Variational Monte Carlo Calculations of A≤4 Nuclei with an Artificial Neural-Network Correlator Ansatz

Second-order optimization strategies for neural network quantum states

The Variational Monte Carlo (VMC) method has recently seen important advances through the use of neural network quantum states. While more and more sophisticated ansatze have been designed to tackle a wide variety of quantum many-body problems, modest progress has been made on the associated optimization algorithms. In this work, we revisit the Kronecker-Factored Approximate Curvature (KFAC), an optimizer that has been used extensively in a variety of simulations. We suggest improvements in the scaling and the direction of this optimizer and find that they substantially increase its performance at a negligible additional cost. We also reformulate the VMC approach in a game theory framework, to propose a novel optimizer based on decision geometry. We find that on a practical test case for continuous systems, this new optimizer consistently outperforms any of the KFAC improvements in terms of stability, accuracy and speed of convergence. Beyond VMC, the versatility of this approach suggests that decision geometry could provide a solid foundation for accelerating a broad class of machine learning algorithms. This article is part of the theme issue 'The liminal position of Nuclear Physics: from hadrons to neutron stars'.

Improved optimization for the neural-network quantum states and tests on the chromium dimer

The advent of Neural-network Quantum States (NQS) has significantly advanced wave function ansatz research, sparking a resurgence in orbital space variational Monte Carlo (VMC) exploration. This work introduces three algorithmic enhancements to reduce computational demands of VMC optimization using NQS: an adaptive learning rate algorithm, constrained optimization, and block optimization. We evaluate the refined algorithm on complex multireference bond stretches of H2O and N2 within the cc-pVDZ basis set and calculate the ground-state energy of the strongly correlated chromium dimer (Cr2) in the Ahlrichs SV basis set. Our results achieve superior accuracy compared to coupled cluster theory at a relatively modest CPU cost. This work demonstrates how to enhance optimization efficiency and robustness using these strategies, opening a new path to optimize large-scale restricted Boltzmann machine-based NQS more effectively and marking a substantial advancement in NQS's practical quantum chemistry applications.

Ab initio quantum chemistry with neural-network wavefunctions

Deep learning methods outperform human capabilities in pattern recognition and data processing problems and now have an increasingly important role in scientific discovery. A key application of machine learning in molecular science is to learn potential energy surfaces or force fields from ab initio solutions of the electronic Schrödinger equation using data sets obtained with density functional theory, coupled cluster or other quantum chemistry (QC) methods. In this Review, we discuss a complementary approach using machine learning to aid the direct solution of QC problems from first principles. Specifically, we focus on quantum Monte Carlo methods that use neural-network ansatzes to solve the electronic Schrödinger equation, in first and second quantization, computing ground and excited states and generalizing over multiple nuclear configurations. Although still at their infancy, these methods can already generate virtually exact solutions of the electronic Schrödinger equation for small systems and rival advanced conventional QC methods for systems with up to a few dozen electrons.

Variational Neural-Network Ansatz for Continuum Quantum Field Theory

Physicists dating back to Feynman have lamented the difficulties of applying the variational principle to quantum field theories. In nonrelativistic quantum field theories, the challenge is to parametrize and optimize over the infinitely many n-particle wave functions comprising the state's Fock-space representation. Here we approach this problem by introducing neural-network quantum field states, a deep learning ansatz that enables application of the variational principle to nonrelativistic quantum field theories in the continuum. Our ansatz uses the Deep Sets neural network architecture to simultaneously parametrize all of the n-particle wave functions comprising a quantum field state. We employ our ansatz to approximate ground states of various field theories, including an inhomogeneous system and a system with long-range interactions, thus demonstrating a powerful new tool for probing quantum field theories.

Stochastic representation of many-body quantum states

The quantum many-body problem is ultimately a curse of dimensionality: the state of a system with many particles is determined by a function with many dimensions, which rapidly becomes difficult to efficiently store, evaluate and manipulate numerically. On the other hand, modern machine learning models like deep neural networks can express highly correlated functions in extremely large-dimensional spaces, including those describing quantum mechanical problems. We show that if one represents wavefunctions as a stochastically generated set of sample points, the problem of finding ground states can be reduced to one where the most technically challenging step is that of performing regression-a standard supervised learning task. In the stochastic representation the (anti)symmetric property of fermionic/bosonic wavefunction can be used for data augmentation and learned rather than explicitly enforced. We further demonstrate that propagation of an ansatz towards the ground state can then be performed in a more robust and computationally scalable fashion than traditional variational approaches allow.

