Hilbert curve vs Hilbert space: exploiting fractal 2D covering to increase tensor network efficiency

TL;DR

This paper introduces a new mapping technique using the Hilbert curve to improve the efficiency and accuracy of tensor network simulations for 2D quantum systems, outperforming traditional snake curve mappings.

Contribution

It demonstrates that the Hilbert curve mapping better preserves locality, leading to enhanced numerical precision in tensor network simulations of large 2D quantum lattices.

Findings

Hilbert curve mapping improves numerical precision for large 2D systems

Hilbert mapping outperforms snake curve in tensor network simulations

Enhanced efficiency in simulating 2D quantum Ising models

Abstract

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.