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

TL;DR

This paper evaluates the superposition-of-product-states (SPS) ansatz for quantum spin systems, highlighting its advantages over tensor networks in terms of information extraction, geometry independence, and parallelizability, across 1D to 3D models.

Contribution

It introduces and assesses the SPS ansatz as a versatile, geometry-independent variational method capable of accurately finding ground states in complex quantum spin systems.

Findings

SPS can accurately extract information from quantum states.

SPS achieves high accuracy in ground state searches for various models.

SPS is structurally independent of system geometry and highly parallelizable.

Abstract

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.