  The challenge of quantum many-body problems comes from the difficulty to
represent large-scale quantum states, which in general requires an
exponentially large number of parameters. Recently, a connection has been made
between quantum many-body states and the neural network representation
(\textit{arXiv:1606.02318}). An important open question is what characterizes
the representational power of deep and shallow neural networks, which is of
fundamental interest due to popularity of the deep learning methods. Here, we
give a rigorous proof that a deep neural network can efficiently represent most
physical states, including those generated by any polynomial size quantum
circuits or ground states of many body Hamiltonians with polynomial-size gaps,
while a shallow network through a restricted Boltzmann machine cannot
efficiently represent those states unless the polynomial hierarchy in
computational complexity theory collapses.
