Neural network quantum states analysis of the Shastry-Sutherland model

Neural-network quantum states (NQS) has emerged as a powerful application of quantum-inspired deep learning for variational Monte Carlo methods, offering a competitive alternative to existing techniques for identifying ground states of quantum problems. A significant advancement toward improving the practical scalability of NQS has been the incorporation of autoregressive models, most recently transformers, as variational ansätze. Transformers learn sequence information with greater expressiveness than recurrent models, but at the cost of increased time complexity with respect to sequence length. We explore the use of the retentive network (RetNet), a recurrent alternative to transformers, as an ansatz for solving electronic ground state problems in ab initio quantum chemistry. Unlike transformers, RetNets overcome this time complexity bottleneck by processing data in parallel during training, and recurrently during inference. We give a simple computational cost estimate of the RetNet and directly compare it with similar estimates for transformers, establishing a clear threshold ratio of problem-to-model size past which the RetNet’s time complexity outperforms that of the transformer. Though this efficiency comes at the expense of decreased expressiveness relative to the transformer, we overcome this gap through training strategies that leverage the autoregressive structure of the model—namely, variational neural annealing. Our findings support the RetNet as a means of improving the time complexity of NQS without sacrificing accuracy. We provide further evidence that the ablative improvements of neural annealing extend beyond the RetNet architecture, suggesting it would serve as an effective general training strategy for autoregressive NQS.

To distill the essential elements of nuclear binding, we seek the simplest Hamiltonian capable of modeling atomic nuclei with percent-level accuracy. A critical aspect of this endeavor consists of accurately solving the quantum many-body problem without incurring an exponential computing cost with the number of nucleons. We address this challenge by leveraging a variational MonteCarlo method based on a highly expressive neural-network quantum state ansatz. In addition to computing binding energies and charge radii of nuclei with up to A=20 nucleons, by evaluating their magnetic moments, we demonstrate that neural-network quantum states are able to correctly capture the self-emerging nuclear shell structure. To this end, we introduce a novel computational protocol based on adding an external magnetic field to the nuclear Hamiltonian, which allows the neural network to learn the preferred polarization of the nucleus within the given magnetic field.

Neural network quantum states such as ansatz wave functions have shown a great deal of promise for finding the ground state of spin models. Recently, work has focused on extending this idea to mixed states for simulating the dynamics of open systems. Most approaches so far have used a purification ansatz where a copy of the system Hilbert space is added, which when traced out gives the correct density matrix. Here we instead present an extension of the restricted Boltzmann machine which directly represents the density matrix in Liouville space. This allows the compact representation of states which appear in mean-field theory. We benchmark our approach on two different versions of the dissipative transverse-field Ising model, which show our ansatz is able to compete with other state-of-the-art approaches. Published by the American Physical Society 2024

Neural-network architectures have been increasingly used to represent quantum many-body wave functions. These networks require a large number of variational parameters and are challenging to optimize using traditional methods, as gradient descent. Stochastic reconfiguration (SR) has been effective with a limited number of parameters, but becomes impractical beyond a few thousand parameters. Here, we leverage a simple linear algebra identity to show that SR can be employed even in the deep learning scenario. We demonstrate the effectiveness of our method by optimizing a Deep Transformer architecture with 3 × 105 parameters, achieving state-of-the-art ground-state energy in the J1–J2 Heisenberg model at J2/J1 = 0.5 on the 10 × 10 square lattice, a challenging benchmark in highly-frustrated magnetism. This work marks a significant step forward in the scalability and efficiency of SR for neural-network quantum states, making them a promising method to investigate unknown quantum phases of matter, where other methods struggle.

Motivated by the Carleo’s work (Science, 2017, 355: 602), we focus on finding the neural network quantum statesapproximation of the unknown ground state of a given Hamiltonian H in terms of the best relative error and explore the influences of sum, tensor product, local unitary of Hamiltonians on the best relative error. Besides, we illustrate our method with some examples.

Solving quantum many-body problems is one of the fundamental challenges in quantum chemistry. While neural network quantum states (NQS) have emerged as a promising computational tool, its training process incurs exponentially growing computational demands, becoming prohibitively expensive for large-scale molecular systems and creating fundamental scalability barriers for real-world applications. To address the above challenges, we present QChem-Trainer, a highperformance NQS training framework for ab initio electronic structure calculations. First, we propose a scalable sampling parallelism strategy with multi-layer workload division and hybrid sampling scheme, which breaks the scalability barriers in large-scale NQS training. Second, we introduce multi-level energy calculation parallelism, enabling more efficient local energy computation in NQS training. Last, we employ cachecentric optimization for transformer-based ansatz and incorporate it with sampling parallelism strategy, which further accelerates the NQS training and achieves stable memory footprints at scale. Experiments demonstrate that QChem-Trainer provides up to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$8.41 \times$</tex> speedup in NQS training and attains a parallel efficiency of up to 95.8% when scaling to 1,536 nodes.

We propose a new variational scheme based on the neural-network quantum states to simulate the stationary states of open quantum many-body systems. Using the high expressive power of the variational ansatz described by the restricted Boltzmann machines, which we dub as the neural stationary state ansatz, we compute the stationary states of quantum dynamics obeying the Lindblad master equations. The mapping of the stationary-state search problem into finding a zero-energy ground state of an appropriate Hermitian operator allows us to apply the conventional variational Monte Carlo method for the optimization. Our method is shown to simulate various spin systems efficiently, i.e., the transverse-field Ising models in both one and two dimensions and the XYZ model in one dimension.

