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

Bridgeman, Jacob C.; Chubb, Christopher T.

doi:10.1088/1751-8121/aa6dc3

Cited by 322 publications

(323 citation statements)

References 101 publications

(189 reference statements)

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“…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.…”

Section: −2mentioning

confidence: 99%

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Discrete gravity on random tensor network and holographic Rényi entropy

Han

Huang

2017

J. High Energ. Phys.

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In this paper we apply the discrete gravity and Regge calculus to tensor networks and Anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We construct the boundary many-body quantum state |Ψ using random tensor networks as the holographic mapping, applied to the Wheeler-deWitt wave function of bulk Euclidean discrete gravity in 3 dimensions. The entanglement Rényi entropy of |Ψ is shown to holographically relate to the on-shell action of Einstein gravity on a branch cover bulk manifold. The resulting Rényi entropy S n of |Ψ approximates with high precision the Rényi entropy of ground state in 2-dimensional conformal field theory (CFT). In particular it reproduces the correct n dependence. Our results develop the framework of realizing the AdS 3 /CFT 2 correspondence on random tensor networks, and provide a new proposal to approximate the CFT ground state.

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Section: −2mentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

Discrete gravity on random tensor network and holographic Rényi entropy

Han

Huang

2017

J. High Energ. Phys.

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“…[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.…”

Section: Gapless Hamiltonians With a Low Density Of Low-energy Statesmentioning

confidence: 99%

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

Arad

Landau

Vazirani

et al. 2017

Commun. Math. Phys.

144

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Abstract:One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n O(log n) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time isÕ(nM(n)), where M(n) is the time required to multiply two n × n matrices.

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“…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.…”

Section: B γ = 29mentioning

confidence: 59%

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

Czarnik

Corboz

2019

Phys. Rev. B

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We study second order finite temperature phase transitions of the 2D quantum Ising and interacting honeycomb fermions models using infinite projected entangled pair states (iPEPS). We obtain an iPEPS thermal state representation by Variational Tensor Network Renormalization (VTNR). We find that at the critical temperature Tc the iPEPS correlation length is finite for the computationally accessible values of the iPEPS bond dimension D. Motivated by this observation we investigate the application of Finite Correlation Length Scaling (FCLS), which has been previously used for iPEPS simulations of quantum critical points at T = 0, to obtain precise values of Tc and the universal critical exponents. We find that in the vicinity of Tc the behavior of observables follows well the one predicted by FCLS. Using FCLS we obtain Tc and the critical exponents in agreement with Quantum Monte Carlo (QMC) results except for couplings close to the quantum critical points where larger bond dimensions are required.

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

Cited by 322 publications

References 101 publications

Discrete gravity on random tensor network and holographic Rényi entropy

Discrete gravity on random tensor network and holographic Rényi entropy

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

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

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