We study the representational power of Boltzmann machines (a type of neural network) in quantum many-body systems. We prove that any (local) tensor network state has a (local) neural network representation. The construction is almost optimal in the sense that the number of parameters in the neural network representation is almost linear in the number of nonzero parameters in the tensor network representation. Despite the difficulty of representing (gapped) chiral topological states with local tensor networks, we construct a quasilocal neural network representation for a chiral $p$-wave superconductor. These results demonstrate the power of Boltzmann machines.

中文翻译：

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张量网络和手性状态的神经网络表示

我们研究了玻尔兹曼机（一种神经网络）在量子多体系统中的表征能力。我们证明任何（本地）张量网络状态都有一个（本地）神经网络表示。在神经网络表示中的参数数量几乎与张量网络表示中的非零参数数量成线性关系的意义上，该构造几乎是最优的。尽管难以用局部张量网络表示（有隙的）手性拓扑状态，但我们为手性构造了一个拟局部神经网络表示$磷$-波超导体。这些结果证明了玻尔兹曼机的威力。