Understanding properties of quantum matter is an outstanding challenge in science. In this paper, we demonstrate how machine-learning methods can be successfully applied for the classification of various regimes in single-particle and many-body systems. We realize neural network algorithms that perform a classification between regular and chaotic behavior in quantum billiard models with remarkably high accuracy. We use the variational autoencoder for autosupervised classification of regular/chaotic wave functions, as well as demonstrating that autoencoders could be used as a tool for detection of anomalous quantum states, such as quantum scars. By taking this method further, we show that machine-learning techniques allow us to pin down the transition from integrability to many-body quantum chaos in Heisenberg XXZ spin chains. For both cases, we confirm the existence of universal W shapes that characterize the transition. Our results pave the way for exploring the power of machine-learning tools for revealing exotic phenomena in quantum many-body systems.
Understanding properties of quantum matter is an outstanding challenge in science. In this paper, we demonstrate how machine-learning methods can be successfully applied for the classification of various regimes in single-particle and many-body systems. We realize neural network algorithms that perform a classification between regular and chaotic behavior in quantum billiard models with remarkably high accuracy. We use the variational autoencoder for autosupervised classification of regular/chaotic wave functions, as well as demonstrating that autoencoders could be used as a tool for detection of anomalous quantum states, such as quantum scars. By taking this method further, we show that machine-learning techniques allow us to pin down the transition from integrability to many-body quantum chaos in Heisenberg XXZ spin chains. For both cases, we confirm the existence of universal W shapes that characterize the transition. Our results pave the way for exploring the power of machine-learning tools for revealing exotic phenomena in quantum many-body systems.
Quantum many-body scars are rare eigenstates hidden within the chaotic spectra of many-body systems, representing a weak violation of the eigenstate thermalization hypothesis (ETH). Identifying these scars, as well as other non-thermal states in complex quantum systems, remains a significant challenge. Besides exact scar states, the nature of other non-thermal states lacking simple analytical characterization remains an open question. In this study, we employ tools from quantum machine learning—specifically, (enhanced) quantum convolutional neural networks (QCNNs), to explore hidden non-thermal states in chaotic many-body systems. Our simulations demonstrate that QCNNs achieve over 99% single-shot measurement accuracy in identifying all known scars. Furthermore, we successfully identify new non-thermal states in models such as the xorX model, the PXP model, and the far-coupling Su-Schrieffer-Heeger model. In the xorX model, some of these non-thermal states can be approximately described as spin-wave modes of specific quasiparticles. We further develop effective tight-binding Hamiltonians within the quasiparticle subspace to capture key features of these many-body eigenstates. Finally, we validate the performance of QCNNs on IBM quantum devices, achieving single-shot measurement accuracy exceeding 63% under real-world noise and errors, with the aid of error mitigation techniques. Our results underscore the potential of QCNNs to uncover hidden non-thermal states in quantum many-body systems.
Most interacting many-body systems in physics are not analytically solvable. Instead, numerical methods are needed for the study of these complex and high-dimensional problems. At present, there are many interesting problems in strongly correlated systems that remain unsolved with current methods. At the heart of this problem is finding an efficient representation that incorporates symmetries, correlations and general features. \n \nIn the context of computer science, machine learning techniques have had astonishing success at reducing the dimensionality of data. The leading method is through the use of artificial neural networks. These networks have been enormously successful at sifting through vast amounts of data to find patterns and regularities. In a sense, neural networks are themselves a statistical system whose properties are adjusted to mimic the features of the data. By finding an effective low-dimensional representation of the data, machine learning has greatly subdued the curse of dimensionality found in many real-world problems. \n \nIn this Thesis, we apply several machine learning techniques to the study of interacting many-body systems in classical and quantum statistical physics. We explore supervised classification of phases of matter with an emphasis on physical interpretation of the net- work. In doing so, we design a custom network architecture that possesses rotational symmetry as an inductive bias. We further investigate connections between the renormalization group and deep learning through applying a super-resolving neural network to the classical Ising model. Towards experimental efforts, we also repurpose generative machine learning to quantum state tomography for the calibration and testing of quantum devices. We conclude with a latent variable model inspired by near-term quantum algorithms. This maps to a variational Monte Carlo ansatz that produces samples efficiently for interacting quantum systems.
