Entanglement area law for shallow and deep quantum neural network states

Jia, Zhih-Ahn; Lu, Wei; Wu, Yu-Chun; Guo, Guang-Can

doi:10.1088/1367-2630/ab8262

Cited by 14 publications

(11 citation statements)

References 54 publications

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“…Because by construction our ansatz represents a product of terms with local support, it can be seen as an effective implementation of a quasi-product states introduced in [22]. This suggests that the entanglement entropy of the state can obey at most an area law.…”

Section: Physical Properties Of the Gnn Ansatzmentioning

confidence: 99%

Learning ground states of quantum Hamiltonians with graph networks

Kochkov¹,

Pfaff²,

Sánchez‐González³

et al. 2021

Preprint

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Solving for the lowest energy eigenstate of the many-body Schrodinger equation is a cornerstone problem that hinders understanding of a variety of quantum phenomena. The difficulty arises from the exponential nature of the Hilbert space which casts the governing equations as an eigenvalue problem of exponentially large, structured matrices. Variational methods approach this problem by searching for the best approximation within a lower-dimensional variational manifold. In this work we use graph neural networks to define a structured variational manifold and optimize its parameters to find high quality approximations of the lowest energy solutions on a diverse set of Heisenberg Hamiltonians. Using graph networks we learn distributed representations that by construction respect underlying physical symmetries of the problem and generalize to problems of larger size. Our approach achieves state-of-the-art results on a set of quantum many-body benchmark problems and works well on problems whose solutions are not positive-definite. The discussed techniques hold promise of being a useful tool for studying quantum many-body systems and providing insights into optimization and implicit modeling of exponentially-sized objects.

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Section: Physical Properties Of the Gnn Ansatzmentioning

confidence: 99%

Learning ground states of quantum Hamiltonians with graph networks

Kochkov¹,

Pfaff²,

Sánchez‐González³

et al. 2021

Preprint

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“…For one-dimensional many-body states, two thoroughly studied, popular but different structures exist-multiscale entanglement renormalization ansatz (MERA) [36] and matrix product states (MPS) [37], of which the EE scales with the subsystem size or not at all, respectively [35]. For most pertinent studies, MPS has been proven to be efficient enough to be applicable to a variety of tasks [38][39][40][41]. However, our experiments show that, regarding our entanglement-structured design of the new tensorized LSTM architecture, LSTM-MERA performs even better than LSTM-MPS in general without increasing the number of parameters.…”

Section: Introductionmentioning

confidence: 99%

Entanglement-Structured LSTM Boosts Chaotic Time Series Forecasting

Meng

Yang

2021

Entropy

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Traditional machine-learning methods are inefficient in capturing chaos in nonlinear dynamical systems, especially when the time difference Δt between consecutive steps is so large that the extracted time series looks apparently random. Here, we introduce a new long-short-term-memory (LSTM)-based recurrent architecture by tensorizing the cell-state-to-state propagation therein, maintaining the long-term memory feature of LSTM, while simultaneously enhancing the learning of short-term nonlinear complexity. We stress that the global minima of training can be most efficiently reached by our tensor structure where all nonlinear terms, up to some polynomial order, are treated explicitly and weighted equally. The efficiency and generality of our architecture are systematically investigated and tested through theoretical analysis and experimental examinations. In our design, we have explicitly used two different many-body entanglement structures—matrix product states (MPS) and the multiscale entanglement renormalization ansatz (MERA)—as physics-inspired tensor decomposition techniques, from which we find that MERA generally performs better than MPS, hence conjecturing that the learnability of chaos is determined not only by the number of free parameters but also the tensor complexity—recognized as how entanglement entropy scales with varying matricization of the tensor.

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“…Specifically, in this paper, we consider a neural-network inspired variational Ansatz for the quantum wave function [44]. Machine learning techniques have been proved fruitful to the field of many-body quantum physics [44][45][46][47][48][49][50][51][52][53][54][55]. Specifically, exciting progress has been made in identifying quantum phases and transitions among them, either symmetry-broken phases [47][48][49][50]56] or topological phases [57], establishing connections to renormalization group techniques [58,59] and perturbation theory [60].…”

Section: Introductionmentioning

confidence: 99%

“…In addition, machine learning ideas have also been used in measuring quantum entanglement and wave function tomography [55,61,62]. We focus on a particular neural network variational state, referred to as restricted Boltzmann machines (RBM) quantum states which have been proven [54,63] to capture both an area-law as well as volume-law worth of entanglement depending on the locality of the neural network. In addition, recent work [52] showed that RBM states can also be used to parametrize quantum wave functions with non-Abelian symmetries.…”

Section: Introductionmentioning

confidence: 99%

Entanglement transitions from restricted Boltzmann machines

Medina,

Vasseur,

Serbyn

2021

Preprint

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The search for novel entangled phases of matter has lead to the recent discovery of a new class of "entanglement transitions", exemplified by random tensor networks and monitored quantum circuits.Most known examples can be understood as some classical ordering transitions in an underlying statistical mechanics model, where entanglement maps onto the free energy cost of inserting a domain wall. In this paper, we study the possibility of entanglement transitions driven by physics beyond such statistical mechanics mappings. Motivated by recent applications of neural networkinspired variational Ansätze, we investigate under what conditions on the variational parameters these Ansätze can capture an entanglement transition. We study the entanglement scaling of shortrange restricted Boltzmann machine (RBM) quantum states with random phases. For uncorrelated random phases, we analytically demonstrate the absence of an entanglement transition and reveal subtle finite size effects in finite size numerical simulations. Introducing phases with correlations decaying as 1/r α in real space, we observe three regions with a different scaling of entanglement entropy depending on the exponent α. We study the nature of the transition between these regions, finding numerical evidence for critical behavior. Our work establishes the presence of long-range correlated phases in RBM-based wave functions as a required ingredient for entanglement transitions.

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Entanglement area law for shallow and deep quantum neural network states

Cited by 14 publications

References 54 publications

Learning ground states of quantum Hamiltonians with graph networks

Learning ground states of quantum Hamiltonians with graph networks

Entanglement-Structured LSTM Boosts Chaotic Time Series Forecasting

Entanglement transitions from restricted Boltzmann machines

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