The vehicle routing problem with time windows (VRPTW) is a classic optimization problem that arises in many different areas, such as logistics and transportation. The goal of the VRPTW is to find the shortest possible route for a fleet of vehicles to visit a set of destinations. In recent years, there has been growing interest in using variational quantum algorithms (VQAs), to find approximate solutions to problems that can be formulated as quadratic unconstrained binary optimization (QUBO) problems. In this work, we formulate the VRPTW as a QUBO and apply a quantum variational approach to the VRPTW using our earlier suggested encoding scheme described in [1] to reduce drastically the number of qubits required. We evaluate our approach on a set of VRPTW instances ranging from 11 to 3964 routes constructed with data provided by researchers from ExxonMobil. We compare the solutions obtained with standard full encoding approaches for which the max problems size possible in NISQ era are of the order of 20-30 routes. We run our algorithms in simulators as well as cloud quantum hardware provided by IBMQ, AWS (Rigetti) and IonQ and benchmark our results against each other as well as on the simulators. We show that our approach can find approximate solutions to the VRPTW that are comparable to the solutions found by quantum algorithms using the full encoding. Our results suggest that our unique encoding approach, provides a promising approach to drastically reducing the number of qubits required to find decent approximate solutions for industry-based optimization problems.

翻译：带时间窗的车队路径问题（VRPTW）是一个经典优化问题，广泛出现在物流与运输等不同领域。VRPTW的目标是为车队寻找访问一组目的地的最短可能路径。近年来，利用变分量子算法（VQA）近似求解可表述为二次无约束二进制优化（QUBO）问题的方案日益受到关注。本文中，我们将VRPTW表述为QUBO问题，并采用我们此前在文献[1]中提出的编码方案，对VRPTW应用量子变分方法，从而显著减少所需的量子比特数量。我们基于埃克森美孚研究人员提供的数据，对一组包含11至3964条路线的VRPTW实例进行了算法评估。我们将所得结果与标准全编码方法（在NISQ时代其最大可处理问题规模约为20-30条路线）的解进行对比。我们在模拟器以及IBMQ、AWS（Rigetti）和IonQ提供的云端量子硬件上运行算法，并将各平台结果及模拟器结果进行基准测试。结果表明，我们的方法能够找到与采用全编码的量子算法所得解相媲美的VRPTW近似解。我们的研究表明，这种独特的编码方案为大幅减少求解工业级优化问题所需量子比特数量以获取良好近似解提供了一条有前景的途径。