发表机构
AI and Quantum Technologies Department, ITECAM, Spain（ITECAM 人工智能与量子技术部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对二维下料问题，比较了坐标基集合打包与无坐标序列对两种QUBO编码，证明后者存在结构性限制，并给出选择实用规则。
AI 中文摘要
二维下料问题（2D-CSP）是一个 NP 难问题，对制造业和物流业具有直接的经济和环境影响。我们将其固定板材变体（允许自由重复使用零件，并包含完整的不重叠和包含约束）编码为用于量子退火的二次无约束二元优化（QUBO）问题，并比较了来自相反编码范式的两种公式。第一种是基于坐标的集合打包模型，每个候选放置位置对应一个二元变量。其变量数量随板材面积和分辨率线性增长，但其基态在构造上即为几何可行的最大面积打包。第二种是无坐标的序列对模型，其变量数量与板材分辨率和尺寸无关。我们证明了这种紧凑性存在结构性限制：对于二维包含问题，任何交互度有界且惩罚项在几何可行布局上消失的无坐标 QUBO，都不可能具有几何可行的基态，因为包含约束是一种最长路径约束，而有界度的惩罚项无法在长度超过其交互阶数的链上强制执行该约束。我们在多种子模拟退火、模拟量子退火以及精确整数规划基线下评估了这两种公式。硬件实验包括 D-Wave 小图嵌入、校准的直接 QPU 扫描以及 Leap 混合求解器（采用惩罚形式和约束原生形式），在包含六个实例的活动中对每个实例进行了校准。混合求解器返回了我们的证书配置，其能量与十三个小数位绑定，同时溢出了板材。我们不声称量子加速。我们的贡献在于一个不可能性结果，刻画了紧凑打包 QUBO 的局限性，以及一个用于在两种公式之间进行选择的实用规则。
英文摘要
The two-dimensional Cutting Stock Problem (2D-CSP) is an NP-hard problem with direct economic and environmental impacts on manufacturing and logistics. We encode its fixed-plate variant, with free piece repetition and full non-overlap and containment constraints, as a Quadratic Unconstrained Binary Optimization (QUBO) problem for quantum annealing and compare two formulations from opposite encoding paradigms. The first was a coordinate-based set-packing model with one binary variable per candidate placement. Its variable count grows linearly with plate area and resolution, but its ground state is, by construction, a geometrically feasible maximum-area packing. The second is a coordinate-free sequence-pair model whose variable count is independent of plate resolution and size. We prove that this compactness has a structural limit: no coordinate-free QUBO of bounded interaction degree whose penalties vanish on every geometrically feasible layout can have a geometrically feasible ground state for 2D containment, because containment is a longest-path constraint that bounded-degree penalties cannot enforce on chains longer than their interaction order. We evaluate both formulations under multi-seed simulated annealing, simulated quantum annealing, and an exact integer-programming baseline. Hardware experiments include D-Wave minor embedding, a calibrated direct-QPU sweep, and Leap hybrid solvers in both penalty and constraint-native form, across a six-instance campaign with per-instance calibration. The hybrid solver returns our certificate configuration, tying its energy to thirteen decimal places while overflowing the plate. We claim no quantum speedup. Our contribution is an impossibility result characterizing the limits of compact packing QUBOs, and a practical rule for choosing between the two formulations.
Comments15 pages, 4 figures, 6 tables. To be published in IEEE
