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Open AccessFeature PaperArticle

Variational Autoencoder Reconstruction of Complex Many-Body Physics

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Center for Energy Science and Technology, Skolkovo Institute of Science and Technology, 3 Nobel Street, Skolkovo, 121205 Moscow Region, Russia
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Moscow Institute of Physics and Technology, Institutskii Per. 9, Dolgoprudny, 141700 Moscow Region, Russia
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Department of Applied Physics, Stanford University 348 Via Pueblo Mall, Stanford, CA 94305, USA
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Valiev Institute of Physics and Technology of Russian Academy of Sciences, Nakhimovskii Pr. 34, 117218 Moscow, Russia
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Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina St. 8, 119991 Moscow, Russia
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(11), 1091; https://doi.org/10.3390/e21111091
Received: 9 October 2019 / Revised: 5 November 2019 / Accepted: 6 November 2019 / Published: 7 November 2019
(This article belongs to the Special Issue Simulation with Entropy Thermodynamics)
Thermodynamics is a theory of principles that permits a basic description of the macroscopic properties of a rich variety of complex systems from traditional ones, such as crystalline solids, gases, liquids, and thermal machines, to more intricate systems such as living organisms and black holes to name a few. Physical quantities of interest, or equilibrium state variables, are linked together in equations of state to give information on the studied system, including phase transitions, as energy in the forms of work and heat, and/or matter are exchanged with its environment, thus generating entropy. A more accurate description requires different frameworks, namely, statistical mechanics and quantum physics to explore in depth the microscopic properties of physical systems and relate them to their macroscopic properties. These frameworks also allow to go beyond equilibrium situations. Given the notably increasing complexity of mathematical models to study realistic systems, and their coupling to their environment that constrains their dynamics, both analytical approaches and numerical methods that build on these models show limitations in scope or applicability. On the other hand, machine learning, i.e., data-driven, methods prove to be increasingly efficient for the study of complex quantum systems. Deep neural networks, in particular, have been successfully applied to many-body quantum dynamics simulations and to quantum matter phase characterization. In the present work, we show how to use a variational autoencoder (VAE)—a state-of-the-art tool in the field of deep learning for the simulation of probability distributions of complex systems. More precisely, we transform a quantum mechanical problem of many-body state reconstruction into a statistical problem, suitable for VAE, by using informationally complete positive operator-valued measure. We show, with the paradigmatic quantum Ising model in a transverse magnetic field, that the ground-state physics, such as, e.g., magnetization and other mean values of observables, of a whole class of quantum many-body systems can be reconstructed by using VAE learning of tomographic data for different parameters of the Hamiltonian, and even if the system undergoes a quantum phase transition. We also discuss challenges related to our approach as entropy calculations pose particular difficulties. View Full-Text
Keywords: complex systems thermodynamics; machine learning; quantum phase transition; Ising model; variational autoencoder complex systems thermodynamics; machine learning; quantum phase transition; Ising model; variational autoencoder
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Luchnikov, I.A.; Ryzhov, A.; Stas, P.-J.; Filippov, S.N.; Ouerdane, H. Variational Autoencoder Reconstruction of Complex Many-Body Physics. Entropy 2019, 21, 1091.

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