Approximating local properties by tensor network states with constant bond dimension

Abstract

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 $\hat{I}´$. In one dimension, we prove that an area law for the Rényi entanglement entropy $R_{\hat{I}}\pm$ with index $\hat{I}\pm <1$ implies a matrix product state representation with bond dimension $\mathrm{poly}(1/\hat{I}´)$. For (at most constant-fold degenerate) ground states of one-dimensional gapped Hamiltonians, a bond dimension almost linear in $1/\hat{I}´$ suffices. In two dimensions, an area law for $R_{\hat{I}}\pm (\hat{I}\pm <1)$ implies a projected entangled pair state representation with bond dimension $e^{O(1/\hat{I}´)}$. 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.

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