Review of Tensor Network Contraction Approaches

Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. This review is designed to give an insightful and practical introduction to TN contraction algorithms. Starting from basic concepts and definitions, we first explain the relations between TN and physical systems, including the TN representations of classical partitions, non-trivial quantum states, time evolution simulations, etc. These problems, which are challenging to solve, can be reduced to TN contraction problems. Then, we present two different but closely-related kinds of algorithms for computing TN contractions: those inspired by numerical renormalization group and those based on self-consistent eigenvalue problems. Their physical implications and practical implementations are discussed in detail. Particularly, the mathematical connections between tensor network encoding and multi-linear algebra are presented. The readership is expected to range from beginners to specialists, with two main goals. One goal is to provide a systematic introduction of TN contraction algorithms (motivations, implementations, relations, implications, etc.), for those who want to further develop TN algorithms. The other goal is to provide a practical guidance to those, who want to learn and to use TN algorithms to solve practical problems. We expect that the review will be useful to anyone devoted to the interdisciplinary sciences with related numerics.