Superposition of Product States in 1D to 3D Quantum Spin Systems Enables Accurate Information Extraction

Quantum spin systems present a significant challenge to physicists seeking to understand complex material properties, and accurately modelling these systems requires increasingly sophisticated computational techniques. Apimuk Sornsaeng from Singapore University of Technology and Design, Itai Arad from National University of Singapore, and Dario Poletti, also from Singapore University of Technology and Design, investigate a novel approach using a superposition-of-product-states (SPS) ansatz, a mathematical framework related to established tensor network methods. This research demonstrates that SPS offers advantages in extracting information and is adaptable to various system geometries, unlike some traditional methods, while also being easily implemented on parallel computing architectures. The team’s work reveals that SPS can achieve high accuracy in finding the ground state of complex spin models, including both one- and three-dimensional systems with interactions ranging from short to long range, representing a valuable step forward in simulating quantum materials.

Tensor Networks Simulate Quantum Spin Systems

Researchers are exploring the capabilities of superposition of product states (SPS), a method for approximating the ground state of quantum spin systems in one, two, and three dimensions. The team systematically investigated SPS by applying it to the Heisenberg model on various lattice structures and sizes, assessing its accuracy and efficiency in capturing the essential physics of these systems. Results demonstrate that SPS achieves surprisingly accurate results in one and two dimensions, even for relatively large systems, but its performance decreases significantly in three dimensions, suggesting the need for more advanced methods to tackle strongly correlated systems in higher dimensions. These findings contribute to a deeper understanding of variational methods for simulating quantum many-body systems and provide valuable insights for developing more efficient algorithms for complex quantum problems.

Tensor networks represent powerful tools for studying many-body quantum systems, particularly those with one-dimensional local interactions. While effective for these systems, their performance is often limited in more complex geometries due to challenges in expressive power and information extraction. This work investigates the performance of a superposition-of-product-states (SPS) ansatz, a variational framework structurally related to canonical polyadic tensor decomposition. Although SPS does not compress information as effectively as traditional tensor networks, it offers advantages including reduced computational cost and the potential for more accurate representation of certain quantum states.

SPS Ansatz Analytical Derivations and Scaling

This work details the analytical derivations and implementation details for a specific variational ansatz, the SPS ansatz, with the goal of demonstrating its efficiency and scalability for solving quantum problems, particularly finding the ground states of many-body systems. The research focuses on deriving formulas for local observables, gradients of the energy, and the 2-Rényi entropy, enabling efficient computation without relying on numerical differentiation or complex simulations. The team also describes how a random coupling system is generated and the resulting network structure is defined. This approach demonstrates that the SPS ansatz, combined with these analytical and implementation techniques, offers a computationally advantageous approach.

Researchers derived analytical expressions for calculating the expectation values of local operators using the SPS ansatz, crucial for optimizing physical quantities during the variational process. These expressions avoid expensive simulations by relating calculations to the ansatz parameters and pre-computed quantities. Furthermore, the team derived analytical expressions for the gradients of the energy, a significant advantage as it avoids slow and inaccurate numerical differentiation or computationally expensive automatic differentiation. The 2-Rényi entropy, a measure of entanglement, was also calculated efficiently using the SPS ansatz, providing insights into the ground state properties and potential issues with the variational algorithm. The ability to derive these analytical expressions demonstrates the power and scalability of the SPS ansatz.

The team generated a random coupling system using an Erdős-Rényi random graph model, creating a network with a specified number of nodes and a connection probability. They used a fixed seed to ensure reproducibility of the random graph generation. Visualizations of the adjacency matrices for different connection probabilities were provided, illustrating the network structure. This approach provides a way to create networks with varying degrees of connectivity, with the connection probability serving as a key control parameter.

This work contributes to the development of more efficient and scalable variational quantum algorithms, offering a potential method for finding the ground states of complex many-body systems and advancing quantum simulation. The random coupling system provides a benchmark for testing the performance of different variational algorithms.

Scalable Variational Ansatz for Quantum Many-Body Systems

This work demonstrates the effectiveness of a superposition-of-product-states (SPS) ansatz as a variational framework for approximating quantum many-body systems. Researchers established a direct link between the SPS ansatz and canonical polyadic tensor decomposition, resulting in a scalable and highly parallelizable method for studying complex quantum states. Investigations into the ansatz’s properties reveal restricted typicality, where local observables exhibit partial stability, and a polynomial scaling of variances with system size, offering insights into its behaviour across random instances.

The team demonstrated the SPS ansatz’s ability to achieve high accuracy in ground-state searches across various models, including one-, two-, and three-dimensional tilted Ising models and random networks. Notably, the method often surpasses the density-matrix renormalization group (DMRG) in speed for three-dimensional and random systems, particularly when utilizing GPU acceleration, while DMRG remains highly efficient for one-dimensional systems. The researchers acknowledge that the ansatz’s performance is limited in the paramagnetic phase, suggesting that incorporating spatial modulation could broaden its expressive power. Future work will focus on extending the ansatz to complex-valued states and applying it to time-evolving systems, potentially expanding its utility in simulating more complex quantum phenomena.