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The Journal of Physical Chemistry Letters

Machine Learning Accelerates Precise Excited-State Potential Energy Surface Calculations on a Quantum Computer

Qianjun Yao, Qun Ji, Xiaopeng Li, Yehui Zhang, Xinyu Chen, Ming‐Gang Ju, Jie Liu, Jinlan Wang

Published Jul 1, 2024• 6 citations
Publisher

Abstract

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.

DOI
10.1021/acs.jpclett.4c01445

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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.
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.
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.
The variational quantum eigensolver (VQE) is a widely employed method to solve electronic structure problems in the current noisy intermediate-scale quantum (NISQ) devices. However, due to inherent noise in the NISQ devices, VQE results on NISQ devices often deviate significantly from the results obtained on noiseless statevector simulators or traditional classical computers. The iterative nature of VQE further amplifies the errors in each loop. Recent works have explored ways to integrate deep neural networks (DNN) with VQE to mitigate iterative errors, albeit primarily limited to the noiseless statevector simulators. In this work, we trained DNN models across various quantum circuits and examined the potential of two DNN-VQE approaches, DNN1 and DNNF, for predicting the ground state energies of small molecules in the presence of device noise. We carefully examined the accuracy of the DNN1, DNNF, and VQE methods on both noisy simulators and real quantum devices by considering different ansatzes of varying qubit counts and circuit depths. Our results illustrate the advantages and limitations of both VQE and DNN-VQE approaches. Notably, both DNN1 and DNNF methods consistently outperform the standard VQE method in providing more accurate ground state energies in noisy environments. However, despite being more accurate than VQE, the energies predicted using these methods on real quantum hardware remain meaningful only at reasonable circuit depths (depth = 15, gates = 21). At higher depths (depth = 83, gates = 112), they deviate significantly from the exact results. Additionally, we find that DNNF does not offer any notable advantage over VQE in terms of speed. Consequently, our study recommends DNN1 as the preferred method for obtaining quick and accurate ground state energies of molecules on current quantum hardware, particularly for quantum circuits with lower depth and fewer qubits.
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.
Abstract Excited states of molecules lie in the heart of photochemistry and chemical reactions. The recent development in quantum computational chemistry leads to inventions of a variety of algorithms that calculate the excited states of molecules on near-term quantum computers, but they require more computational burdens than the algorithms for calculating the ground states. In this study, we propose a scheme of supervised quantum machine learning which predicts the excited-state properties of molecules only from their ground state wavefunction resulting in reducing the computational cost for calculating the excited states. Our model is comprised of a quantum reservoir and a classical machine learning unit which processes the measurement results of single-qubit Pauli operators with the output state from the reservoir. The quantum reservoir effectively transforms the single-qubit operators into complicated multi-qubit ones which contain essential information of the system, so that the classical machine learning unit may decode them appropriately. The number of runs for quantum computers is saved by training only the classical machine learning unit, and the whole model requires modest resources of quantum hardware that may be implemented in current experiments. We illustrate the predictive ability of our model by numerical simulations for small molecules with and without noise inevitable in near-term quantum computers. The results show that our scheme reproduces well the first and second excitation energies as well as the transition dipole moment between the ground states and excited states only from the ground states as inputs. We expect our contribution will enhance the applications of quantum computers in the study of quantum chemistry and quantum materials.