In the world of materials science, important physical phenomena, such as magnetism, superconductivity, and heat transfer, occur as a result of how electrons and atoms within a material interact with one another.

Scientists have developed mathematical descriptions to better understand these behaviors, which are based on the machinery of quantum mechanics. These are pretty straightforward when considering just a few particles, but when the complexity of the system grows, so too does the math.

In the realm of physics, it is not uncommon for different physical systems to have similar mathematical descriptions. One of the most important and interesting examples of this is the Gaudin model, a mathematical theory proposed by a French physicist Michel Gaudin in 1976 describing the behavior of multiple interacting electrons within specific superconductors or qubits interacting with an uncontrolled environment.

Understanding the interactions of these particles with one other and outside electromagnetic fields, as elucidated by this theory, is necessary to operating superconducting and quantum devices, which have emerged as hot topics in both applied and fundamental physics in recent decades.

The Gaudin model has therefore been extensively studied using many traditional methods, but its complete description is still lacking.

Machine learning and the Gaudin model

To improve our understanding of the Gaudin model, a team of Canadian physicists was able to study this mathematical theory using modern machine learning techniques, which consist of training an algorithm to find a correct behavior of the electrons the Gaudin model describes as quickly and efficiently as possible.

“We study the time-evolution of a complex model using a neural-network (machine learning) approach,” explained Victor Wei, a recent graduate of McGill University in Montreal and lead author on a study published in Advanced Physics Research in an email. “There have been several studies similar to this, but the methods have historically been applied to models that may not have practical significance or models that can already be solved efficiently.”

In their study, which analyzed six interacting electrons, the scientists trained an algorithm through thousands of iterations. In this way, they allowed it to identify and refine the quantities that define the solution to the model’s equations, which describe the behavior of either a superconducting or a quantum computing system. This refinement process results in an ever increasing accuracy by building on results from previous runs.

To ensure that the electron’s behavior described by their machine-learning method matched reality, the researchers compared their algorithm’s predictions with established descriptions achieved using other methods. They tested its ability to predict the behavior of a material with a small number of particles, and found excellent agreement between the two descriptions.

This consistency allowed the researchers to conclude that the technique they proposed could be applied to studying the Gaudin model for a large number of interacting particles — where other methods have previously broken down.

“Machine learning is better suited to analyzing the Gaudin model than other methods because the model has a lot of conserved quantities or symmetries to be leveraged, but it is not clear how to leverage them,” said Wei. “[The traditional] methods would find solutions with guaranteed accuracy, but they require exponentially growing computing resources as the quantum system grows larger.

“Machine learning methods, on the other hand, do not have guaranteed accuracy but can often find good approximate solutions efficiently by learning the hidden patterns of the problem.”

Solving real-world problems

The team believe that their results will not remain purely theoretical for too long and in the near future will be used to study the interaction of quantum bits within quantum computers with their environment. Understanding why and how qubits become unstable will help to build the next generation of quantum computers, which currently suffer from inaccuracies due to this instability known as quantum decoherence.

“We believe our research will have practical applications in studying various quantum many-body systems,” Wei said. “We expect our approach of approximating time evolution will significantly extend the applicability of the neural-network-based approach to quantum systems. At the moment, we are cleaning up the programming code and will soon have our method integrated into Netket, one of the largest machine-learning toolboxes for quantum physics.”

The team say they also plan to further refine their findings by studying the Gaudin model for an even larger number of interacting particles, as well as improving their machine learning algorithm.

“We plan to explore our method with systems of progressively larger size beyond the reach of exact methods,” concluded Wei. “As many novel neural-network quantum state ansatzes have been proposed recently, we also plan to improve our current method by incorporating new ideas from the fast-growing machine learning communities.”

Reference: V. Wei, A. Orfi, F. Fehse, and W. A. Coish, Finding the Dynamics of an Integrable Quantum Many-Body System via Machine Learning, Advanced Physics Research (2023), DOI: 10.1002/apxr.202300078

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