Tensor Network States and Geometry

By Glen Evenbly, Guifré Vidal. 2011

336 citations recorded by the source corpus.

Tensor network states are used to approximate ground states of local Hamiltonians on a lattice in D spatial dimensions. Different types of tensor network states can be seen to generate different geometries. Matrix product states (MPS) in D=1 dimensions, as well as projected entangled pair states (PEPS) in D>1 dimensions, reproduce the D-dimensional physical geometry of the lattice model; in contrast, the multi-scale entanglement renormalization ansatz (MERA) generates a (D+1)-dimensional holographic geometry. Here we focus on homogeneous tensor networks, where all the tensors in the network are copies of the same tensor, and argue that certain structural properties of the resulting many-body states are preconditioned by the geometry of the tensor network and are therefore largely independent of the choice of variational parameters. Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for D=1 systems is seen to be determined by the structure of geodesics in the physical and holographic geometries, respectively; whereas the asymptotic scaling of entanglement entropy is seen to always obey a simple boundary law—that is, again in the relevant geometry. This geometrical interpretation offers a simple and unifying framework to understand the structural properties of, and helps clarify the relation between, different tensor network states. In addition, it has recently motivated the branching MERA, a generalization of the MERA capable of reproducing violations of the entropic boundary law in D>1 dimensions.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bc3ce3c72bc5518644d8ce574cc6d31d8cd8245e

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Citations in the corpus

Space from Hilbert Space: Recovering Geometry from Bulk Entanglement (2016, background): “…Effective Theory 22 B. Energy and Gravity 24 VI. Discussion 27 Acknowledgements 28 A. Redundancy-Constraint and Coarse-graining 29 B. Entanglement Perturbations and Coarse Curvature 30 References 32 I. INTRODUCTION Quantum-mechanical theories are generally thought of as theories of something.”
The density matrix renormalization group for ab initio quantum chemistry (2014, background): “and therefore only algebraically with increasing lattice distance ∆x [55, 59].”
Entanglement is not enough (2014, background): “Tensor networks are thought of as existing in bulk space–at a fixed instant which represent the state at that time [14, 18, 19].”
Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, background): “First of all, the construction of MERAs (and TTNs) can be associated to a coarse tesselation of an Anti-de Sitter geometry, where the renormalization direction coincides with the radial coordinate (Evenbly and Vidal, 2011; Swingle, 2012).”
Tensor product methods and entanglement optimization for ab initio quantum chemistry (2014, background): “An obvious observation(100) following from the above will be that the minimal dimension ri is the rank of the i-th matricization (67): Theorem 3.”
Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT (2017, background): “ path-integration into a tensor network computation. Tensor networks are a graphical description of wave functionals in quantum many-body systems in terms of networks of quantum entanglement (see e.g.[41,42]). The optimization of tensor network was introduced in [20], called tensor network renormali”
Supervised Learning with Tensor Networks (2016, background): “Tensor methods have also become prominent in the field of physics, especially the use of tensor networks which accurately capture very high-order tensors while avoiding the the curse of dimensionality through a particular geometry of low-order contracted tensors [6].”
Surface/State Correspondence as a Generalized Holography (2015, background): “renormalization ansatz (MERA), introduced in [6].”
Holographic Complexity Equals Bulk Action? (2015, neutral): “[1,2,4–8]).”
The holographic entropy cone (2015, methodology): “The entropy inequalities that we consider in this paper hold for any physical system that satisfies a version of the Ryu-Takayanagi formula or the discrete entropy formula – such as MERA states built from generic tensors [47] and the tensor networks constructed recently in [48].”