  The restricted Boltzmann machine (RBM) is one of the fundamental building
blocks of deep learning. RBM finds wide applications in dimensional reduction,
feature extraction, and recommender systems via modeling the probability
distributions of a variety of input data including natural images, speech
signals, and customer ratings, etc. We build a bridge between RBM and tensor
network states (TNS) widely used in quantum many-body physics research. We
devise efficient algorithms to translate an RBM into the commonly used TNS.
Conversely, we give sufficient and necessary conditions to determine whether a
TNS can be transformed into an RBM of given architectures. Revealing these
general and constructive connections can cross-fertilize both deep learning and
quantum many-body physics. Notably, by exploiting the entanglement entropy
bound of TNS, we can rigorously quantify the expressive power of RBM on complex
data sets. Insights into TNS and its entanglement capacity can guide the design
of more powerful deep learning architectures. On the other hand, RBM can
represent quantum many-body states with fewer parameters compared to TNS, which
may allow more efficient classical simulations.
