The nonequilibrium quantum many-body problem as a paradigm for extreme data science

Freericks, J. K.; Nikolić, Branislav K.

doi:10.1142/s0217979214300217

Cited by 10 publications

(15 citation statements)

References 98 publications

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“…These studies have shown a favorable polynomial scaling with respect to the system's number of particles. Such findings are consistent across many fields [7][8][9][10][11][12]; despite system complexity in the underlying physics, much of observed behavior is compressed, i.e., the dominant dynamics is manifested in significantly lower dimensions. Compression arises also in the search over the quantum control landscape as a favorable scaling of control complexity [13,14].…”

Section: Introductionsupporting

confidence: 63%

Quantum System Compression: A Hamiltonian Guided Walk Through Hilbert Space

Kosut,

Ho,

Rabitz

2020

Preprint

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We present a systematic study of quantum system compression for the evolution of generic many-body problems. The necessary numerical simulations of such systems are seriously hindered by the exponential growth of the Hilbert space dimension with the number of particles. For a constant Hamiltonian system of Hilbert space dimension n whose frequencies range from fmin to fmax, we show via a proper orthogonal decomposition, that for a run-time T , the dominant dynamics are compressed in the neighborhood of a subspace whose dimension is the smallest integer larger than the time-bandwidth product ∆ = (fmax − fmin)T . We also show how the distribution of initial states can further compress the system dimension. Under the stated conditions, the time-bandwidth estimate reveals the existence of an effective compressed model whose dimension is derived solely from system properties and not dependent on the particular implementation of a variational simulator, such as a machine learning system, or quantum device. However, finding an efficient solution procedure is dependent on the simulator implementation, which is not discussed in this paper. In addition, we show that the compression rendered by the proper orthogonal decomposition encoding method can be further strengthened via a multi-layer autoencoder. Finally, we present numerical illustrations to affirm the compression behavior in time-varying Hamiltonian dynamics in the presence of external fields. We also discuss the potential implications of the findings for machine learning tools to efficiently solve the many-body or other high dimensional Schrödinger equations.

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Section: Introductionsupporting

confidence: 63%

Quantum System Compression: A Hamiltonian Guided Walk Through Hilbert Space

Kosut,

Ho,

Rabitz

2020

Preprint

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“…These encompass fundamental questions ranging from the dynamical properties of high-dimensional systems [10,11] to the exact ground-state properties of strongly interacting fermions [12,13]. At the heart of this lack of understanding lyes the difficulty in finding a general strategy to reduce the exponential complexity of the full many-body wave function down to its most essential features [14].In a much broader context, the problem resides in the realm of dimensional reduction and feature extraction. Among the most successful techniques to attack these problems, artificial neural networks play a prominent role [15].…”

mentioning

confidence: 99%

Solving the quantum many-body problem with artificial neural networks

Carleo¹,

Troyer²

2017

Science

1,987

2,165

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The challenge posed by the many-body problem in quantum physics originates from the difficulty of describing the non-trivial correlations encoded in the exponential complexity of the many-body wave function. Here we demonstrate that systematic machine learning of the wave function can reduce this complexity to a tractable computational form, for some notable cases of physical interest. We introduce a variational representation of quantum states based on artificial neural networks with variable number of hidden neurons. A reinforcement-learning scheme is then demonstrated, capable of either finding the ground-state or describing the unitary time evolution of complex interacting quantum systems. We show that this approach achieves very high accuracy in the description of equilibrium and dynamical properties of prototypical interacting spins models in both one and two dimensions, thus offering a new powerful tool to solve the quantum many-body problem.The wave function Ψ is the fundamental object in quantum physics and possibly the hardest to grasp in a classical world. Ψ is a monolithic mathematical quantity that contains all the information on a quantum state, be it a single particle or a complex molecule. In principle, an exponential amount of information is needed to fully encode a generic many-body quantum state. However, Nature often proves herself benevolent, and a wave function representing a physical many-body system can be typically characterized by an amount of information much smaller than the maximum capacity of the corresponding Hilbert space. A limited amount of quantum entanglement, as well as the typicality of a small number of physical states, are then the blocks on which modern approaches build upon to solve the many-body Schrödinger's equation with a limited amount of classical resources.Numerical approaches directly relying on the wave function can either sample a finite number of physically relevant configurations or perform an efficient compression of the quantum state. Stochastic approaches, like quantum Monte Carlo (QMC) methods, belong to the first category and rely on probabilistic frameworks typically demanding a positive-semidefinite wave function. [1][2][3]. Compression approaches instead rely on efficient representations of the wave function, and most notably in terms of matrix product states (MPS) [4][5][6] or more general tensor networks [7,8]. Examples of systems where existing approaches fail are however numerous, mostly due to the sign problem in QMC [9], and to the inefficiency of current compression approaches in high-dimensional systems. As a result, despite the striking success of these methods, a large number of unexplored regimes exist, including many interesting open problems. These encompass fundamental questions ranging from the dynamical properties of high-dimensional systems [10,11] to the exact ground-state properties of strongly interacting fermions [12,13]. At the heart of this lack of understanding lyes the difficulty in finding a general strategy to reduce the exp...

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“…These ideas acquire a very special meaning in the context of the quantum many-body problem where one deals with datasets that are exponentially large. Sophisticated techniques have been developed to tackle this difficult challenge, such as compressing the data by using information theory and machine learning tools [1] very similar in spirit to algorithms to compress images and videos. In our case, datasets are comprised of all possible electronic configurations and cannot be stored in the memory of the largest supercomputer.…”

Section: Introductionmentioning

confidence: 99%

“…The probability distribution of the visible vectors is formulated by first introducing a joint probability distribution for pairs of visible and hidden vectors from an energy function and Boltzmann weighting. Finally, the probability distribution for visible vectors is taken to be the sum of the joint probability distribution over all possible configurations of the hidden vectors: Carleo and Troyer [9] introduced a variational wavefunction for a spin- 1 2 system of N sites, which is inspired by the functional form of RBM. The visible layer corresponds to the spin configurations σ z = (σ z 1 , σ z 2 , · · · , σ z N ).…”

Section: Introductionmentioning

confidence: 99%

Machine learning approach to dynamical properties of quantum many-body systems

Hendry

Feiguin

2019

Phys. Rev. B

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Variational representations of quantum states abound and have successfully been used to guess ground-state properties of quantum many-body systems. Some are based on partial physical insight (Jastrow, Gutzwiller projected, and fractional quantum Hall states, for instance), and others operate as a black box that may contain information about the underlying structure of entanglement and correlations (tensor networks, neural networks) and offer the advantage of a large set of variational parameters that can be efficiently optimized. However, using variational approaches to study excited states and, in particular, calculating the excitation spectrum, remains a challenge. We present a variational method to calculate the dynamical properties and spectral functions of quantum manybody systems in the frequency domain, where the Green's function of the problem is encoded in the form of a restricted Boltzmann machine (RBM). We introduce a natural gradient descent approach to solve linear systems of equations and use Monte Carlo to obtain the dynamical correlation function. In addition, we propose a strategy to regularize the results that improves the accuracy dramatically.As an illustration, we study the dynamical spin structure factor of the one dimensional J1 − J2 Heisenberg model. The method is general and can be extended to other variational forms.

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The nonequilibrium quantum many-body problem as a paradigm for extreme data science

Cited by 10 publications

References 98 publications

Quantum System Compression: A Hamiltonian Guided Walk Through Hilbert Space

Quantum System Compression: A Hamiltonian Guided Walk Through Hilbert Space

Solving the quantum many-body problem with artificial neural networks

Machine learning approach to dynamical properties of quantum many-body systems

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