There is growing interest in solving computer vision problems such as mesh or point set alignment using Adiabatic Quantum Computing (AQC). Unfortunately, modern experimental AQC devices such as D-Wave only support Quadratic Unconstrained Binary Optimisation (QUBO) problems, which severely limits their applicability. This paper proposes a new way to overcome this limitation and introduces QuCOOP, an optimisation framework extending the scope of AQC to composite and binary-parametrised, possibly non-quadratic problems. The key idea of QuCOOP is to iteratively approximate the original objective function by a sequel of local (intermediate) QUBO forms, whose binary parameters can be sampled on AQC devices. We experiment with quadratic assignment problems, shape matching and point set registration without knowing the correspondences in advance. Our approach achieves state-of-the-art results across multiple instances of tested problems1.

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. Our source code is available at 4dqv.mpiinf.mpg.de/CCuantuMM/.

Finding shape correspondences can be formulated as an ${\mathcal{N}}{\mathcal{P}} - hard$ 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 α-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, al-lowing 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.

Matching problems on 3D shapes and images are challenging as they are frequently formulated as combinatorial quadratic assignment problems (QAPs) with permutation matrix constraints, which are NP-hard. In this work, we address such problems with emerging quantum computing technology and propose several reformulations of QAPs as unconstrained problems suitable for efficient execution on quantum hardware. We investigate several ways to inject permutation matrix constraints in a quadratic unconstrained binary optimization problem which can be mapped to quantum hardware. We focus on obtaining a sufficient spectral gap, which further increases the probability to measure optimal solutions and valid permutation matrices in a single run. We perform our experiments on the quantum computer D-Wave 2000Q(211qubits, adiabatic). Despite the observed discrepancy between simulated adiabatic quantum computing and execution on real quantum hardware, our reformulation of permutation matrix constraints increases the robustness of the numerical computations over other penalty approaches in our experiments. The proposed algorithm has the potential to scale to higher dimensions on future quantum computing architectures, which opens up multiple new directions for solving matching problems in 3D computer vision and graphics.