Variational quantum Monte Carlo (QMC) is an ab initio method for solving the electronic Schrödinger equation that is exact in principle, but limited by the flexibility of the available Ansätze in practice. The recently introduced deep QMC approach, specifically two deep-neural-network Ansätze PauliNet and FermiNet, allows variational QMC to reach the accuracy of diffusion QMC, but little is understood about the convergence behavior of such Ansätze. 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 (MF) complete-basis-set limit. Moving to electron correlation, we then perform an extensive hyperparameter scan of a deep Jastrow factor for LiH and H4 and find that variational energies at the fixed-node limit can be obtained with a sufficiently large network. Finally, we benchmark MF and many-body Ansätze on H2O, increasing the fraction of recovered fixed-node correlation energy of single-determinant Slater–Jastrow-type Ansätze by half an order of magnitude compared to previous variational QMC results, and demonstrate that a single-determinant Slater–Jastrow-backflow version of the Ansatz overcomes the fixed-node limitations. This analysis helps understand the superb accuracy of deep variational Ansätze in comparison to the traditional trial wavefunctions at the respective level of theory and will guide future improvements of the neural-network architectures in deep QMC.
Variational quantum Monte Carlo (QMC) is an ab initio method for solving the electronic Schrödinger equation that is exact in principle, but limited by the flexibility of the available Ansätze in practice. The recently introduced deep QMC approach, specifically two deep-neural-network Ansätze PauliNet and FermiNet, allows variational QMC to reach the accuracy of diffusion QMC, but little is understood about the convergence behavior of such Ansätze. 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 (MF) complete-basis-set limit. Moving to electron correlation, we then perform an extensive hyperparameter scan of a deep Jastrow factor for LiH and H4 and find that variational energies at the fixed-node limit can be obtained with a sufficiently large network. Finally, we benchmark MF and many-body Ansätze on H2O, increasing the fraction of recovered fixed-node correlation energy of single-determinant Slater–Jastrow-type Ansätze by half an order of magnitude compared to previous variational QMC results, and demonstrate that a single-determinant Slater–Jastrow-backflow version of the Ansatz overcomes the fixed-node limitations. This analysis helps understand the superb accuracy of deep variational Ansätze in comparison to the traditional trial wavefunctions at the respective level of theory and will guide future improvements of the neural-network architectures in deep QMC.
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 Schrödinger equation that scales favorably for large systems is variational quantum Monte Carlo (QMC). The recently introduced deep QMC approach uses ansatzes represented by deep neural networks and generates 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 extend one such ansatz (PauliNet) to compute electronic excited states. We demonstrate our method on various small atoms and molecules and consistently achieve high accuracy for low-lying states. To highlight the method's potential, we compute the first excited state of the much larger benzene molecule, as well as the conical intersection of ethylene, with PauliNet matching results of more expensive high-level methods.
The solution of Schrödinger equation generates the quantisation properties of the system, which can completely describe the quantum behaviour of microscopic particles in a physical system. However, solving the Schrödinger equation is a non-deterministic polynomial time (NP)-hard problem. Numerical methods for solving the Schrödinger equation require careful selection of basis configurations, and the complete basis set has high accuracy but its complexity increases exponentially in the number of particles. Therefore, it is crucial to find an effective numerical method. To solve this problem, this paper present a deep learning architecture, VMCNet, using the powerful computational efficiency of neural networks to improve the speed of numerical computation. Moreover, this paper proposes a suitable wavefunction ansatz, witch determines the accuracy of neural networks, that achieves more accurate energy solutions of the many-electron Schrödinger equation compared to the conventional single-determinant variational molecular Monte Carlo. We introduce non-local and local blocks to represent quantum mechanical interactions, which contain more physical information about the particle system. The experimental results show that VMCNet can cover more than 93% of the correlation energy of the main group elements (Be–Ne) for the monoatomic system and more than 90% of the correlation energy for the diatomic system.