Abstract The relative motion of three impenetrable particles on a ring, in our case two identical fermions and one impurity, is isomorphic to a triangular quantum billiard. Depending on the ratio κ of the impurity and fermion masses, the billiards can be integrable or non-integrable (also referred to in the main text as chaotic). To set the stage, we first investigate the energy level distributions of the billiards as a function of 1/κ ∈ [0, 1] and find no evidence of integrable cases beyond the limiting values 1/κ = 1 and 1/κ = 0. Then, we use machine learning tools to analyze properties of probability distributions of individual quantum states. We find that convolutional neural networks can correctly classify integrable and non-integrable states. The decisive features of the wave functions are the normalization and a large number of zero elements, corresponding to the existence of a nodal line. The network achieves typical accuracies of 97%, suggesting that machine learning tools can be used to analyze and classify the morphology of probability densities obtained in theory or experiment.
Quantum computing has the potential to revolutionize numerous aspects of science and engineering, and extensive research in the past decades has been undertaken to exhibit quantum advantage over classical computers. Nevertheless, building robust quantum computers involves enormous technological challenges, and therefore many attempts are devoted to closing the gaps by developing classical simulation algorithms for problems in many body physics. Among these, the most notable are machine learning algorithms. This paper contributes to the development of these toolkits by studying the power of deep generative neural networks, including variational autoencoders and normalizing flows, to represent the distributions generated by quantum computing. The results show that these modern generative networks are only capable of partially capturing features of the quantum system including the overall distribution shapes, but missing the granularities such as local spikes and annihilations in the distributions This incomplete information reconstruction might be due to the inherent limitation of the classical algorithms when encountering the complex quantum states with the exponential growing Hilbert space. The results also show that variational autoencoders perform slightly better than normalizing flows for many cases, and the complimentary strength between the expressibility of variational autoencoders and normalizing flows is observed and worth further exploration.
The central research objects are bosonic many-body systems and their instabilities. One powerful approach to probing these instabilities in quantum systems is through Out-of-Time-Ordered Correlators (OTOCs), and they are the focus of the first chapter. We investigate a transition phenomenon governed primarily by local instabilities, which differentiates integrable dynamics from generic chaotic behavior. This transition provides a promising route to reliably detect quantum chaos in many-body systems, motivating us to study the onset of chaos as a system evolves from integrable to mixed and fully chaotic regimes. Under the assumption that chaos is properly identified, we exploit its characteristic exponential sensitivity and mixing properties in the second part of our work. Here, we propose a novel protocol for Controlling Many-Body Quantum Chaos, utilizing chaos as a resource to achieve exponentially fast targeting of macroscopically occupied single-particle states, connected via chaotic heteroclinic trajectories. This theoretical framework offers a coherent targeting procedure for state preparation in experimental platforms. Finally, in the third part, we explore the manifestation of quantum chaos in low-energy states of many-body systems relevant for quantum computation, as system size increases. Recent advances in Neural Quantum States (NQS), leveraging machine learning progress driven by the surge of artificial intelligence, have significantly extended the numerical boundaries of many-body simulations, such that we now reach system sizes with a hundred sites. These developments open new opportunities for understanding chaos in large-scale quantum systems.
The relative motion of three impenetrable particles on a ring, in our case two identical fermions and one impurity, is isomorphic to a triangular quantum billiard. Depending on the ratio κ of the impurity and fermion masses, the billiards can be integrable or non-integrable (also referred to in the main text as chaotic). To set the stage, we first investigate the energy level distributions of the billiards as a function of 1/κ ∈ [0, 1] and find no evidence of integrable cases beyond the limiting values 1/κ = 1 and 1/κ = 0. Then, we use machine learning tools to analyze properties of probability distributions of individual quantum states. We find that convolutional neural networks can correctly classify integrable and non-integrable states. The decisive features of thewave functions are the normalization and a large number of zero elements, corresponding to the existence of a nodal line. The network achieves typical accuracies of 97%, suggesting that machine learning tools can be used to analyze and classify the morphology of probability densities obtained in theory or experiment.