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Tensor network is a fundamental mathematical tool with a huge range
of applications in physics, such as condensed matter physics,
statistic physics, high energy physics, and quantum information
sciences. This open access book aims to explain the tensor network
contraction approaches in a systematic way, from the basic
definitions to the important applications. This book is also useful
to those who apply tensor networks in areas beyond physics, such as
machine learning and the big-data analysis. Tensor network
originates from the numerical renormalization group approach
proposed by K. G. Wilson in 1975. Through a rapid development in
the last two decades, tensor network has become a powerful
numerical tool that can efficiently simulate a wide range of
scientific problems, with particular success in quantum many-body
physics. Varieties of tensor network algorithms have been proposed
for different problems. However, the connections among different
algorithms are not well discussed or reviewed. To fill this gap,
this book explains the fundamental concepts and basic ideas that
connect and/or unify different strategies of the tensor network
contraction algorithms. In addition, some of the recent progresses
in dealing with tensor decomposition techniques and quantum
simulations are also represented in this book to help the readers
to better understand tensor network. This open access book is
intended for graduated students, but can also be used as a
professional book for researchers in the related fields. To
understand most of the contents in the book, only basic knowledge
of quantum mechanics and linear algebra is required. In order to
fully understand some advanced parts, the reader will need to be
familiar with notion of condensed matter physics and quantum
information, that however are not necessary to understand the main
parts of the book. This book is a good source for non-specialists
on quantum physics to understand tensor network algorithms and the
related mathematics.
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