ABSTRACTBy using the variational Monte Carlo (VMC) method, we calculated the 1sσg-state energies, the dissociation energies, and the binding energies of the hydrogen molecule and its molecular ion in the presence of an aligned magnetic field regime between 0 and 10 a.u. The present calculations are based on using two types of compact and accurate trial wave functions, which are put forward for consideration in calculating energies in the absence of a magnetic field. The obtained results are compared with the most recent accurate values. We conclude that the applications of the VMC method can be successfully extended to cover the case of molecules under the effect of a magnetic field.

In this work, we study two examples of frustrated magnetic systems whose degrees of freedom are spins (or pseudo-spins) on two-dimensional lattices. The first part presents the results obtained for the quantum compass model on a square lattice. This model is a minimal model for the orbital degrees of freedom of transition metal compounds. We use exact diagonalizations, quantum Monte Carlo simulations as well as a perturbative approach in order to study the ground state degeneracy in the thermodynamic limit and the type of order in the ground state. The second part of this work is devoted to the study of the magnetic properties of SrCu2(BO3)2, which is one of the few quasi two-dimensional quantum magnets whose magnetization curve as a function of the magnetic field shows plateaus. The magnetic degrees of freedom are described by a spin 1/2 Heisenberg model on the Shastry-Sutherland lattice in the presence of an external magnetic field. In order to study the magnetic properties at zero-temperature, we use perturbative continuous unitary transformations to derive an effective Hamiltonian in which the triplets Sz = 1 are particles moving on a background of singlets. This Hamiltonian is characterized by a kinetic energy dominated by correlated hopping, which let a particle hop only if there is another particle nearby. In order to better understand the effect of the correlated hopping, we start by studying a minimal model. By using exact diagonalizations, quantum Monte Carlo and a semi-classical approximation we show that the correlated hopping can stabilize a phase characterized by a condensation of pairs of particles and strongly favor supersolid phases. Next, we determine the zero-temperature phase diagram of the effective model by using a classical approximation. A comparison to previous theoretical works and experimental measurements shows that the Shastry-Sutherland model cannot reproduce the series of magnetization plateaus measured experimentally. This suggest that the residual interactions, which were neglected, could be crucial for SrCu2(BO3)2.

Electron-correlation effects in the fullerene molecule and its ions are investigated in the framework of the Hubbard model. The variational Monte Carlo method and the Gutzwiller wave function are used. Most attention is paid to the case of intermediate interactions, but also the strong coupling limit, where the Hubbard model reduces to the antiferromagnetic Heisenberg model, is considered for the fullerene molecule. In this case we obtain a very low variational ground state energy. Futher, we have calculated the main spin correlation functions in the ground state. Only short-range order is found. The pairing energy of two electrons added to a fullerene molecule or to a fullerene ion is also calculated. Contrary to the results obtained by second-order perturbation theory, pair binding is not found.

We discuss the relations between restricted Boltzmann machine (RBM) states and the matrix product states (MPS) for the ground states of 1D translational invariant stabilizer codes. A generic translational invariant and finitely connected RBM state can be expressed as an MPS, and the matrices of the resulting MPS are of rank 1. We dub such an MPS as an RBM-MPS. This provides a necessary condition for exactly realizing a quantum state as an RBM state, if the quantum state can be written as an MPS. For generic 1D stabilizer codes having a non-degenerate ground state with periodic boundary condition, we obtain an expression for the lower bound of their MPS bond dimension, and an upper bound for the rank of their MPS matrices. In terms of RBM, we provide an algorithm to derive the RBM for the cocycle Hamiltonians whose MPS matrices are proved to be of rank 1. Moreover, the RBM-MPS produced by our algorithm has the minimal bond dimension. A family of examples is provided to explain the algorithm. We finally conjecture that these features hold true for all the 1D stabilizer codes having a non-degenerate ground state with periodic boundary condition, as long as their MPS matrices are of rank 1.

Restricted Boltzmann Machines (RBMs) and models derived from them have been successfully used as basic building blocks in deep artificial neural networks for automatic features extraction, unsupervised weights initialization, but also as density estimators. Thus, their generative and discriminative capabilities, but also their computational time are instrumental to a wide range of applications. Our main contribution is to look at RBMs from a topological perspective, bringing insights from network science. Firstly, here we show that RBMs and Gaussian RBMs (GRBMs) are bipartite graphs which naturally have a small-world topology. Secondly, we demonstrate both on synthetic and real-world datasets that by constraining RBMs and GRBMs to a scale-free topology (while still considering local neighborhoods and data distribution), we reduce the number of weights that need to be computed by a few orders of magnitude, at virtually no loss in generative performance. Thirdly, we show that, for a fixed number of weights, our proposed sparse models (which by design have a higher number of hidden neurons) achieve better generative capabilities than standard fully connected RBMs and GRBMs (which by design have a smaller number of hidden neurons), at no additional computational costs.

Low-energy quantum fluctuations and frustrated magnetism in rare-earth-based Shastry-Sutherland lattices: Insights on the CaCo2Al8 structure type antiferromagnets

We propose an efficient method for nonperturbative calculation of Green’s function in acorrelated electron system. The key idea of the method is to project out irrelevantoperators having zero norm in the ground state, which we refer to as effective projectiontheory. We apply the method to a mesoscopic Anderson model and show that for agiven wavefunction ansatz, equations of motion can be closed only by relevantoperators, allowing easy calculation of the zero-temperature Green’s function. Itturns out that the resulting Green’s functions reproduce exact limits of bothweak and strong interactions. The accuracy is also verified for small systems bycomparison with exact diagonalization results, revealing that effective projection theorycaptures the essential correlated features in the entire regime of interactions.

Magnetization plateaus and ground-state phase diagrams of the S=1 Ising model on the Shastry Sutherland lattice