  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.
