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 $\delta$.

In one dimension, we prove that an area law for the Rényi entanglement entropy $R_\alpha$ with index $\alpha<1$ implies a matrix product state representation with bond dimension $\operatorname{poly}(1/\delta)$.

For (at most constant-fold degenerate) ground states of one-dimensional gapped Hamiltonians, a bond dimension almost linear in $1/\delta$ suffices.

In two dimensions, an area law for $R_\alpha(\alpha<1)$ implies a projected entangled pair state representation with bond dimension $e^{O(1/\delta)}$.

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.