We present a novel mapping for studying 2D many-body quantum systems by solving an effective, one-dimensional long-range model in place of the original two-dimensional short-range one. In particular, we address the problem of choosing an efficient mapping from the 2D lattice to a 1D chain that optimally preserves the locality of interactions within the TN structure. By using Matrix Product States (MPS) and Tree Tensor Network (TTN) algorithms, we compute the ground state of the 2D quantum Ising model in transverse field with lattice size up to $64\times64$, comparing the results obtained from different mappings based on two space-filling curves, the snake curve and the Hilbert curve. We show that the locality-preserving properties of the Hilbert curve leads to a clear improvement of numerical precision, especially for large sizes, and turns out to provide the best performances for the simulation of 2D lattice systems via 1D TN structures.

中文翻译：

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希尔伯特曲线与希尔伯特空间：利用分形 2D 覆盖来提高张量网络效率

我们通过求解一个有效的一维长程模型代替原始的二维短程模型，提出了一种用于研究二维多体量子系统的新映射。特别是，我们解决了选择从 2D 晶格到 1D 链的有效映射的问题，该映射可最佳地保留 TN 结构内相互作用的局部性。通过使用矩阵乘积状态 (MPS) 和树张量网络 (TTN) 算法，我们计算横向场中二维量子 Ising 模型的基态，晶格尺寸高达 $64\times64$，比较基于不同映射获得的结果两条空间填充曲线，蛇形曲线和希尔伯特曲线。我们表明，希尔伯特曲线的局部保留特性导致数值精度的明显提高，特别是对于大尺寸，