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Jan Hermann, James Spencer, Kenny Choo, Antonio Mezzacapo, W. M. C. Foulkes, David Pfau, Giuseppe Carleo, Frank Noé

Machine learning and specifically deep-learning methods have outperformed human capabilities in many pattern recognition and data processing problems, in game playing, and now also play an increasingly important role in scientific discovery. A key application of machine learning in the molecular sciences is to learn potential energy surfaces or force fields from ab-initio solutions of the electronic Schr\"odinger equation using datasets obtained with density functional theory, coupled cluster, or other quantum chemistry methods. Here we review a recent and complementary approach: using machine learning to aid the direct solution of quantum chemistry problems from first principles. Specifically, we focus on quantum Monte Carlo (QMC) methods that use neural network ansatz functions in order to solve the electronic Schr\"odinger equation, both in first and second quantization, computing ground and excited states, and generalizing over multiple nuclear configurations. Compared to existing quantum chemistry methods, these new deep QMC methods have the potential to generate highly accurate solutions of the Schr\"odinger equation at relatively modest computational cost.

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Obtaining accurate ground and low-lying excited states of electronic systems is crucial in a multitude of important applications. One ab initio method for solving the electronic Schr\"odinger equation that scales favorably for large systems and whose accuracy is limited only by the choice of wavefunction ansatz employed is variational quantum Monte Carlo (QMC). The recently introduced deep QMC approach, using a new class of ansatzes represented by deep neural networks, has been shown to generate nearly exact ground-state solutions for molecules containing up to a few dozen electrons, with the potential to scale to much larger systems where other highly accurate methods are not feasible. In this paper, we advance one such ansatz (PauliNet) to compute electronic excited states through a simple variational procedure. We demonstrate our method on a variety of small atoms and molecules where we consistently achieve high accuracy for low-lying states. To highlight the method's potential for larger systems, we show that for the benzene molecule, PauliNet is on par with significantly more expensive high-level electronic structure methods in terms of the excitation energy and outperforms them in terms of absolute energies.

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Variational quantum Monte Carlo (QMC) is an ab-initio method for solving the electronic Schr\"odinger equation that is exact in principle, but limited by the flexibility of the available ansatzes in practice. The recently introduced deep QMC approach, specifically two deep-neural-network ansatzes PauliNet and FermiNet, allows variational QMC to reach the accuracy of diffusion QMC, but little is understood about the convergence behavior of such ansatzes. Here, we analyze how deep variational QMC approaches the fixed-node limit with increasing network size. First, we demonstrate that a deep neural network can overcome the limitations of a small basis set and reach the mean-field complete-basis-set limit. Moving to electron correlation, we then perform an extensive hyperparameter scan of a deep Jastrow factor for LiH and H$_4$ and find that variational energies at the fixed-node limit can be obtained with a sufficiently large network. Finally, we benchmark mean-field and many-body ansatzes on H$_2$O, increasing the fraction of recovered fixed-node correlation energy by half an order of magnitude compared to previous VMC results. This analysis helps understanding the superb performance of deep variational ansatzes, and will guide future improvements of the neural network architectures in deep QMC.

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The electronic Schr\"odinger equation describes fundamental properties of molecules and materials, but cannot be solved exactly for larger systems than the hydrogen atom. Quantum Monte Carlo is a suitable method when high-quality approximations are sought, and its accuracy is in principle limited only by the flexibility of the used wave-function ansatz. Here we develop a deep-learning wave-function ansatz, dubbed PauliNet, which has the Hartree-Fock solution built in as a baseline, incorporates the physics of valid wave functions, and is trained using variational quantum Monte Carlo (VMC). Our deep-learning method achieves higher accuracy than comparable state-of-the-art VMC ansatzes for atoms, diatomic molecules and a strongly-correlated hydrogen chain. We anticipate that this method can reveal new physical insights and provide guidance for the design of molecules and materials where highly accurate quantum-mechanical solutions are needed, such as in transition metals and other strongly correlated systems.

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