Generalized Transfer Matrix States from Artificial Neural Networks

We propose and investigate a new family of quantum states, coined generalized transfer matrix states (GTMS), which bridges between tensor network states and states derived from artificial neural networks (ANNs). In particular, we show by means of a constructive embedding that the class of GTMS contains generic matrix product states while at the same time being capable of capturing more long-ranged quantum correlations that go beyond the area-law entanglement properties of tensor networks. While generic deep ANNs are hard to contract, meaning that the corresponding state amplitude can not be exactly evaluated, the GTMS network is shown to be analytically contractible using transfer matrix methods. With numerical simulations, we demonstrate how the GTMS network learns random matrix product states in a supervised learning scheme, and how augmenting the network by long-ranged couplings leads to the onset of volume-law entanglement scaling. We argue that this capability of capturing long-range quantum correlations makes GTMS a promising candidate for the study of critical and dynamical quantum many-body systems.