  The resemblance between the methods used in quantum-many body physics and in
machine learning has drawn considerable attention. In particular, tensor
networks (TNs) and deep learning architectures bear striking similarities to
the extent that TNs can be used for machine learning. Previous results used
one-dimensional TNs in image recognition, showing limited scalability and
flexibilities. In this work, we train two-dimensional hierarchical TNs to solve
image recognition problems, using a training algorithm derived from the
multi-scale entanglement renormalization ansatz. This approach introduces
mathematical connections among quantum many-body physics, quantum information
theory, and machine learning. While keeping the TN unitary in the training
phase, TN states are defined, which encode classes of images into quantum
many-body states. We study the quantum features of the TN states, including
quantum entanglement and fidelity. We find these quantities could be properties
that characterize the image classes, as well as the machine learning tasks.
