Attempts to apply Neural Networks (NN) to a wide range of research problems have been ubiquitous and plentiful in recent literature. Particularly, the use of deep NNs for understanding complex physical and chemical phenomena has opened a new niche of science where the analysis tools from Machine Learning (ML) are combined with the computational concepts of the natural sciences. Reports from this unification of ML have presented evidence that NNs can learn classical Hamiltonian mechanics. This application of NNs to classical physics and its results motivate the following question: Can NNs be endowed with inductive biases through observation as means to provide insights into quantum phenomena? In this work, this question is addressed by investigating possible approximations for reconstructing the Hamiltonian of a quantum system in an unsupervised manner by using only limited information obtained from the system's probability distribution.
Attempts to apply Neural Networks (NN) to a wide range of research problems have been ubiquitous and plentiful in recent literature. Particularly, the use of deep NNs for understanding complex physical and chemical phenomena has opened a new niche of science where the analysis tools from Machine Learning (ML) are combined with the computational concepts of the natural sciences. Reports from this unification of ML have presented evidence that NNs can learn classical Hamiltonian mechanics. This application of NNs to classical physics and its results motivate the following question: Can NNs be endowed with inductive biases through observation as means to provide insights into quantum phenomena? In this work, this question is addressed by investigating possible approximations for reconstructing the Hamiltonian of a quantum system in an unsupervised manner by using only limited information obtained from the system's probability distribution.
One of the main reasons why the physics of quantum many-body systems is hard \nlies in the curse of dimensionality: \nThe number of states of such systems increases exponentially with the number of degrees of freedom involved. \nAs a result, computations for realistic systems become intractable, \nand even numerical methods are limited to comparably small system sizes. \nMany efforts in modern physics research are therefore concerned with \nfinding efficient representations of quantum states and \nclever approximations schemes that would allow them to characterize physical systems of interest. \n \nMeanwhile, Deep Learning (DL) has solved many non-scientific problems \nthat have been unaccessible to conventional methods for a similar reason. \nThe concept underlying DL is to extract knowledge from data \nby identifying patterns and regularities. \nThe remarkable success of DL has excited many physicists about \nthe prospect of leveraging its power to solve intractable problems in physics. \nAt the same time, DL turned out to be an interesting complex many-body \nproblem in itself. \nIn contrast to its widespread empirical applications, \nthe theoretical foundation of DL is strongly underdeveloped. \nIn particular, as long as its decision-making process and result interpretability remain opaque, \nDL can not claim the status of a scientific tool. \n \nIn this thesis, I explore the interface between DL and quantum many-body physics, \nand investigate DL both as a tool and as a subject of study. \nThe first project presented here is a theory-based study of \na fundamental open question about the role of width and the number of parameters in deep neural networks. \nIn this work, we consider a DL setup for the image recognition task on standard benchmarking datasets. \nWe combine controlled experiments with a theoretical analysis, including analytical calculations for a toy model. \nThe other three works focus on the application of Restricted Boltzmann Machines as generative models \nfor the task of wavefunction reconstruction from measurement data on a quantum many-body system. \nFirst, we implement this approach as a software package, making it available as a tool for experimentalists. \nFollowing the idea that physics problems can be used to characterize DL tools, \nwe then use our extensive knowledge of this setup to conduct a systematic study of \nhow the RBM complexity scales with the complexity of the physical system. \nFinally, in a follow-up study we focus on the effects of parameter pruning techniques \non the RBM and its scaling behavior.
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.
Quantum many-body systems are of great interest for many research areas, including physics, biology, and chemistry. However, their simulation is extremely challenging, due to the exponential growth of the Hilbert space with system size, making it exceedingly difficult to parameterize the wave functions of large systems by using exact methods. Neural networks and machine learning, in general, are a way to face this challenge. For instance, methods like tensor networks and neural quantum states are being investigated as promising tools to obtain the wave function of a quantum mechanical system. In this tutorial, we focus on a particularly promising class of deep learning algorithms. We explain how to construct a Physics-Informed Neural Network (PINN) able to solve the Schrödinger equation for a given potential, by finding its eigenvalues and eigenfunctions. This technique is unsupervised, and utilizes a novel computational method in a manner that is barely explored. PINNs are a deep learning method that exploit automatic differentiation to solve integro-differential equations in a mesh-free way. We show how to find both the ground and the excited states. The method discovers the states progressively by starting from the ground state. We explain how to introduce inductive biases in the loss to exploit further knowledge of the physical system. Such additional constraints allow for a faster and more accurate convergence. This technique can then be enhanced by a smart choice of collocation points in order to take advantage of the mesh-free nature of the PINN. The methods are made explicit by applying them to the infinite potential well and the particle in a ring, a challenging problem to be learned by an artificial intelligence agent due to the presence of complex-valued eigenfunctions and degenerate states
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.
Part of the challenge for quantum many-body problems comes from the difficulty of representing large-scale quantum states, which in general requires an exponentially large number of parameters. Neural networks provide a powerful tool to represent quantum many-body states. An important open question is what characterizes the representational power of deep and shallow neural networks, which is of fundamental interest due to the popularity of deep learning methods. Here, we give a proof that, assuming a widely believed computational complexity conjecture, a deep neural network can efficiently represent most physical states, including the ground states of many-body Hamiltonians and states generated by quantum dynamics, while a shallow network representation with a restricted Boltzmann machine cannot efficiently represent some of those states.One of the challenges in studies of quantum many-body physics is finding an efficient way to record the large system wavefunctions. Here the authors present an analysis of the capabilities of recently-proposed neural network representations for storing physically accessible quantum states.
Over the past years, machine learning has emerged as a powerful computational tool to tackle complex problems over a broad range of scientific disciplines. In particular, artificial neural networks have been successfully deployed to mitigate the exponential complexity often encountered in quantum many-body physics, the study of properties of quantum systems built out of a large number of interacting particles. In this Article, we overview some applications of machine learning 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. We present supervised machine learning with convolutional neural networks to learn a phase transition, unsupervised learning with restricted Boltzmann machines to perform quantum tomography, and variational Monte Carlo with recurrent neural-networks for approximating the ground state of a many-body Hamiltonian. We briefly review the key ingredients of each algorithm and their corresponding neural-network implementation, and show numerical experiments for a system of interacting Rydberg atoms in two dimensions.