Tensor Network Contractions

Abstract

PrefaceTensor 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. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarsegraining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multilinear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches. v AcknowledgementsWe are indebted to

Quantum Cellular Automata, Tensor Networks, and Area Laws

“…Our work opens up several questions and possibilities. The identification of QCA with PEPU allows one to use the established techniques based on TN for numerical simulations of their action [34,35]. This also gives us a very natural framework to investigate the classification of (symmetry-protected) topological (SPT) phases for QCA [18] in higher dimensions, with possible implications for the classification of Floquet SPT phases [13,[36][37][38].…”

Review on novel methods for lattice gauge theories

“…Alternatively, a TN (without open indices) may be used to represent the partition function of a certain (classical or quantum) model. In this case, the numerical contraction of the network using different algorithms results in an approximation to the desired value [207].…”

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“…Recently, TN [20][21][22][23][24] is rapidly developed into a powerful quantum-inspired computational tool for machine learning, which brings new possibilities and wide perspectives to process real-life data, such as images and texts, in the quantum processes based on many-qubit (or many-body) states representing the probability distribution of the data that Alice considers to send, (2) encode the specific piece of information to be sent by projecting | , (3) decode the information as a generative process by the projected Born machine. [15,[25][26][27][28][29][30][31][32].…”