Motivated by recent progress in quantum hardware and algorithms researchers have developed quantum heuristics for optimization problems, aiming for advantages over classical methods. To date, quantum hardware is still error-prone and limited in size such that quantum heuristics cannot be scaled to relevant problem sizes and are often outperformed by their classical counterparts. Moreover, if provably optimal solutions are desired, one has to resort to classical exact methods. As however quantum technologies may improve considerably in future, we demonstrate in this work how quantum heuristics with limited resources can be integrated in large-scale exact optimization algorithms for NP-hard problems. To this end, we consider vehicle routing as prototypical NP-hard problem. We model the pricing and separation subproblems arising in a branch-price-and-cut algorithm as quadratic unconstrained binary optimization problems. This allows to use established quantum heuristics like quantum annealing or the quantum approximate optimization algorithm for their solution. A key feature of our algorithm is that it profits not only from the best solution returned by the quantum heuristic but from all solutions below a certain cost threshold, thereby exploiting the inherent randomness is quantum algorithms. Moreover, we reduce the requirements on quantum hardware since the subproblems, which are solved via quantum heuristics, are smaller than the original problem. We provide an experimental study comparing quantum annealing to simulated annealing and to established classical algorithms in our framework. While our hybrid quantum-classical approach is still outperformed by purely classical methods, our results reveal that both pricing and separation may be well suited for quantum heuristics if quantum hardware improves.

翻译：摘要：受量子硬件与算法最新进展的驱动，研究者针对优化问题开发了量子启发式算法，旨在获得相较于经典方法的优势。目前量子硬件仍存在易错性与规模限制，导致量子启发式算法无法扩展至相关问题规模，且常被经典对应方法超越。此外，若需可证明的最优解，必须依赖于经典精确方法。然而，随着未来量子技术可能显著改进，本研究展示了如何在面向NP难题的大规模精确优化算法中集成资源受限的量子启发式算法。为此，我们以车辆路径问题作为典型NP难题。我们将分支定价切割算法中的定价子问题与分离子问题建模为二次无约束二元优化问题，从而可应用量子退火或量子近似优化算法等现有量子启发式算法进行求解。本算法的关键特征在于：不仅利用量子启发式算法返回的最优解，还利用所有低于特定成本阈值的解，从而充分利用量子算法固有的随机性。此外，由于量子启发式算法求解的子问题规模小于原始问题，我们降低了对量子硬件的要求。我们通过实验研究，在框架内比较了量子退火与模拟退火算法及经典算法的性能。尽管我们的混合量子-经典方法仍逊于纯经典方法，但结果表明，若量子硬件性能提升，定价与分离子问题可能非常适合采用量子启发式算法。