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：受近期量子硬件与算法进展的驱动，研究者已针对优化问题开发出量子启发式算法，旨在获得超越经典方法的优势。当前量子硬件仍存在易错性且规模有限，使得量子启发式算法无法扩展至实际应用规模的问题，其性能常逊于经典算法。此外，若需获得可证明的最优解，则必须依赖经典精确算法。然而考虑到量子技术未来可能取得显著进步，本研究展示了如何将资源受限的量子启发式算法整合至针对NP难问题的大规模精确优化算法中。为此，我们以车辆路径问题作为典型NP难问题进行研究。我们将分支定价切割算法中出现的定价与分离子问题建模为二次无约束二进制优化问题，这使得能够使用量子退火或量子近似优化算法等成熟的量子启发式方法进行求解。本算法的关键特征在于：它不仅利用量子启发式算法返回的最优解，还收集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特性。此外，由于通过量子启发式求解的子问题规模小于原始问题，这降低了对量子硬件的要求。我们通过实验研究比较了量子退火、模拟退火以及现有经典算法在本框架中的表现。虽然当前混合量子-经典方法的性能仍落后于纯经典方法，但实验结果表明：若未来量子硬件得到改进，定价与分离子问题均可能非常适合采用量子启发式算法求解。