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Tensor networks and machine learning for approximating and optimizing functions in quantum physics

Moritz August

Published Jan 1, 2018• 0 citations
OpenAlex

Abstract

In this thesis, we explore the intersection of informatics and mathematics to address numerical problems in quantum physics. We introduce, analyze and evaluate novel methods for the approximation of physical quantities and the optimization of performance criteria in quantum control. These methods are based on techniques from the fields of tensor networks, numerical analysis and machine learning. Furthermore, we present work on the relation between machine learning and tensor network methods.

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In this thesis, we explore the intersection of informatics and mathematics to address numerical problems in quantum physics. We introduce, analyze and evaluate novel methods for the approximation of physical quantities and the optimization of performance criteria in quantum control. These methods are based on techniques from the fields of tensor networks, numerical analysis and machine learning. Furthermore, we present work on the relation between machine learning and tensor network methods.
This thesis is devoted to the application of tensor-network methods to problems in quantum many-body physics in one and two dimensions and in machine learning. It contributes to the theory of infinite dimensional tensor networks giving rise to critical or fractional quantum Hall lattice models. The power of artificial neural networks at representing such states is investigated both analytically and numerically, and connections with tensor networks are uncovered, with applications both in the study of quantum states and in machine learning.
Tensor network algorithms seek to minimize correlations to compress the classical data representing quantum states. Tensor network algorithms and similar tools—called tensor network methods—form the backbone of modern numerical methods used to simulate many-body physics and have a further range of applications in machine learning. Finding and contracting tensor network states is a computational task, which may be accelerated by quantum computing. We present a quantum algorithm that returns a classical description of a rank-r tensor network state satisfying an area law and approximating an eigenvector given black-box access to a unitary matrix. Our work creates a bridge between several contemporary approaches, including tensor networks, the variational quantum eigensolver (VQE), quantum approximate optimization algorithm (QAOA), and quantum computation.
Tensor network states are ubiquitous in the investigation of quantum many-body (QMB) physics. Their advantage over other state representations is evident from their reduction in the computational complexity required to obtain various quantities of interest, namely observables. Additionally, they provide a natural platform for investigating entanglement properties within a system. In this dissertation, we develop various novel algorithms and optimizations to tensor networks for the investigation of QMB systems, including classical and quantum circuits. Specifically, we study optimizations for the two-dimensional Ising model in a transverse field, we create an algorithm for the kkk-SAT problem, and we study the entanglement properties of random unitary circuits. In addition to these applications, we reinterpret renormalization group principles from QMB physics in the context of machine learning to develop a novel algorithm for the tasks of classification and regression, and then utilize machine learning architectures for the time evolution of operators in QMB systems.
Tensor networks are a popular and computationally efficient approach to simulate general quantum systems on classical computers and, in a broader sense, a framework for dealing with high-dimensional numerical problems. This paper presents a broad but not exhaustive literature review of state-of-the-art applications of tensor networks and related topics across many research domains, including: machine learning, mathematical optimization, materials science, quantum chemistry and quantum circuit simulation. This review aims to clarify which classes of relevant applications have been proposed for which class of tensor networks, as well as to highlight main application results and limitations compared with other classical or quantum simulation methods. We intend this review to be a high-level tour on tensor network applications that is easy to read by non-experts, so basic technical details of tensor networks are summarized.
Tensor networks have demonstrated significant value for machine learning in a myriad of different applications. However, optimizing tensor networks using standard gradient descent has proven to be difficult in practice. Tensor networks suffer from initialization problems resulting in exploding or vanishing gradients and require extensive hyperparameter tuning. Efforts to overcome these problems usually depend on specific network architectures, or ad hoc prescriptions. In this paper we address the problems of initialization and hyperparameter tuning, making it possible to train tensor networks using established machine learning techniques. We introduce a `copy node' method that successfully initializes arbitrary tensor networks, in addition to a gradient based regularization technique for bond dimensions. We present numerical results that show that the combination of techniques presented here produces quantum inspired tensor network models with far fewer parameters, while improving generalization performance.
Tensor Networks are a relatively new machine learning approach. The architectures proposed initially are inspired by approaches from quantum many-body physics simulations. One common layout is the matrix product state (MPS) also known as a tensor train optimized with gradient descent techniques. We introduce a global normalization condition, so that the MPS represents a quantum state. We investigate two optimization methods that find the locally optimal tensors and compare them regarding their effectiveness. One is based on gradient descent and the other on an adaptation of DMRG.