Artificial neural networks and machine learning have now reached a new era
after several decades of improvement where applications are to explode in many
fields of science, industry, and technology. Here, we use artificial neural
networks to study an intriguing phenomenon in quantum physics--- the
topological phases of matter. We find that certain topological states, either
symmetry-protected or with intrinsic topological order, can be represented with
classical artificial neural networks. This is demonstrated by using three
concrete spin systems, the one-dimensional (1D) symmetry-protected topological
cluster state and the 2D and 3D toric code states with intrinsic topological
orders. For all three cases we show rigorously that the topological ground
states can be represented by short-range neural networks in an \textit{exact}
and \textit{efficient} fashion---the required number of hidden neurons is as
small as the number of physical spins and the number of parameters scales only
\textit{linearly} with the system size. For the 2D toric-code model, we find
that the proposed short-range neural networks can describe the excited states
with abelain anyons and their nontrivial mutual statistics as well. In
addition, by using reinforcement learning we show that neural networks are
capable of finding the topological ground states of non-integrable Hamiltonians
with strong interactions and studying their topological phase transitions. Our
results demonstrate explicitly the exceptional power of neural networks in
describing topological quantum states, and at the same time provide valuable
guidance to machine learning of topological phases in generic lattice models.

本文利用人工神经网络和机器学习研究了量子物理中的拓扑相，证明了短程神经网络能够精确有效地表示拓扑相的基态，能够描述受强相互作用影响非可积哈密顿量的拓扑相变，从而为通用格模型的拓扑相机器学习提供了指导。