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arXiv:1910.03957 · Entropy

Variational Autoencoder Reconstruction of Complex Many-Body Physics

I. A. Luchnikov, Alexander Ryzhov, Pieter-Jan Stas, Sergey N. Filippov, Henni Ouerdane

Published Nov 7, 2019• 39 citations
arXivHTMLPDF

Abstract

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.

DOI
10.3390/e21111091

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Analog quantum simulators provide access to many-body dynamics beyond the reach of classical computation. However, extracting physical insights from experimental data is often hindered by measurement noise, limited observables, and incomplete knowledge of the underlying microscopic model. Here, we develop a machine learning approach based on a variational autoencoder (VAE) to analyze interference measurements of tunnel-coupled one-dimensional Bose gases, which realize the sine-Gordon quantum field theory. Trained in an unsupervised manner, the VAE learns a minimal latent representation that strongly correlates with the equilibrium control parameter of the system. Applied to nonequilibrium protocols, the latent space uncovers signatures of frozen-in solitons following rapid cooling, and reveals anomalous postquench dynamics not captured by conventional correlation-based methods. These results demonstrate that generative models can extract physically interpretable variables directly from noisy and sparse experimental data, providing complementary probes of equilibrium and nonequilibrium physics in quantum simulators. More broadly, our work highlights how machine learning can supplement established field-theoretical techniques, paving the way for scalable, data-driven discovery in quantum many-body systems.
Learning many-body quantum states and quantum phase transitions remains a major challenge in quantum many-body physics. Classical machine learning methods offer certain advantages in addressing these difficulties. In this work, we propose a novel framework that bypasses the need to measure physical observables by directly learning the parameters of parameterized quantum circuits. By integrating the attention mechanism from large language models (LLMs) with a variational autoencoder (VAE), we efficiently capture hidden correlations within the circuit parameters. These correlations allow us to extract information about quantum phase transitions in an unsupervised manner. Moreover, our VAE acts as a classical representation of parameterized quantum circuits and the corresponding many-body quantum states, enabling the efficient generation of quantum states associated with specific phases. We apply our framework to a variety of quantum systems and demonstrate its broad applicability, with particularly strong performance in identifying topological quantum phase transitions.
Abstract Learning many-body quantum states and quantum phase transitions remains a major challenge in quantum many-body physics. Classical machine learning methods offer certain advantages in addressing these difficulties. In this work, we propose a novel framework that bypasses the need to measure physical observables by directly learning the parameters of parameterized quantum circuits. By integrating the attention mechanism from large language models (LLMs) with a variational autoencoder (VAE), we efficiently capture hidden correlations within the circuit parameters. These correlations allow us to extract information about quantum phase transitions in an unsupervised manner. Moreover, our VAE acts as a classical representation of parameterized quantum circuits and the corresponding many-body quantum states, enabling the efficient generation of quantum states associated with specific phases. We apply our framework to a variety of quantum systems and demonstrate its broad applicability, with particularly strong performance in identifying topological quantum phase transitions.
This thesis investigates emergent phenomena in strongly correlated quantum matter, with a central focus on developing first-principles deep learning methods for quantum many-body systems. A common theme throughout this work is the search for microscopic mechanisms that generate unexpected collective behavior, together with computational frameworks capable of describing such behavior in regimes where conventional approaches become inadequate. The first part of the thesis studies a doped Mott insulator in a magnetic field and uncovers a new superconducting state, which we call \emph{magnonic superconductivity}. In this state, magnons mediate pairing, and the superconducting order is carried by composite objects binding electron pairs to spin excitations. This mechanism gives rise to a partially spin-polarized superconducting phase beyond the conventional paradigm of electron pairing and further suggests the possibility of higher-charge bound states, including charge-6e6e6e superconductivity. The main thrust of the thesis is the use of deep neural networks as variational, first-principles representations of many-body wave functions. In contrast to the usual data-driven role of machine learning, the neural-network ans\"atze developed here do not learn from labeled datasets; instead, they are optimized directly from the microscopic Hamiltonian through variational Monte Carlo. Within this framework, we first develop self-attention-based neural-network wave functions for correlated electrons in solids and show that they accurately describe moir\'e systems with minimal human bias, capture interaction-driven phase transitions, and exhibit favorable scaling of representational complexity with system size. We then introduce a general physics-informed initialization strategy that reconstructs complex many-body wave functions from probability density and probability current. This approach enables highly accurate pretraining of neural wave functions, achieves near-exact overlaps for topological and paired quantum states, and allows efficient simulations of fractional quantum Hall systems with realistic interactions at unprecedented sizes. Finally, we introduce QERNEL, an efficient foundational neural variational ansatz designed to learn a family of many-body ground states within a single conditional model. By combining efficient context conditioning with modern permutation-equivariant building blocks, QERNEL achieves substantially improved expressivity at low computational cost. Applied to interacting electrons in semiconductor moir\'e heterobilayers, it captures distinct many-body phases and resolves the sharp phase transition between them in systems as large as 150 electrons. These results show that context-aware neural wave functions can move beyond Hamiltonian-by-Hamiltonian solution toward genuinely foundational models of quantum matter. Taken together, the results of this thesis show that first-principles deep learning can serve not merely as a numerical tool, but as a general framework for quantum many-body physics: one that bridges microscopic Hamiltonians, expressive wave-function ansätze, and scalable computation, while opening new paths to the discovery and characterization of strongly correlated quantum matter.
Interpretable machine learning is rapidly becoming a crucial tool for scientific discovery. Among existing approaches, variational autoencoders (VAEs) have shown promise in extracting the hidden physical features of some input data, with no supervision nor prior knowledge of the system at study. Yet the ability of VAEs to create meaningful, interpretable representations relies on their accurate approximation of the underlying probability distribution of their input. When dealing with quantum data, VAEs must hence account for its intrinsic randomness and complex correlations. While VAEs have been previously applied to quantum data, they have often neglected its probabilistic nature, hindering the extraction of meaningful physical descriptors. Here we demonstrate that two key modifications enable VAEs to learn physically meaningful latent representations: a decoder capable of faithfully reproducing the probability distributions of quantum states and a probabilistic loss tailored to this task. Using benchmark quantum spin models, we identify regimes where standard methods fail while the representations learned by our approach remain meaningful and interpretable. Applied to experimental data from Rydberg atom arrays, the model autonomously uncovers the phase structure without access to prior labels, Hamiltonian details, or knowledge of relevant order parameters, highlighting its potential as an unsupervised and interpretable tool for the study of quantum systems.
Over the past few years, machine learning has emerged as a powerful computational tool to tackle complex problems in a broad range of scientific disciplines.In particular, artificial neural networks have been successfully used to mitigate the exponential complexity often encountered in quantum many-body physics, the study of properties of quantum systems built from a large number of interacting particles.In this article, we review some applications of neural networks in condensed matter physics and quantum information, with particular emphasis on hands-on tutorials serving as a quick start for a newcomer to the field.The prerequisites of this tutorial are basic probability theory and calculus, linear algebra, basic notions of neural networks, statistical physics, and quantum mechanics.The reader is introduced to supervised machine learning with convolutional neural networks to learn a phase transition, unsupervised learning with restricted Boltzmann machines to perform quantum tomography, and the variational Monte Carlo method with recurrent neural networks for approximating the ground state of a many-body Hamiltonian.For each algorithm, we briefly review the key ingredients and their corresponding neuralnetwork implementation, and show numerical experiments for a system of interacting Rydberg atoms in two dimensions.