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

Giovanni Cataldi1,2,3, Ashkan Abedi4, Giuseppe Magnifico1,2,3, Simone Notarnicola1,2,3, Nicola Dalla Pozza4, Vittorio Giovannetti5, and Simone Montangero1,2,3

1Dipartimento di Fisica e Astronomia ``G. Galilei'', Università di Padova, I-35131 Padova, Italy.
2Padua Quantum Technologies Research Center, Università degli Studi di Padova.
3Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Padova, I-35131 Padova, Italy.
4Scuola Normale Superiore, I-56127 Pisa, Italy.
5NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy.

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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.

Simulating quantum many-body systems is a very challenging problem since the memory required to exactly describe the state of the system grows exponentially with the number of degrees of freedom. Therefore, it is important to search for approximate and efficient representations of the quantum states of these systems. In the last decades, tensor networks have demonstrated remarkable capabilities to tackle this problem by efficiently retaining in memory only the relevant features of the state of the system.
In our work, we focus on two-dimensional systems where the search to improve tensor network representations is still ongoing. In particular, a standard strategy is to map the two-dimensional problem onto an effective one-dimensional one and then analyze it via established one-dimensional tensor network algorithms. Here, we show that the choice of the mapping drastically affects the numerical precision. In particular, we focus on two tensor network geometries and compare the results obtained with two possible mappings. We find that the mapping based on the Hilbert space-filling curve, thanks to its locality-preserving properties, guarantees a better precision with respect to the more standard mapping based on the snake curve.

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