With the ability to solve the many-body Schrödinger equation accurately, in principle all physics and chemistry could be derived from first principles. However, exact wave functions of realistic and interesting systems are in general unavailable because they are non-deterministic polynomial-hard to compute (Troyer, 2005). This implies that we need to rely on approximations. The variational Monte Carlo (VMC) method is widely used for ground state studies, but requires a trial wave function ansatz which must trade off between efficiency and accuracy. The method also has many common features with machine learning algorithms, and as neural networks have shown impressive power as function approximators, the idea is to use a neural network as the trial wave function guess. For fermionic systems, like electronic structure systems, the wave function needs to obey Fermi-Dirac statistics. This is typically achieved using a Slater determinant. As a neural network hardly can model this feature, our approach is to replace the single-particle functions in the Slater determinant with restricted Boltzmann machines, getting the RBM ansatz. In addition, we add further correlations via so-called Jastrow factors (Drummond, 2004). The de facto standard trial wave function ansatz for electronic structure calculations is the Slater-Jastrow ansatz, which was implemented as a reference. Our primary focus is on closed-shell circular quantum dots, where we compute the ground state energy and electron density of two-dimensional systems with up to 90 electrons and three-dimensional systems with up to 70 electrons. The energy obtained by the RBM ansatz was reasonably close to experimental results, and it gradually became closer as we added more complex correlation factors. For our most complicated Jastrow factor, the energy was found to be lower than the energy provided by the Slater-Jastrow ansatz for small dots, but for larger dots it was slightly higher. However, the one-body density profile reveals that the RBM ansatz gives more distinctly located electrons compared to the Slater-Jastrow ansatz. This can be explained by the way the RBM ansatz models the correlations. From the two-body density profile, we also observe that the repulsive interactions get more significant as we add a Jastrow factor. Based on the electron densities and the energy distribution between kinetic and potential energy, it is certain that the various methods provide different electron configurations. For low-frequency dots, the electron density becomes more localized with an additional radial peak compared to high-frequency dots. This is reminiscent of what is known as Wigner localization (Ghosal, 2007). The computational time consumption was found to be favorable for the RBM ansatz for small systems and the Slater-Jastrow ansatz for large systems. This can be explained by the exploding number of variational parameters in the RBM ansatz as the system sizes increase. The RBM ansatz with Jastrow factors were notably more computationally intensive than the other ansätze, and evidently, there is no reason to use the simplest Jastrow factor when more complicated Jastrow factors are available.
Given access to accurate solutions of the many-electron Schrödinger equation, nearly all chemistry could be derived from first principles.Exact wave functions of interesting chemical systems are out of reach because they are NP-hard to compute in general, but approximations can be found using polynomially scaling algorithms.The key challenge for many of these algorithms is the choice of wave function approximation, or Ansatz, which must trade off between efficiency and accuracy.Neural networks have shown impressive power as accurate practical function approximators and promise as a compact wave-function Ansatz for spin systems, but problems in electronic structure require wave functions that obey Fermi-Dirac statistics.Here we introduce a novel deep learning architecture, the Fermionic neural network, as a powerful wave-function Ansatz for many-electron systems.The Fermionic neural network is able to achieve accuracy beyond other variational quantum Monte Carlo Ansatz on a variety of atoms and small molecules.Using no data other than atomic positions and charges, we predict the dissociation curves of the nitrogen molecule and hydrogen chain, two challenging strongly correlated systems, to significantly higher accuracy than the coupled cluster method, widely considered the most accurate scalable method for quantum chemistry at equilibrium geometry.This demonstrates that deep neural networks can improve the accuracy of variational quantum Monte Carlo to the point where it outperforms other ab initio quantum chemistry methods, opening the possibility of accurate direct optimization of wave functions for previously intractable many-electron systems.
Simulating strongly correlated fermionic systems remains a fundamental challenge in quantum physics, largely due to the sign problem in quantum Monte Carlo (QMC) methods. We present a neural network-based variational Monte Carlo (NN-VMC) approach, leveraging a flexible neural network ansatz to represent the many-body wavefunction. Focusing on quantum dots with up to 30 electrons, we demonstrate that NN-VMC significantly reduces variational bias and achieves ground-state energies surpassing those of fixed-node diffusion Monte Carlo (DMC). A key feature is that the neural network adaptively learns and optimizes nodal structures during energy minimization. We provide qualitative insights into the nodal structure of fermionic wavefunctions by comparing the nodal structures generated by NN-VMC with those obtained from traditional trial functions. Additionally, we reveal spin-resolved radial distributions and electron density profiles, highlighting the versatility and accuracy of NN-VMC. This work underscores the potential of machine learning to advance quantum simulations and deepen our understanding of strongly correlated systems.
Solutions to the Schrodinger equation can be used to predict the electronic structure of molecules and materials and therefore infer their complex physical and chemical properties. Variational Quantum Monte Carlo (VMC) is a technique that can be used to solve the weak form of the Schrodinger equation. Applying VMC to systems with N electrons involves evaluating the determinant of an N by N matrix. The evaluation of this determinant scales as O(N3) and is the main computational cost in the VMC process. In this work, we investigate an alternative VMC technique based on the Vandermonde determinant. The Vandermonde determinant is a product of pairwise differences and so evaluating it scales as O(N2). Therefore, this approach reduces the computational cost by a factor of N. The Vandermonde determinant was implemented in PyTorch and the performance was assessed in approximating the ground state energy of various quantum systems against existing techniques. The performance is evaluated in a variety of systems, starting with the one-dimensional particle in a box, and then considering more complicated atomic systems with multiple particles. The Vandermonde determinant was also implemented in PauliNet, a deep-learning architecture for VMC. The new method is shown to be computationally efficient, and results in a speed-up as large as 5X. In these cases, the new ansatz obtains a reasonable approximation for wavefunctions of atomic systems, but does not reach the accuracy of the Hartree-Fock method that relies on the Slater determinant. It is observed that while the use of neural networks in VMC can result in highly accurate solutions, further work is necessary to determine an appropriate balance between computational time and accuracy.