Exploring the performance of superposition of product states: from 1D to 3D quantum spin systems

Tensor networks (TNs) are one of the best available tools to study many-body quantum systems. TNs are particularly suitable for one-dimensional local Hamiltonians, while their performance for generic geometries is mainly limited by two aspects: the limitation in expressive power and the approximate extraction of information. Here we investigate the performance of superposition-of-product-states (SPS) ansatz, a variational framework structurally related to canonical polyadic tensor decomposition. The ansatz does not compress information as effectively as tensor networks, but it has the advantages (i) of allowing accurate extraction of information, (ii) of being structurally independent of the geometry of the system, (iii) of being readily parallelizable, and (iv) of allowing analytical shortcuts. We first study the typical properties of the SPS ansatz for spin-1/2 systems, including its entanglement entropy, and its trainability. We then use this ansatz for ground state search in tilted Ising models - including one-dimensional and three-dimensional with short- and long-range interaction, and a random network - demonstrating that SPS can attain high accuracy.