Exploring the performance of superposition of product states: From one-dimensional to three-dimensional quantum spin systems

Tensor networks are one of the best available tools to study many-body quantum systems. Tensor networks 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 the superposition-of-product-states (SPS) ansatz, a variational framework structurally related to canonical polyadic tensor decomposition. While the ansatz is generally not as expressive as tensor networks, it has the properties that it (i) is structurally independent of the geometry of the system, (ii) allows accurate extraction of information, (iii) is readily parallelizable, and (iv) allows 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 a ground-state search in tilted Ising models, including one-dimensional and three-dimensional models with short- and long-range interaction, and a random network, demonstrating that the SPS ansatz can attain high accuracy.