Efficient Representation of Matrx Product State with Restricted Boltzmann Machine

One of the fundamental problems in many-body physics is the lack of an efficient representational ansatz for highly entangled quantum states. Tensor network state is potentially one of such ansatzes, especially in one-dimensional(1D) case, as its 1D form, matrix product state, has been proven an efficient representation of ground states of gapped 1D systems and seen a lot of applications in both numerical and analytical work. On the other hand, Restricted Boltzmann Machine (RBM), a probabilistic model widely used in machine learning, has recently drawn a lot of attentions as a successful variational ansatz in computing some many-body ground states. Here we prove that RBM can efficiently represent almost all matrix product states asymptotically thus serving as a new ansatz for quantum many-body states. We also give numerical experimental results as a support to our claim and concrete examples for useful many-body highly entangled states.