Hand-waving and interpretive dance: an introductory course on tensor networks

Abstract

View the article online for updates and enhancements. AbstractThe curse of dimensionality associated with the Hilbert space of spin systems provides a significant obstruction to the study of condensed matter systems. Tensor networks have proven an important tool in attempting to overcome this difficulty in both the numerical and analytic regimes.These notes form the basis for a seven lecture course, introducing the basics of a range of common tensor networks and algorithms. In particular, we cover: introductory tensor network notation, applications to quantum information, basic properties of matrix product states, a classification of quantum phases using tensor networks, algorithms for finding matrix product states, basic properties of projected entangled pair states, and multiscale entanglement renormalisation ansatz states.The lectures are intended to be generally accessible, although the relevance of many of the examples may be lost on students without a background in many-body physics/quantum information. For each lecture, several problems are given, with worked solutions in an ancillary file.

Discrete gravity on random tensor network and holographic Rényi entropy

“…Tetrahedron edges are geodesics in the hyperbolic space, and have edge lengths L . The set of edge lengths {L } on the triangulation defines a discrete metric of Regge 1 The set of | a are shown to form an overcomplete basis in the boundary Hilbert space if there is no restriction on the link variables a [32]. Whether | a with a as tiling of the spatial slice still form an over-complete basis is an interesting question but doesn't affect the present analysis.…”

“…It is originated in condense matter physics because tensor network states efficiently compute ground states of many-body quantum systems [1,2]. In addition, the tensor network has wide applications to quantum information theory by its relation to error correcting codes and quantum entanglement [3].…”

Rigorous RG Algorithms and Area Laws for Low Energy Eigenstates in 1D

“…[8,28] for background material on MPS and their use in variational algorithms) for a basis of (a good approximation to) the ground space (resp. low energy subspace) of the Hamiltonian.…”

Finite correlation length scaling with infinite projected entangled pair states at finite temperature

“…Similar values for T c and β are obtained with VTNR using an optimization with [79]. In (b) the m /m collapse (18) gives Tc = 0.609 (4). In (c) β estimation based on a data collapse using Eq.…”