Quantum States and Neural Networks: Efficient Conversion via Tensor Decomposition

The pursuit of efficient methods for characterising complex quantum systems represents a significant challenge in modern physics, particularly when seeking the ground state, the lowest energy configuration, of many-body systems. Researchers now demonstrate a novel approach to initialise neural network quantum states using tensor network states, offering a potentially faster route to approximating these ground states. Ryui Kaneko of Sophia University and Shimpei Goto of the University of Tokyo, alongside colleagues, detail their findings in a paper entitled ‘Seeding neural network quantum states with tensor network states’. Their work centres on converting matrix product states (MPSs), a compact way to represent quantum states, into the wave functions of restricted Boltzmann machines, a type of neural network, through a mathematical technique called canonical polyadic (CP) decomposition. This enables the generation of well-behaved initial states for variational calculations, thereby reducing the computational time required to converge on an accurate solution and systematically improving the proximity of the initial state to the actual ground state as the complexity of the decomposition increases. The researchers validate their method using the transverse-field Ising model, a standard model in condensed matter physics, and suggest its applicability to systems with more intricate wave function structures.

Ryui Kaneko of Sophia University and Shimpei Goto of the University of Tokyo present a method for efficiently approximating the conversion of matrix product states (MPSs) into restricted Boltzmann machine (RBM) wave functions, offering a potential solution to challenges in simulating complex quantum many-body problems. Their research introduces a technique utilising a multinomial hidden unit, achieved through a canonical polyadic (CP) decomposition of the MPSs, enabling the generation of well-behaved initial neural network states for ground-state calculations in polynomial time with respect to the number of variational parameters. This innovative approach systematically reduces the distance between the initial states and the true ground states by increasing the rank of the CP decomposition, demonstrating an advancement in computational efficiency and accuracy for quantum simulations.

The researchers address a fundamental challenge in quantum many-body physics: accurately representing and simulating the behaviour of complex quantum systems, which often demand exponentially growing computational resources as system size increases. Traditional methods struggle with systems exhibiting complex nodal structures – regions where the wave function changes sign – in their ground-state wave functions, hindering progress in understanding materials and phenomena with intricate quantum properties. Kaneko and Goto’s method offers a potential solution by leveraging machine learning and neural networks to approximate quantum states, establishing a direct connection between matrix product states and restricted Boltzmann machines. A matrix product state is a tensor network representation of a quantum state, particularly effective for one-dimensional systems, while a restricted Boltzmann machine is a type of artificial neural network often used for unsupervised learning.

By representing the quantum state as a neural network, the researchers can utilise powerful optimisation algorithms developed for machine learning to find approximate solutions to complex quantum problems. This conversion is achieved in polynomial time relative to the number of variational parameters, a significant advantage over many traditional methods that suffer from exponential scaling. Polynomial scaling implies that the computational cost increases as a power of the system size, making it far more manageable for larger systems than exponential scaling.

Crucially, the research demonstrates that increasing the ‘rank’ of the CP decomposition systematically reduces the distance between the initial neural network state and the true ground state of the system. The rank represents the number of components used to represent the original quantum state, and a higher rank allows for a more accurate representation, albeit at the cost of increased computational effort. The researchers carefully control this trade-off between accuracy and computational cost, demonstrating the method’s versatility and practicality, validated using the transverse-field Ising model, a well-studied model in condensed matter physics.

This methodology has broader implications for the field of quantum many-body physics, potentially enabling the study of systems with complex nodal structures in their ground-state wave functions, which are notoriously difficult to simulate using traditional methods. The ability to efficiently generate good initial states for variational calculations is crucial for obtaining accurate results, and this method offers a promising pathway to achieve this. Furthermore, the connection between matrix product states and neural networks opens up new possibilities for applying machine learning techniques to solve problems in quantum physics, potentially leading to the development of novel algorithms and insights. This convergence of quantum physics and machine learning promises to accelerate progress in both fields.

The researchers meticulously demonstrate their method’s polynomial scaling behaviour, making it feasible to apply to larger and more complex systems. This is a significant improvement over traditional methods with exponential scaling. The team’s approach builds upon existing work, integrating concepts from tensor network methods, neural networks, and tensor decomposition techniques, including contributions from LeBlanc et al. (2015) and Wu et al. (2024) on the application of tensor network methods and neural networks to quantum state representation, and the insights of Sherman and Kolda (2020) on tensor decomposition techniques for data analysis. By combining these established techniques, Kaneko and Goto have developed a novel and efficient method for approximating quantum states, pushing the boundaries of computational quantum physics.

The researchers envision a wide range of potential applications for their method, including the study of strongly correlated materials, quantum magnetism, and topological phases of matter. These systems exhibit complex quantum behaviour that is difficult to capture using traditional methods, and the new approach offers a promising pathway to overcome these challenges. By efficiently representing and simulating these systems, researchers can gain new insights into their fundamental properties and potentially discover new materials with novel functionalities. The potential impact of this research extends beyond fundamental science, with implications for technological advancements in areas such as quantum computing and materials science.

In conclusion, Kaneko and Goto’s research presents a significant advancement in the field of computational quantum physics, offering a novel and efficient method for approximating quantum states using a combination of matrix product states and restricted Boltzmann machines. Their approach overcomes limitations of traditional methods, enabling the study of complex quantum systems that were previously intractable. The polynomial scaling behaviour of the method, combined with its versatility and potential for broad applications, positions it as a valuable tool for researchers in quantum physics, materials science, and related fields. This work paves the way for future advancements in our understanding of complex quantum phenomena and the development of new quantum technologies.