Machine Learning for Quantum Many-body Physics

Motivated by startling advances in industry applications, we are actively exploring what role modern machine learning (ML) will play in our own field of quantum many-body physics. In the last two years we have made rapid progress demonstrating the ability of a variety of discriminative and generative ML algorithms to make headway on our own favourite problems in condensed matter, cold atoms, and quantum information systems. Despite this, we have yet to completely tie our field's efforts to the core of the disruptive advances that propelled deep learning to the forefront of ML research in 2012. In this talk, I will ask what deep learning can do for computational research in physics, and vice versa. Most tantalizingly, is there an underlying scaling speedup associated with deep learning that can be harnessed for applications in quantum many-body physics?

Efficient and automated classification of phases from minimally processed data is one goal of machine learning in condensed-matter and statistical physics. Supervised algorithms trained on raw samples of microstates can successfully detect conventional phase transitions by learning a bulk feature, e.g. the average magnitization. In this talk I will discuss the challenges and advances in classification of topological phases with neural networks. Specifically, I will present the results we obtained for the Kosterlitz-Thouless (KT) transition in the two-dimensional classical XY model [Phys. Rev. B 97, 045207 (2018)].

In this talk, I will introduce machine learning techniques to the detection of quantum nonlocality in many-body systems, with a focus on the restricted-Boltzmann-machine (RBM) architecture. Using reinforcement learning, I will show that RBM is capable of finding the maximum quantum violations of multipartite Bell inequalities with given measurement settings. Our results build a novel bridge between computer-science-based machine learning and quantum many-body nonlocality, which will benefit future studies in both areas.

Self-learning Monte Carlo (SLMC) method is a general-purpose numerical method to simulate many-body systems. SLMC can efficiently cure the critical slowing down in both bosonic and fermionic systems. Moreover, for fermionic systems, SLMC can generally reduce the computational complexity and speed up simulations even away from the critical points. For example, SLMC is more than 1000 times faster than the conventional method for the double exchange model in 8*8*8 cubic lattice. In this talk, I will give an introduction about the background, basic idea and the design principle of SLMC. Later, I will explicitly show how to use SLMC and its great accelerations in classic systems, free fermions coupled with classical spins systems, and interacting fermion systems. I will also talk about how to explicitly use advanced machine learning techniques in SLMC.

A model of deep unsupervised learning is introduced as a continuum time approximation of stochastic gradient descent. The model gives rise to a highly non-isotropic Fokker-Planck equation with a diffusion matrix corresponding to the covariance matrix of the loss function. We show how the hierarchical structure of the deep network gives rise to many 'sloppy' variables whose contribution to data features is negligible, and a few rigid variables that determine features and correlations in the data. We give arguments indicating that learning of the rigid variables converges rapidly under stochastic gradient descent, while convergence of the sloppy variables is very slow. Finally we point out connections with models of holography, and discuss how insight for tensor network models can help extract hidden features in the data.

Motivated by recent progress in applying techniques from the field of artificial neural networks (ANNs) to quantum many-body physics, we investigate as to what extent the flexibility of ANNs can be used to efficiently study systems that host chiral topological phases such as fractional quantum Hall (FQH) phases. With benchmark examples, we demonstrate that training ANNs of restricted Boltzmann machine type in the framework of variational Monte Carlo can numerically solve FQH problems to good approximation. Furthermore, we show by explicit construction how n-body correlations can be kept at an exact level with ANN wave-functions exhibiting polynomial scaling with power n in system size. Using this construction, we analytically represent the paradigmatic Laughlin wave-function as an ANN state.

The application of artificial neural network to central questions in the theory of quantum matter is a rapidly developing field. So far, much more progress has been in using AI to solve computational bottle-necks either through speed up or through an efficient representation of the wave function or probability. In this talk, I will emphasize the advantage of AI-human intelligence synergy in dealing with both computational and experimental data. In the computational front, I will discuss the use of quantum loop topography that bridges theoretical concepts such as response function with the artificial neural network. In the experimental front, I will discuss how we can use AI to gain new theoretical insight from experimental data.

In this contribution I will focus on the possibilities of using machine learning methods to classify and/or predict the dynamics of a quantum system. In particular, the classification of (dynamical) phases from time traces of observables has proven to be useful in the study of many-body localized systems. On the other hand, the prediction of quantum dynamics using machine learning is an open problem for which I will present one possible approach.

Neural networks hold the promise for progress on some of the computationally most demanding problems in condensed matter physics, such as the investigation of the many-body localization transition. The main challenges in this case are the lack of a good and universally accepted order parameter (instead a whole zoo of quantities has been proposed to study and pin down the transition) and the need for an immense amount of disorder averaging. We follow a more radical ansatz to work around these problems. We avoid the definition of any order parameter or observable and instead exploit a state-of-the-art technique in deep learning and unsupervised learning to find the phase transition as the point at which the eigenfunctions of the Hamiltonian undergo a qualitative change. Domain adversarial neural networks work directly on the eigenstates of a quantum system, which is a spin-1/2 Heisenberg chain in a random magnetic field in this case. We enforce the extraction of invariant features from the states with the adversarial approach and classify the states by studying the properties of invariant features with unsupervised machine learning methods. Through the use of convolutional filters, our method is scalable and needs much less disorder averaging than traditional numerical methods. This reduces the numerical effort for mapping out the phase diagram by a factor of ~50. Since our method is unsupervised, it is capable of uncovering physics that other deep learning networks are blind to. We show that our approach overcomes a weakness of transfer learning and opens the door to more accurate prediction of phase transitions. Moreover, we demonstrate the importance of an automatically extracted invariant representation of the physical data to study complex problems.

