Adiabatic Quantum Graph Matching with Permutation Matrix Constraints

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 (2^11 qubits, 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

QuAnt: Quantum Annealing with Learnt Couplings

Modern quantum annealers can find high-quality solutions to combinatorialoptimisation objectives given as quadratic unconstrained binary optimisation(QUBO) problems. Unfortunately, obtaining suitable QUBO forms in computervision remains challenging and currently requires problem-specific analyticalderivations. Moreover, such explicit formulations impose tangible constraintson solution encodings. In stark contrast to prior work, this paper proposes tolearn QUBO forms from data through gradient backpropagation instead of derivingthem. As a result, the solution encodings can be chosen flexibly and compactly.Furthermore, our methodology is general and virtually independent of thespecifics of the target problem type. We demonstrate the advantages of learntQUBOs on the diverse problem types of graph matching, 2D point cloud alignmentand 3D rotation estimation. Our results are competitive with the previousquantum state of the art while requiring much fewer logical and physicalqubits, enabling our method to scale to larger problems. The code and the newdataset will be open-sourced.<br

Q-{M}atch: {I}terative Shape Matching via Quantum Annealing

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