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arXiv:1803.10908 · Physical review. E

Matrix product operators for sequence-to-sequence learning

Chu Guo, Zhanming Jie, Wei Lu, Dario Poletti

Published Oct 5, 2018• 48 citations
arXivHTMLPDF

Abstract

The method of choice to study one-dimensional strongly interacting many-body quantum systems is based on matrix product states and operators. Such a method allows one to explore the most relevant and numerically manageable portion of an exponentially large space. It also allows one to accurately describe correlations between distant parts of a system, an important ingredient to account for the context in machine learning tasks. Here we introduce a machine learning model in which matrix product operators are trained to implement sequence-to-sequence prediction, i.e., given a sequence at a time step, it allows one to predict the next sequence. We then apply our algorithm to cellular automata (for which we show exact analytical solutions in terms of matrix product operators) and to nonlinear coupled maps. We show advantages of the proposed algorithm when compared to conditional random fields and a bidirectional, long, short-term-memory neural network. To highlight the flexibility of the algorithm, we also show that it can readily perform classification tasks.

DOI
10.1103/physreve.98.042114

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The method of choice to study one-dimensional strongly interacting many-body quantum systems is based on matrix product states and operators. Such a method allows one to explore the most relevant and numerically manageable portion of an exponentially large space. It also allows one to accurately describe correlations between distant parts of a system, an important ingredient to account for the context in machine learning tasks. Here we introduce a machine learning model in which matrix product operators are trained to implement sequence-to-sequence prediction, i.e., given a sequence at a time step, it allows one to predict the next sequence. We then apply our algorithm to cellular automata (for which we show exact analytical solutions in terms of matrix product operators) and to nonlinear coupled maps. We show advantages of the proposed algorithm when compared to conditional random fields and a bidirectional, long, short-term-memory neural network. To highlight the flexibility of the algorithm, we also show that it can readily perform classification tasks.
Algorithms developed to solve many-body quantum problems, like tensor networks, can turn into powerful quantum-inspired tools to tackle issues in the classical domain. This work focuses on matrix product operators, a prominent numerical technique to study many-body quantum systems, especially in one dimension. It has been previously shown that such a tool can be used for classification, learning of deterministic sequence-to-sequence processes, and generic quantum processes. We further develop a matrix product operator algorithm to learn probabilistic sequence-to-sequence processes and apply this algorithm to probabilistic cellular automata. This new approach can accurately learn probabilistic cellular automata processes in different conditions, even when the process is a probabilistic mixture of different chaotic rules. In addition, we find that the ability to learn these dynamics is a function of the bitwise difference between the rules and whether one is much more likely than the other. Published by the American Physical Society 2024
Matrix product state has become the algorithm of choice when studying one-dimensional interacting quantum many-body systems, which demonstrates to be able to explore the most relevant portion of the exponentially large quantum Hilbert space and find accurate solutions. Here we propose a quantum inspired K-means clustering algorithm which first maps the classical data into quantum states represented as matrix product states, and then minimize the loss function using the variational matrix product states method in the enlarged space. We demonstrate the performance of this algorithm by applying it to several commonly used machine learning datasets and show that this algorithm could reach higher prediction accuracies and that it is less likely to be trapped in local minima compared to the classical K-means algorithm.
The matrix product state has been demonstrated to be able to explore the most relevant portion of the exponentially large quantum Hilbert space and find accurate solutions for one-dimensional interacting quantum many-body systems. Inspired by this success, here we propose a clustering algorithm based on the matrix product state, which first maps the classical data into quantum states represented as matrix product states, and then minimizes the loss function using a variational matrix product states algorithm in the enlarged space. We demonstrate this algorithm by applying it to several commonly used machine learning data sets, showing that this algorithm could reach higher learning precision and that it is less likely to be trapped in local minima compared to the standard K-means algorithm. We also show that this algorithm can achieve state-of-the-art learning precision on popular computer vision data sets when used in combination with better initialization schemes.
Machine learning (ML) architectures such as convolutional neural networks (CNNs) have garnered considerable recent attention in the study of quantum many-body systems. However, advanced ML approaches such as gated recurrent neural networks (RNNs) have seldom been applied to such contexts. Here we demonstrate that a special class of RNNs known as long short-term memory (LSTM) networks is capable of learning and accurately predicting the time evolution of one-dimensional (1D) Ising model with simultaneous transverse and parallel magnetic fields, as quantitatively corroborated by relative entropy measurements and magnetization between the predicted and exact state distributions. In this unsupervised learning task, the many-body state evolution was predicted in an autoregressive way from an initial state, without any guidance or knowledge of any Hamiltonian. Our work paves the way for future applications of advanced ML methods in quantum many-body dynamics without relying on the explicit form of the Hamiltonian.
Matrix-product states (MPS) have proven to be a versatile ansatz for modeling quantum many-body physics. For many applications, and particularly in one-dimension, they capture relevant quantum correlations in many-body wave functions while remaining tractable (polynomial scaling) to store and manipulate on a classical computer. This has motivated researchers to also apply the MPS ansatz to machine learning (ML) problems where capturing complex correlations in datasets is also a key requirement. Here, for the first time, we develop an MPS-based algorithm, MPSTime, for learning a joint probability distribution underlying a time-series dataset, and show how it can be used to tackle important time-series ML problems, including classification and imputation. MPSTime can efficiently learn complicated time-series probability distributions directly from data, requires only moderate maximum MPS bond dimension χ max (with values for our applications ranging between χ max = 20 − 160 ), and can be trained for both classification and imputation tasks under a single logarithmic loss function. Using synthetic and publicly available real-world datasets—spanning applications in medicine, energy, and astronomy—we demonstrate performance competitive with state-of-the-art ML approaches, but with the key advantage of encoding the full joint probability distribution learned from the data, which is useful for analyzing and interpreting its underlying structure. This manuscript is supplemented with the release of a publicly available code package that implements our approach, and can be used to reproduce all presented results. The effectiveness of the MPS-based ansatz for capturing complex correlation structures in time-series data makes it a powerful foundation for tackling challenging time-series analysis problems across science, industry, and medicine.
This is a set of lectures on tensor networks with a strong emphasis on the core algorithms involving Matrix Product States (MPS) and Matrix Product Operators (MPO). Compared to other presentations, particular care has been given to disentangle aspects of tensor networks from the quantum many-body problem: MPO/MPS algorithms are presented as a way to deal with linear algebra on extremely (exponentially) large matrices and vectors, regardless of any particular application. The lectures include well-known algorithms to find eigenvectors of MPOs (the celebrated DMRG), solve linear problems, and recent learning algorithms that allow one to map a known function into an MPS (the Tensor Cross Interpolation, or TCI, algorithm). The lectures end with a discussion of how to represent functions and perform calculus with tensor networks using the "quantics" representation. They include the detailed analytical construction of important MPOs such as those for differentiation, indefinite integration, convolution, and the quantum Fourier transform. Three concrete applications are discussed in detail: the simulation of a quantum computer (either exactly or with compression), the simulation of a quantum annealer, and techniques to solve partial differential equations (e.g. Poisson, diffusion, or Gross-Pitaevskii) within the "quantics" representation. The lectures have been designed to be accessible to a first-year PhD student and include detailed proofs of all statements.