We apply the artificial neural network in a supervised manner to map out the quantum phase diagram of disordered topological superconductor in class DIII. Given the disorder that keeps the discrete symmetries of the ensemble as a whole, translational symmetry which is broken in the quasiparticle distribution individually is recovered statistically by taking an ensemble average. By using this, we classify the phases by the artificial neural network that learned the quasiparticle distribution in the clean limit and show that the result is totally consistent with the calculation by the transfer matrix method or noncommutative geometry approach. If all three phases, namely the Z2, trivial, and the thermal metal phases appear in the clean limit, the machine can classify them with high confidence over the entire phase diagram. If only the former two phases are present, we find that the machine remains confused in the certain region, leading us to conclude the detection of the unknown phase which is eventually identified as the thermal metal phase.

Despite the rapidly growing interest in harnessing machine learning to identify quantum phases in many-body systems, a key challenge is in extracting essential, non-local information and efficiently passing it to the artificial neural network. We show that human intelligence can complement artificial intelligence and pre-select the relevant information from the 'big data.' For instance, we introduce Quantum Loop Topography that consists of operators taking semi-local loop structures and determined by the characteristics of the targeted phases, such as physical responses or quasi-particle statistics. Therefore, Quantum Loop Topography offers a coherent, feature-selection interface between the artificial neural network and the sample states. With this architecture, we demonstrate that even a simple neural network with a single fully-connected hidden layer can be trained to distinguish Chern insulator and fractional Chern insulator, as well as $Z_2$ quantum spin liquid, with high efficiency and fidelity. I’ll also summarize recent progress on tackling other quantum phases, reverse engineering artificial neural networks, and connections with quantum entanglement. Our work paves the route towards powerful applications of machine learning in the study of quantum phases, phase transitions, and beyond.

We discuss how to combine Boltzmann machine learning techniques with conventional variational Monte Carlo methods developed for decades with various physical intuitions, to reach state-of-the-art accurate ground states of quantum spin and fermion models [1,2]. We next discuss how a machine learning technique combined with existing angle-resolved photoemission spectra offers an emergent understanding on a grand challenge at the heart of the superconducting mechanism of the cuprate superconductors, in otherwise inaccessible ways. [1] Y. Nomura, A. S. Darmawan, Y. Yamaji and M. Imada: Phys. Rev. B 96, 205152 (2017) [2] Y. Nomura, G. Carleo and M. Imada: arXiv:1802.09558

In our recent work (https://arxiv.org/abs/1802.05267) we employ reinforcement learning to train a deep artificial neural network that discovers, tabula rasa (i.e., with no human knowledge or guidance), complete quantum-error-correction strategies in a collection of quantum bits subject to decoherence. The network discovers optimal encoding and decoding protocols of the logical qubit as well as intricate correction strategies that are adapted to the measurement outcomes. A key novelty of our work is the development of an immediate reward scheme based on a physically meaningful quantity describing the capacity to protect the quantum information stored in a quantum memory. Our work opens the prospect for developing fully automated quantum-error-correction strategies in complex quantum systems. In the first part of the talk I will explain how the neural network develops on its own the building blocks of a quantum-error-correction strategy: the encoding of the logical qubit, the error detection and correction, and the decoding of the logical qubit at the end of the simulation time. In the second part I will explain how our approach can be used to develop strategies that are fully adapted to the subtleties of the quantum device such as the hardware specifications and the noise. I will also present a detailed analysis of the internal workings of the neural network.

I will discuss a quantum state tomography technique for general quantum mixed states that relies on informationally complete (IC) positive-operator valued measures (POVMs). We learn the resulting probabilistic representation of the state using standard restricted Boltzmann machines RBMs and Autoregressive Distribution Estimators NADEs. We test our ideas using synthetic data generated numerically.

The intersection between machine learning and quantum physics has recently been enjoying a surge of academic interest, in more than one ramification: This applies to notions of quantum machine learning, in the same way as the application of classical machine learning to quantum problems. Of course, notions of learning come in handy in many contexts. In this talk, we will have a look at three such intersection points where notions of machine learning seem particularly promising: (i) This is the application of ideas of reinforcement learning to the decoding problem for topological quantum memories. (ii) Then, we consider methods of compressed sensing and matrix completion as a method contributing to machine learning. (iii) Finally, we will have a look at the problem of finding suitable orbitals in quantum chemistry in variants of density functional theory.

