Jointly matching multiple, non-rigidly deformed 3D shapes is a challenging, $\mathcal{NP}$-hard problem. A perfect matching is necessarily cycle-consistent: Following the pairwise point correspondences along several shapes must end up at the starting vertex of the original shape. Unfortunately, existing quantum shape-matching methods do not support multiple shapes and even less cycle consistency. This paper addresses the open challenges and introduces the first quantum-hybrid approach for 3D shape multi-matching; in addition, it is also cycle-consistent. Its iterative formulation is admissible to modern adiabatic quantum hardware and scales linearly with the total number of input shapes. Both these characteristics are achieved by reducing the $N$-shape case to a sequence of three-shape matchings, the derivation of which is our main technical contribution. Thanks to quantum annealing, high-quality solutions with low energy are retrieved for the intermediate $\mathcal{NP}$-hard objectives. On benchmark datasets, the proposed approach significantly outperforms extensions to multi-shape matching of a previous quantum-hybrid two-shape matching method and is on-par with classical multi-matching methods.

翻译：联合匹配多个非刚性变形的三维形状是一个具有挑战性的$\mathcal{NP}$-难问题。完美匹配必然满足循环一致性：沿着多个形状的逐点对应关系行进，最终必须回到原始形状的起始顶点。然而，现有的量子形状匹配方法不支持多形状场景，更无法保证循环一致性。本文解决了这一开放挑战，提出了首个用于三维形状多匹配的量子混合方法；此外，该方法还具备循环一致性。其迭代公式适用于现代绝热量子硬件，且计算复杂度与输入形状总数呈线性关系。这两个特性均通过将$N$形状问题简化为一系列三形状匹配来实现，这一推导过程是我们的主要技术贡献。借助量子退火技术，针对中间$\mathcal{NP}$-难目标可以检索到低能量的高质量解。在基准数据集上，所提方法显著优于先前量子混合双形状匹配方法的多形状匹配扩展，且与经典多匹配方法性能相当。