，我们检验并讨论了所提求解器的有效性。