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We propose and implement an interpretable machine learning classification model for Explainable AI (XAI) based on expressive Boolean formulas. Potential applications include credit scoring and diagnosis of medical conditions. The Boolean formula defines a rule with tunable complexity (or interpretability), according to which input data are classified. Such a formula can include any operator that can be applied to one or more Boolean variables, thus providing higher expressivity compared to more rigid rule-based and tree-based approaches. The classifier is trained using native local optimization techniques, efficiently searching the space of feasible formulas. Shallow rules can be determined by fast Integer Linear Programming (ILP) or Quadratic Unconstrained Binary Optimization (QUBO) solvers, potentially powered by special purpose hardware or quantum devices. We combine the expressivity and efficiency of the native local optimizer with the fast operation of these devices by executing non-local moves that optimize over subtrees of the full Boolean formula. We provide extensive numerical benchmarking results featuring several baselines on well-known public datasets. Based on the results, we find that the native local rule classifier is generally competitive with the other classifiers. The addition of non-local moves achieves similar results with fewer iterations, and therefore using specialized or quantum hardware could lead to a speedup by fast proposal of non-local moves., Comment: 28 pages, 16 figures, 4 tables

We propose and analyze the use of Bayesian optimization techniques to design quantum annealing schedules with minimal user and resource requirements. We showcase our scheme with results for two paradigmatic spin models. We find that Bayesian optimization is able to identify schedules resulting in fidelities several orders of magnitude better than standard protocols for both quantum and reverse annealing, as applied to the $p$-spin model. We also show that our scheme can help improve the design of hybrid quantum algorithms for hard combinatorial optimization problems, such as the maximum independent set problem, and illustrate these results via experiments on a neutral atom quantum processor available on Amazon Braket., Comment: 20 pages, 15 figures

We provide a comprehensive reply to the comment written by Stefan Boettcher [arXiv:2210.00623] and argue that the comment singles out one particular non-representative example problem, entirely focusing on the maximum cut problem (MaxCut) on sparse graphs, for which greedy algorithms are expected to perform well. Conversely, we highlight the broader algorithmic development underlying our original work, and (within our original framework) provide additional numerical results showing sizable improvements over our original data, thereby refuting the comment's original performance statements. Furthermore, it has already been shown that physics-inspired graph neural networks (PI-GNNs) can outperform greedy algorithms, in particular on hard, dense instances. We also argue that the internal (parallel) anatomy of graph neural networks is very different from the (sequential) nature of greedy algorithms, and (based on their usage at the scale of real-world social networks) point out that graph neural networks have demonstrated their potential for superior scalability compared to existing heuristics such as extremal optimization. Finally, we conclude highlighting the conceptual novelty of our work and outline some potential extensions., Comment: Manuscript: 2 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:2302.03602

We provide a comprehensive reply to the comment written by Chiara Angelini and Federico Ricci-Tersenghi [arXiv:2206.13211] and argue that the comment singles out one particular non-representative example problem, entirely focusing on the maximum independent set (MIS) on sparse graphs, for which greedy algorithms are expected to perform well. Conversely, we highlight the broader algorithmic development underlying our original work, and (within our original framework) provide additional numerical results showing sizable improvements over our original results, thereby refuting the comment's performance statements. We also provide results showing run-time scaling superior to the results provided by Angelini and Ricci-Tersenghi. Furthermore, we show that the proposed set of random d-regular graphs does not provide a universal set of benchmark instances, nor do greedy heuristics provide a universal algorithmic baseline. Finally, we argue that the internal (parallel) anatomy of graph neural networks is very different from the (sequential) nature of greedy algorithms and emphasize that graph neural networks have demonstrated their potential for superior scalability compared to existing heuristics such as parallel tempering. We conclude by discussing the conceptual novelty of our work and outline some potential extensions., Comment: Manuscript: 3 pages, 2 figures

We solve robot trajectory planning problems at industry-relevant scales. Our end-to-end solution integrates highly versatile random-key algorithms with model stacking and ensemble techniques, as well as path relinking for solution refinement. The core optimization module consists of a biased random-key genetic algorithm. Through a distinct separation of problem-independent and problem-dependent modules, we achieve an efficient problem representation, with a native encoding of constraints. We show that generalizations to alternative algorithmic paradigms such as simulated annealing are straightforward. We provide numerical benchmark results for industry-scale data sets. Our approach is found to consistently outperform greedy baseline results. To assess the capabilities of today's quantum hardware, we complement the classical approach with results obtained on quantum annealing hardware, using qbsolv on Amazon Braket. Finally, we show how the latter can be integrated into our larger pipeline, providing a quantum-ready hybrid solution to the problem., Comment: 17 pages, 6 figures

We show how graph neural networks can be used to solve the canonical graph coloring problem. We frame graph coloring as a multi-class node classification problem and utilize an unsupervised training strategy based on the statistical physics Potts model. Generalizations to other multi-class problems such as community detection, data clustering, and the minimum clique cover problem are straightforward. We provide numerical benchmark results and illustrate our approach with an end-to-end application for a real-world scheduling use case within a comprehensive encode-process-decode framework. Our optimization approach performs on par or outperforms existing solvers, with the ability to scale to problems with millions of variables., Comment: Manuscript: 8 pages, 5 figures, 2 tables. Supplemental Material: 1 page, 2 tables

Combinatorial optimization problems are pervasive across science and industry. Modern deep learning tools are poised to solve these problems at unprecedented scales, but a unifying framework that incorporates insights from statistical physics is still outstanding. Here we demonstrate how graph neural networks can be used to solve combinatorial optimization problems. Our approach is broadly applicable to canonical NP-hard problems in the form of quadratic unconstrained binary optimization problems, such as maximum cut, minimum vertex cover, maximum independent set, as well as Ising spin glasses and higher-order generalizations thereof in the form of polynomial unconstrained binary optimization problems. We apply a relaxation strategy to the problem Hamiltonian to generate a differentiable loss function with which we train the graph neural network and apply a simple projection to integer variables once the unsupervised training process has completed. We showcase our approach with numerical results for the canonical maximum cut and maximum independent set problems. We find that the graph neural network optimizer performs on par or outperforms existing solvers, with the ability to scale beyond the state of the art to problems with millions of variables., Comment: Manuscript: 13 pages, 5 figures, 1 table. Supplemental Material: 1 page, 1 table