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arXiv:2206.03637

Exploring accurate potential energy surfaces via integrating variational quantum eigensovler with machine learning

Yanxian Tao, Xiongzhi Zeng, Fan Yi, Jie Liu, Zhenyu Li, Jinlong Yang

Published Jun 8, 2022• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2206.03637

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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.
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.
Electronically excited-state problems represent a crucial research field in quantum chemistry, closely related to numerous practical applications in photophysics and photochemistry. The emerging of quantum computing provides a promising computational paradigm to solve the Schrödinger equation for predicting potential energy surfaces (PESs). Here, we present a deep neural network model to predict parameters of the quantum circuits within the framework of variational quantum deflation and subspace search variational quantum eigensolver, which are two popular excited-state algorithms to implement on a quantum computer. The new machine learning-assisted algorithm is employed to study the excited-state PESs of small molecules, achieving highly accurate predictions. We then apply this algorithm to study the excited-state properties of the ArF system, which is essential to a gas laser. Through this study, we believe that with future advancements in hardware capabilities, quantum computing could be harnessed to solve excited-state problems for a broad range of systems.
Reflecting the increasing interest in quantum computing, the variational quantum eigensolver (VQE) has attracted much attentions as a possible application of near-term quantum computers. Although the VQE has often been applied to quantum chemistry, high computational cost is required for reliable results because infinitely many measurements are needed to obtain an accurate expectation value and the expectation value is calculated many times to minimize a cost function in the variational optimization procedure. Therefore, it is necessary to reduce the computational cost of the VQE for a practical task such as estimating the potential energy surfaces (PESs) with chemical accuracy, which is of particular importance for the analysis of molecular structures and chemical reaction dynamics. A hybrid quantum-classical neural network has recently been proposed for surrogate modeling of the VQE [Xia et alet\ alet al, Entropy 22, 828 (2020)]. Using the model, the ground state energies of a simple molecule such as H2 can be inferred accurately without the variational optimization procedure. In this study, we have extended the model by using the subspace-search variational quantum eigensolver procedure so that the PESs of the both ground and excited state can be inferred with chemical accuracy. We also demonstrate the effects of sampling noise on performance of the pre-trained model by using IBM's QASM backend.
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) of molecules with respect to their nuclear positions is a primary tool in understanding chemical reactions from first principles. However, obtaining this information is complicated by the fact that sampling a large number of ground states over a high-dimensional PES can require a vast number of state preparations. In this work, we propose using a generative quantum machine learning model to prepare quantum states at arbitrary points on the PES. The model is trained using quantum data consisting of ground-state wavefunctions associated with different classical nuclear coordinates. Our approach uses a classical neural network to convert the nuclear coordinates of a molecule into quantum parameters of a variational quantum circuit. The model is trained using a fidelity loss function to optimize the neural network parameters. We show that gradient evaluation is efficient and numerically demonstrate our method's ability to prepare wavefunctions on the PES of hydrogen chains, water, and beryllium hydride. In all cases, we find that a small number of training points are needed to achieve very high overlap with the groundstates in practice. From a theoretical perspective, we further prove limitations on these protocols by showing that if we were able to learn across an avoided crossing using a small number of samples, then we would be able to violate Grover's lower bound. Additionally, we prove lower bounds on the amount of quantum data needed to learn a locally optimal neural network function using arguments from quantum Fisher information. This work further identifies that quantum chemistry can be an important use case for quantum machine learning.
A faithful description of chemical processes requires exploring extended regions of the molecular potential energy surface (PES), which remains challenging for strongly correlated systems. Transferable deep-learning variational Monte Carlo (VMC) offers a promising route by efficiently solving the electronic Schrödinger equation jointly across molecular geometries at consistently high accuracy, yet its stochastic nature renders direct exploration of molecular configuration space nontrivial. Here, we present a framework for highly accurate ab initio exploration of PESs that combines transferable deep-learning VMC with a cost-effective estimation of energies, forces, and Hessians. By continuously sampling nuclear configurations during VMC optimization of electronic wave functions, we obtain transferable descriptions that achieve zero-shot chemical accuracy within chemically relevant distributions of molecular geometries. Throughout the subsequent characterization of molecular configuration space, the PES is evaluated only sparsely, with local approximations constructed by estimating VMC energies and forces at sampled geometries and aggregating the resulting noisy data using Gaussian process regression. Our method enables accurate and efficient exploration of complex PES landscapes, including structure relaxation, transition-state searches, and minimum-energy pathways, for both ground and excited states. This opens the door to studying bond breaking, formation, and large structural rearrangements in systems with pronounced multi-reference character.