The Binary Paint Shop Problem (BPSP) is an $\mathsf{APX}$-hard optimisation problem in automotive manufacturing: given a sequence of $2n$ cars, comprising $n$ distinct models each appearing twice, the task is to decide which of two colours to paint each car so that the two occurrences of each model are painted differently, while minimising consecutive colour swaps. The key performance metric is the paint swap ratio, the average number of colour changes per car, which directly impacts production efficiency and cost. Prior work showed that the Quantum Approximate Optimisation Algorithm (QAOA) at depth $p=7$ achieves a paint swap ratio of $0.393$, outperforming the classical Recursive Greedy (RG) heuristic with an expected ratio of $0.4$ [Phys. Rev. A 104, 012403 (2021)]. More recently, the classical Recursive Star Greedy (RSG) heuristic was conjectured to achieve an expected ratio of $0.361$. In this study, we develop the theoretical foundations for applying QAOA to BPSP through a reduction of BPSP to weighted MaxCut, and use this framework to benchmark two state-of-the-art low-depth QAOA variants, eXpressive QAOA (XQAOA) and Recursive QAOA (RQAOA), at $p=1$ (denoted XQAOA$_1$ and RQAOA$_1$), against the strongest classical heuristics known to date. Across instances ranging from $2^7$ to $2^{12}$ cars, XQAOA$_1$ achieves an average ratio of $0.357$, surpassing RQAOA$_1$ and all classical heuristics, including the conjectured performance of RSG. Surprisingly, RQAOA$_1$ shows diminishing performance as size increases: despite using provably optimal QAOA$_1$ parameters at each recursion, it is outperformed by RSG on most $2^{11}$-car instances and all $2^{12}$-car instances. To our knowledge, this is the first study to report RQAOA$_1$'s performance degradation at scale. In contrast, XQAOA$_1$ remains robust, indicating strong potential to asymptotically surpass all known heuristics.

翻译：二元喷漆车间问题（BPSP）是汽车制造中一个$\mathsf{APX}$-难优化问题：给定包含$2n$辆汽车的序列（其中$n$种不同车型各出现两次），需要为每辆车分配两种颜色之一，使得每种车型的两次出现被涂装为不同颜色，同时最小化连续颜色切换次数。关键性能指标是喷漆切换率（即每辆车的平均颜色变更次数），该指标直接影响生产效率和成本。先前研究表明，深度$p=7$的量子近似优化算法（QAOA）可实现$0.393$的喷漆切换率，优于经典递归贪婪（RG）启发式算法的期望比率$0.4$[Phys. Rev. A 104, 012403 (2021)]。近期研究推测经典递归星型贪婪（RSG）启发式算法可实现$0.361$的期望比率。本研究通过将BPSP归约为加权最大割问题，建立了QAOA应用于BPSP的理论基础，并在此框架下对两种最先进的低深度QAOA变体——表达型QAOA（XQAOA）与递归QAOA（RQAOA）在$p=1$时（记作XQAOA$_1$和RQAOA$_1$）与当前已知最强经典启发式算法进行基准测试。在$2^7$至$2^{12}$辆汽车的实例范围内，XQAOA$_1$实现了$0.357$的平均比率，超越了RQAOA$_1$及所有经典启发式算法（包括RSG的推测性能）。值得注意的是，RQAOA$_1$的性能随规模增大而衰减：尽管每次递归都使用可证明最优的QAOA$_1$参数，但在大多数$2^{11}$辆汽车实例和全部$2^{12}$辆汽车实例中均被RSG超越。据我们所知，这是首次报道RQAOA$_1$在大规模场景下的性能退化现象。相比之下，XQAOA$_1$始终保持稳健性能，显示出渐近超越所有已知启发式算法的强大潜力。