Machine Learning Meets Condensed Matter
Why machine learning?
Machine learning is one of today’s most rapidly cutting-edge growing fields of research, with unprecedented promises to offer solutions to existing engineering and research problems. The powerful combination of recent development and practical progress of computing architectures has made possible a large number of successful machine learning applications, in various fields such as automated translation, image and voice recognition, or game-playing. Recent advancements in machine learning and deep learning with important applications in diverse fields such as high energy physics, condensed matter, astronomy [1] or industry have deepen the understanding and further the progress of the field, leading to the recent development of result-driven techniques and advanced algorithms with specific agenda.
While traditional computing algorithms are reaching their limits in simulation capabilities and spending computational resources, condensed matter physics and quantum many-body research require alternative techniques of investigation, problem solving, diagnosis and discovery.
Neural networks and machine learning methods in general, have finally reached the next stage of development after several decades of significant progress in diverse fields of science, industry, and technology. What does this mean for condensed-matter physics? The key question here is: how can industry-standard machine learning algorithms help condensed matter physics research? In particular, machine learning techniques are recently employed for studying classical and quantum many-body tasks encountered in condensed matter, quantum information, and related fields of physics. Some of the existing techniques employed today by machine learning methods may lend themselves in the future to fundamental research, to an extent, with specific focus on condensed matter and quantum many-body physics topics.
Quantum many-body simulations of recent models such as predicting quantum phase transition or exotic emergent phenomena, while conceptually simple, still require a large number of quantum states, leading to an exponentially large number of parameters and therefore becoming computationally difficult, since the solution time can grow exponentially with the size of the task.
So, why machine learning? Recently, machine learning methods were proven to be extremely useful in diverse areas of condensed matter research, reproducing existing results generated with other techniques with smaller computational cost and less effort. Deep learning also offers a powerful tool to efficiently represent quantum many-body states, including the ground states of many-body Hamiltonians or quantum dynamics states.
Condensed matter physics studies microscopic scale interactions of all types of matter at quantum and atom levels, describing them in terms of mesoscopic and macroscopic structure and properties. Condensed matter systems are quite difficult to simulate with traditional computational techniques, predicting approximate solutions hard to test. As condensed matter tasks always deal with massive amounts of interacting particles, these problems become well-suited candidates for solving with machine learning methods, due to big data requirements.
In the past few years, condensed matter physicists started to employ artificial intelligence techniques and especially machine learning algorithms and neural networks, to recognize patterns in the behavior dynamics of many-body systems. Condensed-matter physics deals with different properties and phases of matter under varying conditions, as well as the behavior of these phases using different laws of physics, especially quantum mechanics. Various constructive connections between these fields can cross-fertilize both machine learning and quantum many-body physics.
Such methods can be used together with conventional computing algorithms, such as Quantum Monte Carlo algorithms or Tensor networks, like Matrix Product States or MERA, running on supercomputers, for studying collections of particles in a material. Tensor networks are a recent advanced technique that are gaining traction and find new applications in both machine learning (Neural networks, Deep learning) and diverse subfields of physics (MERA, for example) that require identifying and extrapolating patterns from data.
There are also several classical computer science optimization problems, such as Boolean satisfiability and the travelling salesman problem, which are significantly difficult, having been framed under the generic umbrella term of NP-hard problems. Most optimization problems can be formulated as the problem of finding the ground state of a classical Ising-like Hamiltonian from many-body theory.
Machine learning can find patterns in a black box, as we don’t actually understand how these patterns are detected. Built heavily on statistics, machine learning methods are powerful tools for recognition and search of patterns and regularities in data. With the exponential growth in the volume of data to be transferred, stored or processed, new methods of machine learning become important. The technique of pattern recognition helps detecting arrangements of any potential features or properties that may provide information about a given data set. This is achieved by classifying the data based on the existing knowledge and on the statistical features extracted from different patterns and their representation.
New solutions to old problems
There are numerous applications of machine learning and neural networks in condensed matter physics. Important open questions of fundamental interest in quantum many body systems may find their answers and insights into the powerful shallow or deep learning architectures that exhibit a complexity that scales similar to the quantum many-body problem. Recent work also suggested [2] that machine learning algorithms are similar and have a common denominator with the “renormalization group”, an mathematical apparatus used in particle and condensed matter physics that maps a microscopic picture onto a macroscopic one.
Several approaches that employ supervised, unsupervised and reinforcement learning methods were developed in the recent years [3]. A recent emerging subset of machine learning is deep neural learning or deep neural networks, using neural networks capable of unsupervised learning from data that is unstructured or unlabeled. Examples of such tools are Generative Adversarial Networks, Boltzmann Machines, Variational Autoencoders, and Convolutional Neural Networks [4].
Notably, Restricted Boltzmann machines (RBMs) stand out as as a versatile tool originated in statistical physics and high predictive power for theoretical condensed matter physics models and quantum information theory simulations. RBMs are, indeed, one of the fundamental techniques of deep learning, with various applications in dimensional reduction, feature extraction, and recommender systems through modeling of probability distributions associated with wide variety of datasets [5].
Strongly correlated quantum many-body physics requires challenging, high-demanding computational resources for the study of the many-body quantum wavefunction, which exhibits an exponentially scaling complexity. High-performance computational tools such as quantum Monte Carlo and density matrix renormalization group (DMRG) methods have been employed in the recent years to solve problems in condensed-matter physics[6] [7] [8], with important connections to quantum information sciences [9] [10], ranging from numerical solutions and quantum simulators of simple models to of thermalization and quantum quenches, and much more.
A number of efficient algorithms were rigorously developed to quantify and translate RBMs into tensor network states, with the purpose of employing powerful deep learning architectures in future quantum many-body physics research, such as studying the entanglement entropy bound or the area law. Furthermore, RBMs can produce more efficient classical simulations, due to their higher power in representing quantum many-body states [11] with fewer parameters than tensor network states,
Looking forward
Condensed matter physics community has already taken advantage, diving into recent explorations of existing predictive algorithms underlying machine learning and neural networks, building an impressive consensus among various predictions and similarities between the two sciences, to steer the next steps in the progress of physics. The currently growing mutually beneficial relation between the fields of condensed matter, statistical physics, and machine learning has opened a new window into future approaches, powerful toy models and computational/data analysis methods being migrated towards the theoretical physics community,
The recent predictive, representational and computational power of machine learning in processing and simulating large data sets in quantum many body physics and condensed matter systems, inspired from a range of real-world problems, such as computer vision and natural language processing, offer a successful, compelling high-profile and efficient tool contributing to the advancements of physical sciences and beyond.
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[10] Junwei Liu, Yang Qi, Zi Yang Meng, and Liang Fu, Phys. Rev. B 95, 041101(R) (2017) doi
[11] Biamonte, Wittek, Pancotti, Rebentrost, Wiebe, Lloyd, Nature 549, 195–202 (2017) doi
