Neural network quantum states provide a novel representation of the many-body states of interacting quantum systems and open up a promising route to solve frustrated quantum spin models that evade other numerical approaches. Yet its capacity to describe complex magnetic orders with large unit cells has not been demonstrated, and its performance in a rugged energy landscape has been questioned. Here we apply restricted Boltzmann machines (RBMs) and stochastic gradient descent to seek the ground states of a compass spin model on the honeycomb lattice, which unifies the Kitaev model, Ising model and the quantum 120° model with a single tuning parameter. We report calculation results on the variational energy, order parameters and correlation functions. The phase diagram obtained is in good agreement with the predictions of tensor network ansatz, demonstrating the capacity of RBMs in learning the ground states of frustrated quantum spin Hamiltonians. The limitations of the calculation are discussed. A few strategies are outlined to address some of the challenges in machine learning frustrated quantum magnets.
Neural network quantum states provide a novel representation of the many-body states of interacting quantum systems and open up a promising route to solve frustrated quantum spin models that evade other numerical approaches. Yet its capacity to describe complex magnetic orders with large unit cells has not been demonstrated, and its performance in a rugged energy landscape has been questioned. Here we apply restricted Boltzmann machines (RBMs) and stochastic gradient descent to seek the ground states of a compass spin model on the honeycomb lattice, which unifies the Kitaev model, Ising model and the quantum 120° model with a single tuning parameter. We report calculation results on the variational energy, order parameters and correlation functions. The phase diagram obtained is in good agreement with the predictions of tensor network ansatz, demonstrating the capacity of RBMs in learning the ground states of frustrated quantum spin Hamiltonians. The limitations of the calculation are discussed. A few strategies are outlined to address some of the challenges in machine learning frustrated quantum magnets.
The simulation of interacting many-body quantum systems is a central chal- lenge in condensed matter physics due to the exponential growth of the Hilbert space with system size. In this project, we implement a Neural Network Quantum State (NQS) ansatz based on a Complex Restricted Boltzmann Machine (RBM) to solve for the ground state of the 1D Transverse Field Ising Model (TFIM). Using Variational Monte Carlo (VMC) optimization via Stochastic Reconfiguration (SR), we successfully train the network to approximate the ground state wave function across the paramagnetic-ferromagnetic phase transition. We benchmark our results against Exact Diagonalization (ED) for small systems (N = 12), demonstrating high accuracy in ground state energy estimation. Furthermore, we investigate quantum correlations and order parameters, specifically the transverse magnetization mx and longitudinal spin-spin correlations Czz(r), highlighting the RBM’s ability to capture long-range order and critical behavior near the quantum critical point hx/J = 1.
Variational Monte Carlo with neural network quantum states has proven to be a promising avenue for evaluating the ground state energy of spin Hamiltonians. Based on anecdotal evidence, it has been claimed repeatedly in the literature that neural network quantum state simulations are insensitive to the sign problem. We present a detailed and systematic study of restricted Boltzmann machine (RBM) based variational Monte Carlo for quantum spin chains, resolving exactly how relevant stoquasticity is in this setting. We show that in most cases, when the Hamiltonian is phase connected with a stoquastic point, the complex RBM state can faithfully represent the ground state, and local quantities can be evaluated efficiently by sampling. On the other hand, we identify a number of new phases that are challenging for the RBM Ansatz, including non-topological robust non-stoquastic phases as well as stoquastic phases where sampling is nevertheless inefficient. Our results suggest that great care needs to be taken with neural network quantum state based variational Monte Carlo when the system under study is highly frustrated.
