From Tensor Network Quantum States to Tensorial Recurrent Neural Networks

Considering the relation between neural networks (NN) and TN, the first works focused on the restricted Boltzmann machines (RBM), which are one of the simplest Tensor networks (TN) have been extensively used to classes of NN. It is impossible to efficiently map an represent the states of quantum many-body physical systems RBM onto a TN, as they correspond to string-bond states [1-3]. Matrix product states (MPS) are possibly with an arbitrary nonlocal geometry [28]. This result was the simplest family of TN, and are suitable to capture later refined to show that an RBM may correspond to an the ground state of 1D gapped Hamiltonians [4, 5]. They MPS with an exponentially large bond dimension, and can be contracted in polynomial time to compute physical only short-range RBM can be mapped onto efficiently quantities exactly, and optimized by density matrix computable entangled plaquette states [31]. Similar results renormalization group (DMRG) [6] when used as variational have been obtained that deep Boltzmann machines ansätze. More powerful TN architectures that with proper constraints can be mapped onto TN that cannot be efficiently contracted in general have been are efficiently computable through transfer matrix methods proposed later, notably projected entangled pair states [32].