We present the \texttt{NeuMC} software package, based on \pytorch, aimed at facilitating the research on neural samplers in lattice field theories. Neural samplers based on normalizing flows are becoming increasingly popular in the context of Monte-Carlo simulations as they can effectively approximate target probability distributions, possibly alleviating some shortcomings of the Markov chain Monte-Carlo methods. Our package provides tools to create such samplers for two-dimensional field theories.

Applications of normalizing flows to the sampling of field configurations in lattice gauge theory have so far been explored almost exclusively in two space-time dimensions. We report new algorithmic developments of gauge-equivariant flow architectures facilitating the generalization to higher-dimensional lattice geometries. Specifically, we discuss masked autoregressive transformations with tractable and unbiased Jacobian determinants, a key ingredient for scalable and asymptotically exact flow-based sampling algorithms. For concreteness, results from a proof-of-principle application to SU(3) lattice gauge theory in four space-time dimensions are reported.

Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany Recent results suggest that flow-based algorithms may provide efficient sampling of field distributions for lattice field theory applications, such as studies of quantum chromodynamics and the Schwinger model. In this work, we provide a numerical demonstration of robust flow-based sampling in the Schwinger model at the critical value of the fermion mass. In contrast, at the same parameters, conventional methods fail to sample all parts of configuration space, leading to severely underestimated uncertainties. Many important physical systems across particle and condensed matter physics can be described in the language of quantum field theory (QFT). Autocorrelations may become especially severe if MCMC updates are configurations, which generates samples by continuously unlikely to generate transitions between modes that are evolving the fields through configuration space via Hamiltonian separated in configuration space.

In these proceedings we present lattice gauge equivariant convolutional neural networks (L-CNNs) which are able to process data from lattice gauge theory simulations while exactly preserving gauge symmetry. We review aspects of the architecture and show how L-CNNs can represent a large class of gauge invariant and equivariant functions on the lattice. We compare the performance of L-CNNs and non-equivariant networks using a non-linear regression problem and demonstrate how gauge invariance is broken for non-equivariant models.

Institute for Theoretical Physics, TU Wien, Austria (Dated: December 25, 2020) We propose Lattice gauge equivariant Convolutional Neural Networks (L-CNNs) for generic machine learning applications on lattice gauge theoretical problems. At the heart of this network structure is a novel convolutional layer that preserves gauge equivariance while forming arbitrarily shaped Wilson loops in successive bilinear layers. We demonstrate that L-CNNs can learn and generalize gauge invariant quantities that traditional convolutional neural networks are incapable of finding. Gauge field theories are an important cornerstone of larger symmetry space is available [33]. This impressive result was transported along a given closed path.

Generative neural samplers offer a complementary approach to Monte Carlo methods for problems in statistical physics and quantum field theory. This work tests the ability of generative neural samplers to estimate observables for real-world low-dimensional spin systems. It maps out how autoregressive models can sample configurations of a quantum Heisenberg chain via a classical approximation based on the Suzuki-Trotter transformation. We present results for energy, specific heat and susceptibility for the isotropic XXX and the anisotropic XY chain that are in good agreement with Monte Carlo results within the same approximation scheme.

In Ref. [11], this approach was demonstrated in the Gauge theories based on SU(N) or U(N) groups describe context of U(1) gauge theory. Here, we develop a class of many aspects of nature. For example, the Standard kernels for SU(N) group elements (and describe a similar Model of nuclear and particle physics is a nonabelian construction for U(N) group elements). We show that if gauge theory with the symmetry group U(1) an invertible transformation acts only on the eigenvalues SU(2) SU(3), candidate theories for physics beyond the of a matrix and is equivariant under permutation of those Standard Model can be defined based on strongly interacting eigenvalues, then it is equivariant under matrix conjugation SU(N) gauge theories [1, 2], SU(N) gauge symmetries and may be used as a kernel. Moreover, by making emerge in various condensed matter systems [3-7], a connection to the maximal torus within the group and and SU(N) and U(N) gauge symmetries feature in the to the Weyl group of the root system, we show that this low energy limit of certain string-theory vacua [8]. In is in fact a universal way to define a kernel for unitary the context of the rapidly-developing area of machinelearning groups.

Neural-Network Quantum States have been recently introduced as an Ansatz for describing the wave function of quantum many-body systems. We show that there are strong connections between Neural-Network Quantum States in the form of Restricted Boltzmann Machines and some classes of Tensor-Network states in arbitrary dimensions. In particular we demonstrate that short-range Restricted Boltzmann Machines are Entangled Plaquette States, while fully connected Restricted Boltzmann Machines are String-Bond States with a nonlocal geometry and low bond dimension. These results shed light on the underlying architecture of Restricted Boltzmann Machines and their efficiency at representing many-body quantum states. String-Bond States also provide a generic way of enhancing the power of Neural-Network Quantum States and a natural generalization to systems with larger local Hilbert space. We compare the advantages and drawbacks of these different classes of states and present a method to combine them together. This allows us to benefit from both the entanglement structure of Tensor Networks and the efficiency of Neural-Network Quantum States into a single Ansatz capable of targeting the wave function of strongly correlated systems. While it remains a challenge to describe states with chiral topological order using traditional Tensor Networks, we show that Neural-Network Quantum States and their String-Bond States extension can describe a lattice Fractional Quantum Hall state exactly. In addition, we provide numerical evidence that Neural-Network Quantum States can approximate a chiral spin liquid with better accuracy than Entangled Plaquette States and local String-Bond States. Our results demonstrate the efficiency of neural networks to describe complex quantum wave functions and pave the way towards the use of String-Bond States as a tool in more traditional machine-learning applications.

Boltzmann machines are physics informed generative models with wide applications in machine learning. They can learn the probability distribution from an input dataset and generate new samples accordingly. Applying them back to physics, the Boltzmann machines are ideal recommender systems to accelerate Monte Carlo simulation of physical systems due to their flexibility and effectiveness. More intriguingly, we show that the generative sampling of the Boltzmann Machines can even discover unknown cluster Monte Carlo algorithms. The creative power comes from the latent representation of the Boltzmann machines, which learn to mediate complex interactions and identify clusters of the physical system. We demonstrate these findings with concrete examples of the classical Ising model with and without four spin plaquette interactions. Our results endorse a fresh research paradigm where intelligent machines are designed to create or inspire human discovery of innovative algorithms.

We study the performance of different message passing algorithms in the two dimensional Edwards Anderson model. We show that the standard Belief Propagation (BP) algorithm converges only at high temperature to a paramagnetic solution. Then, we test a Generalized Belief Propagation (GBP) algorithm, derived from a Cluster Variational Method (CVM) at the plaquette level. We compare its performance with BP and with other algorithms derived under the same approximation: Double Loop (DL) and a two-ways message passing algorithm (HAK). The plaquette-CVM approximation improves BP in at least three ways: the quality of the paramagnetic solution at high temperatures, a better estimate (lower) for the critical temperature, and the fact that the GBP message passing algorithm converges also to non paramagnetic solutions. The lack of convergence of the standard GBP message passing algorithm at low temperatures seems to be related to the implementation details and not to the appearance of long range order. In fact, we prove that a gauge invariance of the constrained CVM free energy can be exploited to derive a new message passing algorithm which converges at even lower temperatures. In all its region of convergence this new algorithm is faster than HAK and DL by some orders of magnitude.