In black-box combinatorial optimization, objective evaluations are often expensive, so high quality solutions must be found under a limited budget. Factorization machine with quantum annealing (FMQA) builds a quadratic surrogate model from evaluated samples and optimizes it on an Ising machine. However, FMQA requires binary decision variables, and for nonbinary structures such as integer permutations, the choice of binary encoding strongly affects search efficiency. If the encoding fails to reflect the original neighborhood structure, small Hamming moves may not correspond to meaningful modifications in the original solution space, and constrained problems can yield many infeasible candidates that waste evaluations. Recent work combines FMQA with a binary autoencoder (bAE) that learns a compact binary latent code from feasible solutions, yet the mechanism behind its performance gains is unclear. Using a small traveling salesman problem as an interpretable testbed, we show that the bAE reconstructs feasible tours accurately and, compared with manually designed encodings at similar compression, better aligns tour distances with latent Hamming distances, yields smoother neighborhoods under small bit flips, and produces fewer local optima. These geometric properties explain why bAE+FMQA improves the approximation ratio faster while maintaining feasibility throughout optimization, and they provide guidance for designing latent representations for black-box optimization.

翻译：在黑箱组合优化中，目标函数评估通常代价高昂，因此必须在有限评估预算内找到高质量解。基于量子退火的因子分解机（FMQA）通过已评估样本构建二次代理模型，并在伊辛机上对其进行优化。然而，FMQA要求决策变量为二元形式，对于整数排列等非二元结构，二元编码方式的选择会显著影响搜索效率。若编码未能反映原始邻域结构，微小的汉明移动可能无法对应原始解空间中有意义的修改，且约束问题可能产生大量不可行候选解，造成评估资源浪费。近期研究将FMQA与二元自编码器（bAE）相结合，后者从可行解中学习紧凑的二元潜在编码，但其性能提升的内在机制尚不明确。本文以小型旅行商问题作为可解释性测试平台，证明bAE能够准确重构可行路径，且在相近压缩率下，相较于人工设计的编码方式，bAE能更好地对齐路径距离与潜在汉明距离，在小规模比特翻转下产生更平滑的邻域结构，并生成更少的局部最优解。这些几何特性解释了为何bAE+FMQA能在整个优化过程中保持可行性的同时更快地提升近似比，并为黑箱优化中潜在表示的设计提供了指导原则。