Finding shape correspondences can be formulated as an NP-hard quadratic assignment problem (QAP) that becomes infeasible for shapes with high sampling density. A promising research direction is to tackle such quadratic optimization problems over binary variables with quantum annealing, which, in theory, allows to find globally optimal solutions relying on a new computational paradigm. Unfortunately, enforcing the linear equality constraints in QAPs via a penalty significantly limits the success probability of such methods on currently available quantum hardware. To address this limitation, this paper proposes Q-Match, i.e., a new iterative quantum method for QAPs inspired by the alpha-expansion algorithm, which allows solving problems of an order of magnitude larger than current quantum methods. It works by implicitly enforcing the QAP constraints by updating the current estimates in a cyclic fashion. Further, Q-Match can be applied for shape matching problems iteratively, on a subset of well-chosen correspondences, allowing us to scale to real-world problems. Using the latest quantum annealer, the D-Wave Advantage, we evaluate the proposed method on a subset of QAPLIB as well as on isometric shape matching problems from the FAUST dataset.

中文翻译：

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Q匹配：通过量子退火的迭代形状匹配

可以将找到形状对应关系公式化为NP困难的二次分配问题（QAP），该问题对于具有高采样密度的形状变得不可行。一个有前途的研究方向是用量子退火解决二进制变量上的此类二次优化问题，从理论上讲，这可以依靠新的计算范式找到全局最优解。不幸的是，通过惩罚在QAP中强制执行线性相等约束会大大限制此类方法在当前可用的量子硬件上的成功概率。为了解决这个限制，本文提出了一种Q-Match，即一种受α扩展算法启发的QAP迭代量子方法，它可以解决比当前量子方法大一个数量级的问题。它通过以循环方式更新当前估算值来隐式强制QAP约束，从而发挥作用。此外，Q-Match可以在精心选择的对应子集上迭代地应用于形状匹配问题，从而使我们能够扩展到实际问题。使用最新的量子退火器D-Wave Advantage，我们在QAPLIB的子集以及FAUST数据集中的等距形状匹配问题上评估了所提出的方法。