Formal Verification of Variational Quantum Circuits

Abstract: Variational quantum circuits (VQCs) are a central component of many quantum machine learning algorithms, offering a hybrid quantum-classical framework that, under certain aspects, can be considered similar to classical deep neural networks. A shared aspect is, for instance, their vulnerability to adversarial inputs, small perturbations that can lead to incorrect predictions. While formal verification techniques have been extensively developed for classical models, no comparable framework exists for certifying the robustness of VQCs. Here, we present the first in-depth theoretical and practical study of the formal verification problem for VQCs. Inspired by abstract interpretation methods used in deep learning, we analyze the applicability and limitations of interval-based reachability techniques in the quantum setting. We show that quantum-specific aspects, such as state normalization, introduce inter-variable dependencies that challenge existing approaches. We investigate these issues by introducing a novel semantic framework based on abstract interpretation, where the verification problem for VQCs can be formally defined, and its complexity analyzed. Finally, we demonstrate our approach on standard verification benchmarks.