Exact and Asymptotically Complete Robust Verifications of Neural Networks via Quantum Optimization

Abstract: Deep neural networks (DNNs) enable high performance across domains but remain vulnerable to adversarial perturbations, limiting their use in safety-critical settings. Here, we introduce two quantum-optimization-based models for robust verification that reduce the combinatorial burden of certification under bounded input perturbations. For piecewise-linear activations (e.g., ReLU and hardtanh), our first model yields an exact formulation that is sound and complete, enabling precise identification of adversarial examples. For general activations (including sigmoid and tanh), our second model constructs scalable over-approximations via piecewise-constant bounds and is asymptotically complete, with approximation error vanishing as the segmentation is refined. We further integrate Quantum Benders Decomposition with interval arithmetic to accelerate solving, and propose certificate-transfer bounds that relate robustness guarantees of pruned networks to those of the original model. Finally, a layerwise partitioning strategy supports a quantum--classical hybrid workflow by coupling subproblems across depth. Experiments on robustness benchmarks show high certification accuracy, indicating that quantum optimization can serve as a principled primitive for robustness guarantees in neural networks with complex activations.