Approximating local properties by tensor network states with constant bond dimension

Classical simulation of quantum many-body systems is a fundamental challenge due to their exponentially large Hilbert spaces. Tensor network states are a powerful ansatz to efficiently represent many physically relevant quantum states. A key question is the bond dimension—which determines the number of parameters in the ansatz—required to approximate all local properties to accuracy δ. In one dimension, we prove that an area law for the Rényi entanglement entropy Rα with index α < 1 implies a matrix product state representation with bond dimension poly(1/δ). For (at most constant-fold degenerate) ground states of one-dimensional gapped Hamiltonians, a bond dimension almost linear in 1/δ suffices. In two dimensions, an area law for Rα(α < 1) implies a projected entangled pair state representation with bond dimension eO(1/δ). In both one and two dimensions, analogous results are obtained for states with logarithmic corrections to the area law. These findings rigorously justify the common practice of using a system-size-independent bond dimension in tensor network simulations.