Two-dimensional Automata of Highly Entangled States

We develop an approach to generate highly entangled ground states of local Hamiltonians. We extend the definition of pushdown automata to two dimensions by adding a stack structure to the two-dimensional online tesselation automata. To classify the complexity of the generated states, we consider the generalization of context-free language (CFL) by adapting the Chomsky normal form to two-dimensional geometries. Such generalization enables us to construct local projectors forming local Hamiltonians. Through the extended Chomsky normal form, we also investigate the relation between general language complexity and Hamiltonian locality. Finally, we interpret the normal form as the local transformation of the random processes, the universality of which can help to classify the entanglement scaling.