Integrating quantum neural networks with the variational quantum eigensolver to calculate nonadiabatic coupling vectors.

Machine learning nonadiabatic coupling vectors (NACVs) is challenging due to the localized value problem and the sign problem. In this study, we integrate quantum neural networks (QNNs) with the variational quantum eigensolver (VQE) to predict NACVs at different molecular geometries. Parameterized quantum circuits provide a compact and expressive representation of wavefunctions, and VQE offers an efficient way of optimizing such circuit-based Ansätze. Instead of optimizing them at all geometries, QNNs are used to learn parameters of the VQE wavefunctions. Then, NACVs are directly computed from the wavefunctions. In order to meet the high fidelity requirement of wavefunctions for accurate NACV calculations, we introduce a bootstrap optimization procedure in pre-training of the model that supplies robust initial parameters obtained by sequentially scanning the potential energy surface (PES). We demonstrate that this QNN-VQE framework effectively circumvents the localized value and sign problems, providing a unified and efficient protocol for the simultaneous determination of PESs and NACVs.