Holography as deep learning

Gan, Wen-Cong; Shu, Fu-Wen

doi:10.1142/s0218271817430209

Cited by 46 publications

(35 citation statements)

References 19 publications

(22 reference statements)

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“…Although many progress of RBM states has been made, the DBM states are less investigated [30]. There are several crucial reasons why we need deep neural network rather than shallow one: (i) the representational power of shallow network is limited, there exist states which can be efficiently represented by deep neural network while the shallow one can not represent [30]; (ii) aAny Boltzmann machine (BM) can be reduced into a DBM, this also makes some limits in usage of shallow BM (with just one hidden layer, viz, RBM) [30]; (iii) the hierarchy structure of deep neural is more suitable for encoding holography [49,55,56] and for procedure such as renormalization [57]. Now let us take a close look at the geometry of a DBM neural network.…”

Section: Entanglement Features Of Deep Neural Network Statesmentioning

confidence: 99%

Entanglement area law for shallow and deep quantum neural network states

Jia

et al. 2020

New J. Phys.

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A study of the artificial neural network representation of quantum many-body states is presented. The locality and entanglement properties of states for shallow and deep quantum neural networks are investigated in detail. By introducing the notion of local quasi-product states, for which the locally connected shallow feed-forward neural network states and restricted Boltzmann machine states are special cases, we show that Rényi entanglement entropies of all these states obey the entanglement area law. Besides, we also investigate the entanglement features of deep Boltzmann machine states and show that locality constraints imposed on the neural networks make the states obey the entanglement area law. Finally, as an application, we apply the notion of Rényi entanglement entropy to understand the power of neural networks, and show that image classification problems can be efficiently solved must obey the area law.

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Section: Entanglement Features Of Deep Neural Network Statesmentioning

confidence: 99%

Entanglement area law for shallow and deep quantum neural network states

Jia

et al. 2020

New J. Phys.

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“…The DBM states are also explored in various works. [29,42,43] In this section, we briefly review the progress in this direction.…”

Section: Representational Power Of Neural Network Statesmentioning

confidence: 99%

“…[28,38,39,41] Furthermore, the deep BM (DBM) states were also investigated under different approaches. [29,42,43] Despite all the progress in applying neural networks in quantum physics, many important topics still remain to be explored. The obvious topics are the exact definition of a quantum neural network state and the mathematics and physics behind the efficiency of quantum neural network states.…”

Section: Introductionmentioning

confidence: 99%

Quantum Neural Network States: A Brief Review of Methods and Applications

et al. 2019

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One of the main challenges of quantum many‐body physics is the exponential growth in the dimensionality of the Hilbert space with system size. This growth makes solving the Schrödinger equation of the system extremely difficult. Nonetheless, many physical systems have a simplified internal structure that typically makes the parameters needed to characterize their ground states exponentially smaller. Many numerical methods then become available to capture the physics of the system. Among modern numerical techniques, neural networks, which show great power in approximating functions and extracting features of big data, are now attracting much interest. In this work, the progress in using artificial neural networks to build quantum many‐body states is reviewed. The Boltzmann machine representation is taken as a prototypical example to illustrate various aspects of the neural network states. The classical neural networks are also briefly reviewed, and it is illustrated how to use neural networks to represent quantum states and density operators. Some physical properties of the neural network states are discussed. For applications, the progress in many‐body calculations based on neural network states, the neural network state approach to tomography, and the classical simulation of quantum computing based on Boltzmann machine states are briefly reviewed.

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“…The resemblance between the two approaches is beyond superficial. At the theoretical level, there is a mapping between deep learning and the renormalization group [15], which in turn connects holography and deep learning [16,17], and also allows to design networks from the perspective of quantum entanglement [18]. In turn, neural networks can represent quantum states [19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Machine learning by unitary tensor network of hierarchical tree structure

et al. 2019

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The resemblance between the methods used in quantum-many body physics and in machine learning has drawn considerable attention. In particular, tensor networks (TNs) and deep learning architectures bear striking similarities to the extent that TNs can be used for machine learning. Previous results used one-dimensional TNs in image recognition, showing limited scalability and flexibilities. In this work, we train two-dimensional hierarchical TNs to solve image recognition problems, using a training algorithm derived from the multi-scale entanglement renormalization ansatz. This approach introduces mathematical connections among quantum many-body physics, quantum information theory, and machine learning. While keeping the TN unitary in the training phase, TN states are defined, which encode classes of images into quantum many-body states. We study the quantum features of the TN states, including quantum entanglement and fidelity. We find these quantities could be properties that characterize the image classes, as well as the machine learning tasks.

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Holography as deep learning

Cited by 46 publications

References 19 publications

Entanglement area law for shallow and deep quantum neural network states

Entanglement area law for shallow and deep quantum neural network states

Quantum Neural Network States: A Brief Review of Methods and Applications

Machine learning by unitary tensor network of hierarchical tree structure

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