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 Renyi 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 Renyi entanglement entropy to understanding the power of neural networks and show that image classification problems which can be efficiently solved must obey the area law.

中文翻译：

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浅层和深层量子神经网络态的纠缠区定律

介绍了对量子多体态的人工神经网络表示的研究。详细研究了浅层和深层量子神经网络的状态的局域性和纠缠特性。通过引入局部准积状态的概念，其中局部连接的浅前馈神经网络状态和受限玻尔兹曼机状态是特例，我们证明了所有这些状态的人一纠缠熵都服从纠缠面积定律。此外，我们还研究了深玻尔兹曼机状态的纠缠特征，并表明施加在神经网络上的局部约束使状态服从纠缠面积定律。最后，作为一个应用程序，