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A study of the artificial neural network representation of quantum many-bodystates is presented. The locality and entanglement properties of states forshallow and deep quantum neural networks are investigated in detail. Byintroducing the notion of local quasi-product states, for which the locallyconnected shallow feed-forward neural network states and restricted Boltzmannmachine states are special cases, we show that R\'{e}nyi entanglement entropiesof all these states obey the entanglement area law. Besides, we alsoinvestigate the entanglement features of deep Boltzmann machine states and showthat locality constraints imposed on the neural networks make the states obeythe entanglement area law. Finally, as an application, we apply the notion ofR\'{e}nyi entanglement entropy to understanding the power of neural networksand show that image classification problems which can be efficiently solvedmust obey the area law.

Entanglement Area Law for Shallow and Deep Quantum Neural Network States