Continuous tensor network renormalization for quantum fields

Hu, Qi; Franco-Rubio, Adrián; Vidal, Guifré

doi:10.48550/arxiv.1809.05176

Cited by 8 publications

(7 citation statements)

References 57 publications

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“…An interesting circle of ideas in this context is the tensor network interpretation of the holographic duality, which suggests that the bulk Cauchy slice should be thought of as a tensor network. A tensor network, in particular the MERA [31,32,[52][53][54][55][56][57][58], is a variational ansatz for the wavefunctions of states in a CFT, which makes key use of the entanglement structure of these states from a position-space renormalization group perspective. In particular, the wavefunction is built as a quantum circuit, with successive layers of local operations called "disentanglers" and "isometries".…”

Section: Surface-state Correspondence and Tensor Networkmentioning

confidence: 99%

On the flow of states under $T\overline{T}$

Kruthoff¹,

Parrikar²

2020

Preprint

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We study the T T deformation of two dimensional quantum field theories from a Hamiltonian point of view, focusing on aspects of the theory in Lorentzian signature. Our starting point is a simple rewriting of the spatial integral of the T T operator, which directly implies the deformed energy spectrum of the theory. Using this rewriting, we then derive flow equations for various quantities in the deformed theory, such as energy eigenstates, operators, and correlation functions. On the plane, we find that the deformation merely has the effect of implementing successive canonical/Bogoliubov transformations along the flow. This leads us to define a class of non-local, "dressed" operators (including a dressed stress tensor) which satisfy the same commutation relations as in the undeformed theory. This further implies that on the plane, the deformed theory retains its symmetry algebra, including conformal symmetry, if the original theory is a CFT. On the cylinder the T T deformation is much more non-trivial, but even so, correlation functions of certain dressed operators are integral transforms of the original ones. Finally, we propose a tensor network interpretation of our results in the context of AdS/CFT.

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Section: Surface-state Correspondence and Tensor Networkmentioning

confidence: 99%

On the flow of states under $T\overline{T}$

Kruthoff¹,

Parrikar²

2020

Preprint

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“…Tensor networks are prominent approaches for studying classical statistical physics and quantum many-body physics problems [1][2][3]. In recent years, its application has expanded rapidly to diverse regions include simulating and designing of quantum circuits [4][5][6][7], quantum error correction [8,9], machine learning [10][11][12][13][14], language modeling [15,16], quantum field theory [17][18][19][20] and holography duality [21,22].…”

Section: Introductionmentioning

confidence: 99%

Differentiable Programming Tensor Networks

et al. 2019

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Differentiable programming is a fresh programming paradigm which composes parameterized algorithmic components and trains them using automatic differentiation. The concept emerges from deep learning but is not only limited to training neural networks. We present theory and practice of programming tensor network algorithms in a fully differentiable way. By formulating the tensor network algorithm as a computation graph, one can compute higher order derivatives of the program accurately and efficiently using automatic differentiation. We present essential techniques to differentiate through the tensor networks contraction algorithms, including numerical stable differentiation for tensor decompositions and efficient backpropagation through fixed point iterations. As a demonstration, we compute the specific heat of the Ising model directly by taking the second order derivative of the free energy obtained in the tensor renormalization group calculation. Next, we perform gradient based variational optimization of infinite projected entangled pair states for quantum antiferromagnetic Heisenberg model and obtain start-of-the-art variational energy and magnetization with moderate efforts. Differentiable programming removes laborious human efforts in deriving and implementing analytical gradients for tensor network programs, which opens the door to more innovations in tensor network algorithms and applications. * wanglei@iphy.ac.cn † txiang@iphy.ac.cn This fact has limited broad adoption of the gradient based optimization of tensor network states to more complex systems. Alternative approaches, such as computing the gradient using numerical derivative has limited accuracy and efficiency, therefore only applies to cases with few variational parameters [36,37]. While deriving the gradient manually using the chain rule is only manageable for purposely designed simple tensor network structures [38].Differentiable programming provides an elegant and efficient solution to these problems by composing the whole tensor network program in a fully differentiable manner. In this paper, we present essential automatic differentiation techniques which compute (higher order) derivatives of tensor network programs efficiently to numeric precision. This progress opens the door to gradient-based (and even Hessian-based) optimization of tensor network states in a general setting. Moreover, computing (higher order) gradients of the output of a tensor network algorithm offer a straightforward approach to compute physical quantities such as the specific heat and magnetic susceptibilities. The differentiable programming approach is agnostic to the detailed lattice geometry, Hamiltonian, and tensor network contraction schemes. Therefore, the approach is general enough to support a wide class of tensor network applications.We will focus on applications which involve two dimensional infinite tensor networks where the differentiable programming techniques offer significant benefits compared to the conventional approaches. We show that after solving m...

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“…However, there are other approaches to the RG flow. Examples include the continuous Tensor Network Renormalization (cTNR) in [56], the generalization of cMERA using Euclidean pathintegrals [24,57], the RG flow for free O(N ) model using Polchinski's exact RG [58]. In some of these approaches the map from the IR physics to the UV is no longer an exact isometry.…”

Section: Summary and Discussionmentioning

confidence: 99%

Renormalization group and approximate error correction

Furuya,

Lashkari,

Moosa

2021

Preprint

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In renormalization group (RG) flow, the low energy states form a code subspace that is approximately protected against the local short-distance errors. We motivate this connection with an example of spin-blocking RG in classical spin models. We consider the continuous multi-scale renormalization ansatz (cMERA) for massive free fields as a concrete example of real-space RG in quantum field theory (QFT) and show that the low-energy coherent states are approximately protected from the errors caused by the high-energy localized coherent operators. In holographic RG flows, we study the phase transition in the entanglement wedge of a single region and argue that one needs to define the price and the distance of the code with respect to the reconstructable wedge.

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Continuous tensor network renormalization for quantum fields

Cited by 8 publications

References 57 publications

On the flow of states under $T\overline{T}$

On the flow of states under $T\overline{T}$

Differentiable Programming Tensor Networks

Renormalization group and approximate error correction

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