Physical systems differring in their microscopic details often display strikingly similar behaviour when probed at macroscopic scales. Those universal properties, largely determining their physical characteristics, are revealed by the powerful renormalization group (RG) procedure, which systematically retains “slow” degrees of freedom and integrates out the rest. However, the important degrees of freedom may be difficult to identify. We demonstrate a machine learning algorithm capable of identifying the relevant degrees of freedom and executing RG steps iteratively without any prior knowledge about the system. We introduce an artificial neural network based on a model-independent, information-theoretic characterization of a real-space RG procedure, performing this task. We apply the algorithm to classical statistical physics problems in one and two dimensions. We demonstrate RG flow and extract the Ising critical exponent. Our results demonstrate that machine learning techniques can extract abstract physical concepts and consequently become an integral part of theory- and model-building.

We propose to generalise classical maximum likelihood learning to density matrices. As the objective function, we propose a quantum likelihood that is related to the cross entropy between density matrices. We apply this learning criterion to the quantum Boltzmann machine (QBM), previously proposed by \cite{amin2016quantum}. We demonstrate for the first time learning a quantum Hamiltonian from quantum statistics. For the anti-ferromagnetic Heisenberg and XYZ model we recover the true ground state wave function and Hamiltonian. The second contribution is to apply quantum learning to learn from classical data. Quantum learning uses in addition to the classical statistics also quantum statistics for learning. These statistics may violate the Bell inequality, as in the quantum case. Maximizing the quantum likelihood yields results that are significantly more accurate than the classical maximum likelihood approach in several cases. We give an example how the QBM can learn a strongly non-linear problem such as the parity problem. The solution shows entanglement, quantified by the entanglement entropy.

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The number of possible materials is practically infinite, while only few hundred thousands of (inorganic) materials are known to exist and for few of them even basic properties are systematically known. In order to speed up the identification and design of new and novel optimal materials for a desired property or process, strategies for quick and well-guided exploration of the materials space are highly needed. A desirable strategy would be to start from a large body of experimental or theoretical data, and by means of “data-analytics” methods, to identify yet unseen patterns or structures in the data. This leads to the identification of maps (or charts) of materials where different regions correspond to materials with different properties. The main challenge on building such maps is to find the appropriate descriptive parameters (called descriptors) that define these regions of interest. Here, I will present methods for the machine-aided identification of descriptors and materials maps applied to the prediction of novel 2D topological insulators, stable perovskites, and of CO$_2$ adsorption-energy and activation on metal-oxide surfaces. I will also describe the infrastructure to perform such analyses online, via the "data-analytics toolkit" within the framework of the Novel-Materials-Discovery (NOMAD) Laboratory.

Computational discovery and design of novel materials requires their accurate simulation on the atomic scale. Numerical approximations to the electronic structure problem enable this in principle, but their applicability is severely limited by their computational cost. In high-throughput settings, machine learning can reduce these costs significantly by rapidly and accurately interpolating between ab initio reference calculations. For this, a numerical representation of atomistic systems that supports interpolation is crucial. Using kernel regression with the many-body tensor representation, I will present empirical evidence for accurate predictions of formation enthalpies on benchmark datasets of crystal structures, including Al-Ga-In sesquioxides and several binary alloys.

Tensor networks are an efficient approach to representing complicated many-body wavefunctions in terms of many smaller tensors, and they lead to powerful algorithms for studying strongly correlated systems. But tensor networks could be applied much more broadly beyond representing wavefunctions. Large tensors resembling wavefunctions appear naturally in certain classes of models studied extensively in machine learning. Decomposing the model parameters as a tensor network leads to interesting algorithms for training models on real-world data which scale better than standard approaches. In addition to training models directly for recognizing labeled data, tensor network coarse-graining can be used to extract statistically significant "features" for subsequent learning tasks. I will also highlight other benefits of the tensor network approach such as the flexibility to blend different approaches and to interpret trained models.

Learning a probability distribution from data and approximating a quantum state are two fundamentally related problems. Techniques designed to represent probability distributions, such as Boltzmann machines, have been applied as Ansatz for a many-body wave function, while tensor networks designed to represent quantum states have been used in machine learning. I will discuss how these distinct approaches are related, with a focus on the relationship between probabilistic graphical models and tensor networks. I will show that the two frameworks can be connected through the definition of generalized tensor networks which allow tensors to be copied. This correspondence has applications for particular models in both quantum physics and machine learning. On the one hand I will discuss the relationship between restricted Boltzmann machines and generalized tensor networks, and show that these models can be used to approximate challenging many-body quantum states such as chiral topological states. On the other hand I will introduce ways to combine tensor networks and neural networks in machine learning and present a benchmark of different network architectures on the tasks of image classification and sound recognition.

Data is a crucial raw material of this century, and the amount of data that has been created in materials science in recent years and is being created every new day is immense. Without a proper infrastructure that allows for collecting and sharing data, the envisioned success of materials science, and in particular, Big-Data driven materials science will be hampered. For the field of computational materials science, the NOMAD (Novel Materials Discovery) Center of Excellence (CoE) has changed the scientific culture towards a comprehensive and FAIR data sharing, opening new avenues for mining Big Data of materials science. Novel data-analytics concepts and tools turn data into knowledge and help the prediction of new materials or the identification of new properties of already known materials.