Tensor networks for complex quantum systems

Originally developed in the context of condensed-matter physics and based on renormalization group ideas, tensor networks have been revived thanks to quantum information theory and the progress in understanding the role of entanglement in quantum many-body systems. Moreover, tensor network states have turned out to play a key role in other scientific disciplines. In this context, here I provide an overview of the basic concepts and key developments in the field. I briefly discuss the most important tensor network structures and algorithms, together with an outline of advances related to global and gauge symmetries, fermions, topological order, classification of phases, entanglement Hamiltonians, holografic duality, artificial intelligence, the 2D Hubbard model, 2D quantum antiferromagnets, conformal field theory, quantum chemistry, disordered systems and many-body localization. Understanding entanglement in many-body systems provided a description of complex quantum states in terms of tensor networks. This Review revisits the main tensor network structures, key ideas behind their numerical methods and their application in fields beyond condensed matter physics. Key pointsTensor networks are mathematical representations of quantum many-body states based on their entanglement structure.Different tensor network structures describe different physical situations, such as low-energy states of gapped 1D systems, 2D systems and scale-invariant systems.Variational methods over families of tensor networks enable the approximation of the low-energy properties of complex quantum Hamiltonians. Other methods also allow the simulation of time evolution, the calculation of low-energy excitations and much more.Symmetric tensor network states enable more efficient simulation methods and the description of fermionic systems, lattice gauge theories, topological order and the classification of phases of quantum matter.Tensor networks, such as the multiscale entanglement renormalization ansatz, have been linked to a possible lattice realization of the holographic principle in quantum gravity.Tensor networks also provide a natural framework for understanding machine learning and probabilistic language models.