Survey of Tensor Networks

Nicola Pancotti, Johnnie Gray

We develop an exact mapping between the one-step replica symmetry breaking cavity method and tensor networks. The two schemes come with complementary mathematical and numerical toolboxes that could be leveraged to improve the respective states of the art. As an example, we construct a tensor-network representation of Survey Propagation, one of the best deterministic k-SAT solvers. The resulting algorithm outperforms any existent tensor-network solver by several orders of magnitude. We comment on the generality of these ideas, and we show how to extend them to the context of quantum tensor networks.

Simone Montangero

Timo Felser, Simone Montangero

Richik Sengupta, Soumik Adhikary, Ivan Valer'evich Oseledets +1 more

A tensor network is a type of decomposition used to express and approximate large arrays of data.A given dataset, quantum state, or higher-dimensional multilinear map is factored and approximated by a composition of smaller multilinear maps.This is reminiscent to how a Boolean function might be decomposed into a gate array: this represents a special case of tensor decomposition, in which the tensor entries are replaced by 0, 1 and the factorisation becomes exact.The associated techniques are called tensor network methods: the subject developed independently in several distinct fields of study, which have more recently become interrelated through the language of tensor networks.The tantamount questions in the field relate to expressability of tensor networks and the reduction of computational overheads.A merger of tensor networks with machine learning is natural.On the one hand, machine learning can aid in determining a factorisation of a tensor network approximating a data set.On the other hand, a given tensor network structure can be viewed as a machine learning model.Herein the tensor network parameters are adjusted to learn or classify a data-set.In this survey we review the basics of tensor networks and explain the ongoing effort to develop the theory of tensor networks in machine learning.1

Marcos Diez García, A. M. Romero

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.

Simone Montangero