  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
allows for some problems a more efficient search in the solution space.
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 implicitly enforces the QAP
constraints by updating the current estimates in a cyclic fashion. Further,
Q-Match can be applied 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.
