Neural networks offer a novel approach to represent wave functions for solving quantum many-body problems. But what kinds of quantum states are efficiently represented by neural networks? To approach this problem, the authors study here entanglement properties of an ensemble of neural network states represented by random restricted Boltzmann machines. Phases with distinct entanglement features are identified and characterized. In particular, the authors show that these neural network states can look typical in their entanglement profile while still being distinguishable from a typical state by their fractal dimensions.
Neural networks offer a novel approach to represent wave functions for solving quantum many-body problems. But what kinds of quantum states are efficiently represented by neural networks? To approach this problem, the authors study here entanglement properties of an ensemble of neural network states represented by random restricted Boltzmann machines. Phases with distinct entanglement features are identified and characterized. In particular, the authors show that these neural network states can look typical in their entanglement profile while still being distinguishable from a typical state by their fractal dimensions.
One of the main challenges of quantum many-body physics is that the dimensionality of the Hilbert space grows exponentially with the system size, which makes it extremely difficult to solve the Schrodinger equations of the system. But typically, many physical systems have a simplified internal structure which makes the parameters needed to characterize their ground states exponentially smaller. This makes many numerical methods possible in capture the physics of the system. Among these modern numerical techniques, neural networks, which show great power in approximating functions and extracting features of the big data, are now attracting many interests. Neural network representation of quantum many-body states shows great potential in solving some traditionally difficult quantum problems involving large number of freedoms. In this work, we briefly review the progress of using the artificial neural network to build quantum many-body ansatz states. We take Boltzmann machine representation as a prototypical example to illustrate various aspects of neural network representation of quantum many-body states. We first briefly review the classical neural networks, then we illustrate how to use neural networks to represent quantum states and density operators. Some physical properties of the neural network states, including entanglement features, representational power, and the relation with tensor network states, are discussed. For applications, we briefly review the progress of many-body calculating based on neural network states, neural network state approach to tomography, and also the classical simulation of quantum computing based on Boltzmann machine states. At the end of the work, some outlooks and open problems are given.
Neural-Network Quantum States have been recently introduced as an Ansatz for describing the wave function of quantum many-body systems. We show that there are strong connections between Neural-Network Quantum States in the form of Restricted Boltzmann Machines and some classes of Tensor-Network states in arbitrary dimensions. In particular we demonstrate that short-range Restricted Boltzmann Machines are Entangled Plaquette States, while fully connected Restricted Boltzmann Machines are String-Bond States with a nonlocal geometry and low bond dimension. These results shed light on the underlying architecture of Restricted Boltzmann Machines and their efficiency at representing many-body quantum states. String-Bond States also provide a generic way of enhancing the power of Neural-Network Quantum States and a natural generalization to systems with larger local Hilbert space. We compare the advantages and drawbacks of these different classes of states and present a method to combine them together. This allows us to benefit from both the entanglement structure of Tensor Networks and the efficiency of Neural-Network Quantum States into a single Ansatz capable of targeting the wave function of strongly correlated systems. While it remains a challenge to describe states with chiral topological order using traditional Tensor Networks, we show that Neural-Network Quantum States and their String-Bond States extension can describe a lattice Fractional Quantum Hall state exactly. In addition, we provide numerical evidence that Neural-Network Quantum States can approximate a chiral spin liquid with better accuracy than Entangled Plaquette States and local String-Bond States. Our results demonstrate the efficiency of neural networks to describe complex quantum wave functions and pave the way towards the use of String-Bond States as a tool in more traditional machine-learning applications.
Abstract 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.
Part of the challenge for quantum many-body problems comes from the difficulty of representing large-scale quantum states, which in general requires an exponentially large number of parameters. Neural networks provide a powerful tool to represent quantum many-body states. An important open question is what characterizes the representational power of deep and shallow neural networks, which is of fundamental interest due to the popularity of deep learning methods. Here, we give a proof that, assuming a widely believed computational complexity conjecture, a deep neural network can efficiently represent most physical states, including the ground states of many-body Hamiltonians and states generated by quantum dynamics, while a shallow network representation with a restricted Boltzmann machine cannot efficiently represent some of those states.One of the challenges in studies of quantum many-body physics is finding an efficient way to record the large system wavefunctions. Here the authors present an analysis of the capabilities of recently-proposed neural network representations for storing physically accessible quantum states.
Modern deep learning has enabled unprecedented achievements in various domains. Nonetheless, employment of machine learning for wave function representations is focused on more traditional architectures such as restricted Boltzmann machines (RBMs) and fully connected neural networks. In this Letter, we establish that contemporary deep learning architectures, in the form of deep convolutional and recurrent networks, can efficiently represent highly entangled quantum systems. By constructing tensor network equivalents of these architectures, we identify an inherent reuse of information in the network operation as a key trait which distinguishes them from standard tensor network-based representations, and which enhances their entanglement capacity. Our results show that such architectures can support volume-law entanglement scaling, polynomially more efficiently than presently employed RBMs. Thus, beyond a quantification of the entanglement capacity of leading deep learning architectures, our analysis formally motivates a shift of trending neural-network-based wave function representations closer to the state-of-the-art in machine learning.
Abstract 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.