AI 中文总结
针对多约束位置优化问题，将其转化为QUBO模型，采用不平衡惩罚处理不等式约束，利用量子近似优化算法及其变体WS - QAOA，并引入线性斜坡参数调度，组合方法提高了解质量和可行性指标，在混合量子 - 经典优化中有潜力。
AI 中文摘要
最大覆盖位置问题（MCLP）是一个NP难组合优化问题，旨在确定能使总覆盖最大化的最优设施布局。它有等式和不等式约束，随着实例规模增长，探索解空间的复杂度显著增加。混合量子 - 经典方法可能是经典优化方法的一个有前途的替代方案。本文将MCLP表述为二次无约束二元优化（QUBO）模型，其中约束嵌入对解质量至关重要。采用不平衡惩罚（UP）替代松弛变量（SV）处理不等式约束，且不增加变量数量。研究聚焦量子近似优化算法（QAOA）及其变体加权和量子近似优化算法（WS - QAOA），后者利用从问题连续松弛导出的有偏初始状态，还引入线性斜坡（LR）参数调度以降低优化复杂度。评估了这些技术单独及组合使用时随电路深度\(p\)和问题规模的性能。结果表明，UP、LR和WS - QAOA的组合方法持续提高了解质量和可行性指标，且随着问题规模增加保持稳健性能，凸显了其在混合量子 - 经典优化框架中的潜力。
英文摘要
The Maximal Covering Location Problem (MCLP) is an NP-hard Combinatorial Optimization Problem (COP) that aims to determine the optimal facility placements that maximize total coverage. It is characterized by both equality and inequality constraints, which ensure correct coverage but significantly increase the complexity of exploring the solution space as instance size grows. Hybrid quantum-classical approaches might offer a promising alternative to classical optimization methods by enabling the exploration of complex energy landscapes through quantum superposition and probabilistic sampling. In this work, the MCLP is formulated as a Quadratic Unconstrained Binary Optimization (QUBO) model, where constraint embedding plays a critical role in solution quality. In particular, Unbalanced Penalization (UP) is employed as an alternative to the Slack Variables (SV) for handling inequality constraints without increasing the number of variables. This study focuses on QAOA and one of its variants, the WS-QAOA, which leverages a biased initial state derived from a continuous relaxation of the problem. Additionally, a linear ramp (LR) parameter schedule is incorporated to reduce optimization complexity. The performance of these techniques is evaluated both individually and in combination, as a function of circuit depth $p$ and problem size. Results show that the combined approach of UP, LR, and WS-QAOA consistently improves solution quality and feasibility metrics, while maintaining robust performance as the problem size increases, highlighting its potential within hybrid quantum-classical optimization frameworks.