Revealing the nature of hidden charm pentaquarks with machine learning

We study the nature of the hidden charm pentaquarks, i.e., the P_c4312,P_c4440 and P_c(4457), with a neural network approach in pionless effective field theory. In this framework, the normal χ² fitting approach cannot distinguish the quantum numbers of the P_c(4440) and P_c(4457). In contrast to that, the neural network-based approach can discriminate them, which still cannot be seen as a proof of the spin of the states since pion exchange is not considered in the approach. In addition, we also illustrate the role of each experimental data bin of the invariant J/ψp mass distribution on the underlying physics in both neural network and fitting methods. Their similarities and differences demonstrate that neural network methods can use data information more effectively and directly. This study provides more insights about how the neural network-based approach predicts the nature of exotic states from the mass spectrum.

Solving the nuclear pairing model with neural network quantum states

We present a variational Monte Carlo method that solves the nuclear many-body problem in the occupation number formalism exploiting an artificial neural network representation of the ground-state wave function. A memory-efficient version of the stochastic reconfiguration algorithm is developed to train the network by minimizing the expectation value of the Hamiltonian. We benchmark this approach against widely used nuclear many-body methods by solving a model used to describe pairing in nuclei for different types of interaction and different values of the interaction strength. Despite its polynomial computational cost, our method outperforms coupled-cluster and provides energies that are in excellent agreement with the numerically exact full configuration-interaction values.

Orbital Mixer: Using Atomic Orbital Features for Basis-Dependent Prediction of Molecular Wavefunctions

Leveraging ab initio data at scale has enabled the development of machine learning models capable of extremely accurate and fast molecular property prediction. A central paradigm of many previous studies focuses on generating predictions for only a fixed set of properties. Recent lines of research instead aim to explicitly learn the electronic structure via molecular wavefunctions, from which other quantum chemical properties can be directly derived. While previous methods generate predictions as a function of only the atomic configuration, in this work we present an alternate approach that directly purposes basis-dependent information to predict molecular electronic structure. Our model, Orbital Mixer, is composed entirely of multi-layer perceptrons (MLPs) using MLP-Mixer layers within a simple, intuitive, and scalable architecture that achieves competitive Hamiltonian and molecular orbital energy and coefficient prediction accuracies compared to the state-of-the-art.

Neural-Network Quantum States for Spin-1 Systems: Spin-Basis and Parameterization Effects on Compactness of Representations

Neural network quantum states (NQS) have been widely applied to spin-1/2 systems, where they have proven to be highly effective. The application to systems with larger on-site dimension, such as spin-1 or bosonic systems, has been explored less and predominantly using spin-1/2 Restricted Boltzmann Machines (RBMs) with a one-hot/unary encoding. Here, we propose a more direct generalization of RBMs for spin-1 that retains the key properties of the standard spin-1/2 RBM, specifically trivial product states representations, labeling freedom for the visible variables and gauge equivalence to the tensor network formulation. To test this new approach, we present variational Monte Carlo (VMC) calculations for the spin-1 anti-ferromagnetic Heisenberg (AFH) model and benchmark it against the one-hot/unary encoded RBM demonstrating that it achieves the same accuracy with substantially fewer variational parameters. Furthermore, we investigate how the hidden unit complexity of NQS depend on the local single-spin basis used. Exploiting the tensor network version of our RBM we construct an analytic NQS representation of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state in the xyz spin-1 basis using only M=2N hidden units, compared to M∼O(N2) required in the Sz basis. Additional VMC calculations provide strong evidence that the AKLT state in fact possesses an exact compact NQS representation in the xyz basis with only M=N hidden units. These insights help to further unravel how to most effectively adapt the NQS framework for more complex quantum systems.

Convergence to the fixed-node limit in deep variational Monte Carlo

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.