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The Journal of Chemical Physics

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

S F Zhang, Zhen Liu, Zhenyu Li

Published May 4, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1063/5.0319519

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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.
The potential energy surface (PES) is crucial for interpreting a variety of chemical reaction processes. However, predicting accurate PESs with high-level electronic structure methods is a challenging task due to the high computational cost. As an appealing application of quantum computing, we show in this work that variational quantum algorithms can be integrated with machine learning (ML) techniques as a promising scheme for exploring accurate PESs. Different from using a ML model to represent the potential energy, we encode the molecular geometry information into a deep neural network (DNN) to represent parameters of the variational quantum eigensolver (VQE), leaving the PES to the wave function ansatz. Once the DNN model is trained, the variational optimization procedure that hinders the application of the VQE to complex systems is avoided, and thus the evaluation of PESs is significantly accelerated. Numerical results demonstrate that a simple DNN model is able to reproduce accurate PESs for small molecules.
The potential energy surface (PES) is crucial for interpreting a variety of chemical reaction processes. However, predicting accurate PESs with high-level electronic structure methods is a challenging task due to the high computational cost. As an appealing application of quantum computing, we show in this work that variational quantum algorithms can be integrated with machine learning (ML) techniques as a promising scheme for exploring accurate PESs. Different from using a ML model to represent the potential energy, we encode the molecular geometry information into a deep neural network (DNN) for representing parameters of the variational quantum eigensolver (VQE), leaving the PES to the wave function ansatz. Once the DNN model is trained, the variational optimization procedure that hinders the application of the VQE to complex systems is avoided and thus the evaluation of PESs is significantly accelerated. Numerical results demonstrate that a simple DNN model is able to reproduce accurate PESs for small molecules.
A longstanding computational challenge is the accurate simulation of many-body particle systems. Especially for deriving key characteristics of high-impact but complex systems such as battery materials and high entropy alloys (HEA). While simple models allow for simulations of the required scale, these methods often fail to capture the complex dynamics that determine the characteristics. A long-theorized approach is to use quantum computers for this purpose, which allows for a more efficient encoding of quantum mechanical systems. In recent years, the field of quantum computing has become significantly more mature. Furthermore, the rise in integration of machine learning with quantum computing further pushes to a near-term advantage. In this work we aim to improve the well-established quantum computing method for calculating the inter-atomic potential, the variational quantum eigensolver, by presenting an auto-encoded VQE with neural-network predictions: NN-AE-VQE. We apply a quantum autoencoder for a compressed quantum state representation of the atomic system, to which a naive circuit ansatz is applied. This reduces the number of circuit parameters to optimize, while still minimal reduction in accuracy. Additionally, we train a classical neural network to predict the circuit parameters to avoid computationally expensive parameter optimization. We demonstrate these methods on a H2 molecule, achieving chemical accuracy. We believe this method shows promise of efficiently capturing highly accurate systems while omitting current bottlenecks of variational quantum algorithms. Finally, we explore options for exploiting the algorithm structure and further algorithm improvements.
The variational quantum eigensolver (VQE) is a hybrid algorithm that has the potential to provide a quantum advantage in practical chemistry problems that are currently intractable on classical computers. VQE trains parameterized quantum circuits using a classical optimizer to approximate the eigenvalues and eigenstates of a given Hamiltonian. However, VQE faces challenges in task-specific design and machine-specific architecture, particularly when running on noisy quantum devices. This can have a negative impact on its trainability, accuracy, and efficiency, resulting in noisy quantum data. We propose variational denoising, an unsupervised learning method that employs a parameterized quantum neural network to improve the solution of VQE by learning from noisy VQE outputs. Our approach can significantly decrease energy estimation errors and increase fidelities with ground states compared to noisy input data for the H2 , LiH, and BeH2 molecular Hamiltonians and the transverse field Ising model. Surprisingly, it only requires noisy data for training. Variational denoising can be integrated into quantum hardware, increasing its versatility as an end-to-end quantum processing for quantum data. Published by the American Physical Society 2024
Variational Quantum Eigensolvers (VQEs) are a leading class of noisy intermediate-scale quantum (NISQ) algorithms with broad applications in quantum physics and quantum chemistry. However, as system size increases, VQE optimization is increasingly hindered by the barren plateau phenomenon, where gradients vanish and the loss function becomes trapped in local minima. While machine learning-based parameter initialization methods have been proposed to address this challenge, they often show limited effectiveness in complex VQE problems. This is primarily due to their inadequate ability to model the intricate correlations embedded in the Hamiltonian structure and the associated ansatz circuits. In this paper, we propose \textit{Qracle}, a graph neural network (GNN)-based parameter initializer for VQEs. \textit{Qracle} systematically encodes both the Hamiltonian and the associated ansatz circuit into a unified graph representation and leverages a GNN to learn a mapping from VQE problem graphs to optimized ansatz parameters. Compared to state-of-the-art initialization techniques, \textit{Qracle} achieves a reduction in initial loss of up to 10.8610.8610.86, accelerates convergence by decreasing optimization steps by up to 64.42%64.42\%64.42%, and improves final performance with up to a 26.43%26.43\%26.43% reduction in Symmetric Mean Absolute Percentage Error (SMAPE).
We introduce a general framework called neural-network- (NN) encoded variational quantum algorithms (VQAs), or NNVQA for short, to address the challenges of implementing VQAs on noisy intermediate-scale quantum (NISQ) computers. Specifically, NNVQA feeds input (such as parameters of a Hamiltonian) from a given problem to a neural network and uses its outputs to parameterize an ansatz circuit for the standard VQA. Combining the strengths of NN and parameterized quantum circuits, NNVQA can accelerate the training process of VQAs and handle a broad family of related problems with varying input parameters with the pretrained NN. To concretely illustrate the merits of NNVQA, we present results on a NN variational quantum eigensolver (VQE) for solving the ground state of parameterized XXZXXZXXZ spin models in one and two dimensions. Our results demonstrate that NNVQE is able to estimate the ground-state energies of parameterized Hamiltonians with high precision without fine tuning, and significantly reduce the overall training cost to estimate ground-state properties across the phases of the given Hamiltonian. We also employ an active learning strategy to further increase the training efficiency while maintaining prediction accuracy. These encouraging results demonstrate that NNVQAs offer an alternative hybrid quantum-classical paradigm to utilize NISQ resources for solving more realistic and challenging computational problems.