  Recently, quantum-state representation using artificial neural networks has
started to be recognized as a powerful tool. However, due to the black-box
nature of machine learning, it is difficult to analyze what machine learns or
why it is powerful. Here, by applying one of the simplest neural networks, the
restricted Boltzmann machine (RBM), to the ground-state representation of the
one-dimensional (1D) transverse-field Ising (TFI) model, we make an attempt to
directly analyze the optimized network parameters. In the RBM optimization, a
zero-temperature quantum state is mapped onto a finite-temperature classical
state of the extended Ising spins that constitute the RBM. We find that the
quantum phase transition from the ordered phase to the disordered phase in the
1D TFI model by increasing the transverse field is clearly reflected in the
behaviors of the optimized RBM parameters and hence in the finite-temperature
phase diagram of the classical RBM Ising system. The present finding of a
correspondence between the neural-network parameters and quantum phases
suggests that a careful investigation of the neural-network parameters may
provide a new route to extracting nontrivial physical insights from the
neural-network wave functions.
