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Artificial Intelligence Reduces A 100,000-Equation Quantum Physics Problem To Only Four Equations

 Physicists have used artificial intelligence to condense a difficult quantum issue that previously required 100,000 equations into a manageable assignment that only requires four equations—all without losing accuracy. The research, which appeared in Physical Review Letters on September 23, may completely alter how scientists approach studying systems with lots of interacting electrons. The method may also help in the development of materials with desirable qualities like superconductivity or use in the production of renewable energy if it is transferable to other issues.


An illustration of a mathematical tool for simulating the motion and behaviour of electrons on a lattice. A single interaction between two electrons is represented by each pixel. Up until recently, around 100,000 equations—one for each pixel—were needed to correctly capture the system. Only four equations remained after the issue was minimised using machine learning. Therefore, just four pixels would be required for a comparable depiction in the compressed form. Credit: Flatiron Institute/Domenico Di Sante

"We start with this huge object of all these coupled-together differential equations; then we're using machine learning to turn it into something so small you can count it on your fingers," says Domenico Di Sante, an assistant professor at the University of Bologna in Italy and the study's principal author, is a visiting research fellow at the Center for Computational Quantum Physics (CCQ) at the Flatiron Institute in New York City.

The challenging issue relates to the motion of electrons on a lattice that resembles a grid. Interaction occurs when two electrons are present at the same lattice location. Scientists may study how electron behavior leads to desired phases of matter, such as superconductivity, in which electrons flow through a material without resistance, using this configuration, known as the Hubbard model, which idealizes several significant classes of materials. Additionally, the model acts as a proving ground for fresh approaches before they are applied to more intricate quantum systems.


But the Hubbard model appears to be rather straightforward. The issue demands a significant amount of computer power, even for a small number of electrons using state-of-the-art computational techniques. That's because interactions between electrons might induce quantum mechanical entanglements in their fates: The two electrons cannot be dealt separately, even when they are far apart on distinct lattice sites, therefore physicists must deal with all of the electrons at once rather than one at a time. The computing problem becomes increasingly more difficult as there are more electrons present because more entanglements form.

Renormalization groups are a tool that may be used to examine a quantum system. The Hubbard model is one example of a system that physicists use this mathematical tool to examine how the behavior of a system varies as scientists alter parameters like temperature or consider the properties on various scales. Unfortunately, there may be tens of thousands, hundreds of thousands, or even millions of individual equations in a renormalization group that must be solved in order to maintain track of all potential couplings between electrons without making any sacrifices. Additionally, the equations are challenging: Each one symbolizes the interaction of two electrons.


Di Sante and his coworkers questioned whether they could utilize a neural network, a machine learning technology, to simplify the renormalization group. The neural network resembles a cross between an anxious switchboard operator and evolution according to the principle of the strongest. The full-size renormalization group is first connected to the machine learning algorithm. In order to locate a smaller set of equations that yield the same result as the original, jumbo-size renormalization group, the neural network adjusts the strengths of those connections. Even with only four equations, the program's output was able to reproduce the physics of the Hubbard model.


"It's essentially a machine that has the power to discover hidden patterns," Di Sante says. "When we saw the result, we said, 'Wow, this is more than what we expected.' We were really able to capture the relevant physics."

It took weeks for the machine learning algorithm to train, which required a lot of computer power. The good news, according to Di Sante, is that they can modify their curriculum to address additional issues without having to start from scratch now that it has been trained. In order to gain extra insights that could otherwise be challenging for physicists to understand, he and his partners are also looking into what machine learning is "learning" about the system.

The main unanswered question is how well the novel method applies to more complicated quantum systems, such as materials with long-range electron interactions. According to Di Sante, there are also intriguing potential for applying the method to other disciplines that work with renormalization groups, such as cosmology, and neurology.

Domenico Di Sante et al, Deep Learning the Functional Renormalization Group, Physical Review Letters (2022).

Journal information: Physical Review Letters

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