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arXiv · 1903.10048

Approximating local properties by tensor network states with constant bond dimension

Yichen Huang
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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 $\delta$. In one dimension, we prove that an area law for the R\'enyi 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.

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Yichen Huang. 2019-03-24. Approximating local properties by tensor network states with constant bond dimension. https://doi.org/10.1109/tit.2026.3694133

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