From tensor network quantum states to tensorial recurrent neural networks

Tensor networks (TN) have been extensively used to represent the states of quantum many-body physical systems. Matrix product states (MPS) are suitable to capture the ground state of 1D gapped Hamiltonians but not 2D ones, and More powerful TNs cannot be efficiently contracted in general. We show that any MPS can be exactly represented by a recurrent neural network (RNN) with a linear memory update, and generalize it to 2D lattices using a multilinear memory update. It supports perfect sampling and wave function evaluation in polynomial time, and can represent an area law of entanglement entropy. Numerical evidence shows that it can encode the wave function using a bond dimension lower by orders of magnitude when compared to MPS, with an accuracy that can be systematically improved by increasing the bond dimension.