The use of artificial neural networks to represent quantum wave functions has recently attracted interest as a way to solve complex many-body problems. The potential of these variational parametrizations has been supported by analytical and numerical evidence in controlled benchmarks. While approaching the end of the early research phase in this field, it becomes increasingly important to show how neural-network states perform for models and physical problems that constitute a clear open challenge for other many-body computational methods. In this paper, we start addressing this aspect, concentrating on a presently unsolved model describing two-dimensional frustrated magnets. Using a fully convolutional neural network model as a variational ans\"atz, we study the frustrated spin-1/$2{J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the square lattice. We demonstrate that the resulting predictions for both ground-state energies and properties are competitive with, and often improve upon, existing state-of-the-art methods. In a relatively small region in the parameter space, corresponding to the maximally frustrated regime, our ans\"atz exhibits comparatively good but not the best performance. The gap between the complexity of the models adopted here and those routinely adopted in deep-learning applications is, however, still substantial, such that further improvements in future generations of neural-network quantum states are likely to be expected.
We investigate the Shastry-Sutherland model (SSM), i.e., spin-1/2 quantum Heisen- berg model on a Shastry-Sutherland lattice, using a newly emerging approach exploiting well-developed machine learning techniques. We utilize neural networks as variational quantum states in quantum Monte Carlo investigations of the ground state properties. We first focus on SSM without an external magnetic field. For small lattices accessible via exact diagonalization, we compare the precision of various architectures based on re- stricted Boltzmann machines (RBM) or group-convolutional neural networks. The most versatile and precise architecture, namely complex-valued RBM, is then applied for larger lattices. Here we investigate the frustrated regime. We show that the RBM is able to represent all three of the major and fundamentally different phases of SSM. Finally, we apply the complex-valued RBM for SSM in a finite external magnetic field. We find that it cannot capture the intriguing steps in magnetization typical for SSM correctly due to its tendency to prefer more ordered states with higher magnetization. 1
Quantum spin liquids (QSLs) are highly elusive states of matter that are char- acterized by long range quantum entanglement and the lack of long range magnetic ordering. There are a variety of candidate materials with QSLs as well as model Hamiltonians where exact QSL ground states can be proven. The Kitaev honeycomb model is among the few models with an analytically derived QSL ground state. Ad- ditional interactions such as the Gamma and Heisenberg interactions are included to match real behaviours in candidate Kitaev materials to form the larger JKΓΓ′model. Due to the size of the Hilbert spaces associated with many-body systems, numerical techniques centre around the use of more efficient representations of the wavefunc- tion that reduce the need for exponential parameterization. Neural Quantum States (NQS) are one of the latest additions to the toolkit, making use of neural networks to represent the mapping between spin configurations and their corresponding wave- function amplitudes. By using variational monte carlo (VMC) to derive the ground state, the problem of finding the associated NQS representation translates to a Ma- chine Learning task of minimizing a cost function through gradient descent. Group Convolutional Neural Networks (GCNNs) are one of the latest NQS architectures which construct wavefunction ansatzes that are symmetric to a group of symmetries and have yielded state-of-the-art results in benchmark models. In this thesis, ground state calculations were performed using GCNNs for the Kitaev and Gamma honey- comb models and ground state properties were investigated. In the Kitaev model, optimization was affected by a complicated learning landscape, limiting the conver- gence towards the ground state. Performance improved with the Gamma model for smaller systems but still fell short of confirming an identical stripy patterned ground state found with iDMRG and iPEPS.
Recently, quantum-state representation using artificial neural networks has started to be recognized as a powerful tool. However, due to the black-box nature of machine learning, it is difficult to analyze what machine learns or why it is powerful. Here, by applying one of the simplest neural networks, the restricted Boltzmann machine (RBM), to the ground-state representation of the one-dimensional (1D) transverse-field Ising (TFI) model, we make an attempt to directly analyze the optimized network parameters. In the RBM optimization, a zero-temperature quantum state is mapped onto a finite-temperature classical state of the extended Ising spins that constitute the RBM. We find that the quantum phase transition from the ordered phase to the disordered phase in the 1D TFI model by increasing the transverse field is clearly reflected in the behaviors of the optimized RBM parameters and hence in the finite-temperature phase diagram of the classical RBM Ising system. The present finding of a correspondence between the neural-network parameters and quantum phases suggests that a careful investigation of the neural-network parameters may provide a new route to extracting nontrivial physical insights from the neural-network wave functions.