Scientists trained a machine learning technique to capture the physics of electrons traveling on a lattice with considerably fewer equations than would normally be needed, all while maintaining accuracy. Physicists utilizing artificial intelligence have reduced a challenging quantum problem that formerly required 100,000 equations to a manageable effort of as few as four equations. All of this was done without sacrificing precision. The findings could change the way scientists analyze systems with multiple interacting electrons. Furthermore, if the approach is applicable to other issues, it could potentially aid in the construction of materials with extremely valuable features such as superconductivity or utility for clean energy generation.

“We start with this massive object with all these coupled-together differential equations, and then we use machine learning to reduce it to something so little you can count it on your fingers,” study lead author Domenico Di Sante explains. He is a visiting research fellow at the Flatiron Institute’s Center for Computational Quantum Physics (CCQ) in New York City and an assistant professor at the University of Bologna in Italy.

The difficult quantum problem concerns how electrons flow on a gridlike lattice. Interaction occurs when two electrons occupy the same lattice location. This configuration, known as the Hubbard model, idealizes numerous major types of materials and allows scientists to study how electron behavior gives rise to highly desired phases of matter, such as superconductivity, in which electrons flow through a material without resistance. The model also serves as a testbed for novel methods before they are applied to more sophisticated quantum systems.

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The Hubbard model, on the other hand, is deceptively simple. The problem necessitates huge computer power even for a small number of electrons and cutting-edge computational methodologies. Because electrons’ fates can get quantum mechanically entangled as they interact. This means that even though the two electrons are far apart on different lattice locations, they cannot be addressed separately. As a result, physicists must deal with all electrons at once rather than one at a time. Additional electrons result in more entanglements, making the challenging computing challenge considerably more difficult.

A renormalization group is one technique for analyzing a quantum system. This is a mathematical instrument used by physicists to examine how the behavior of a system, such as the Hubbard model, varies when researchers adjust properties such as temperature or examine the attributes on different scales. Unfortunately, a renormalization group that tracks all potential electron couplings and makes no sacrifices can contain tens of thousands, hundreds of thousands, or even millions of unique equations that must be solved. Furthermore, the equations are rather difficult: Each one illustrates an electron pair interacting.

Di Sante and his colleagues wondered if they could utilize a neural network, a machine learning technology, to make the renormalization group more controllable. The neural network resembles a frenzied switchboard operator crossed with survival-of-the-fittest evolution. The machine learning program begins by making connections within the full-size renormalization group. The neural network then adjusts the strength of those connections until it discovers a small set of equations that produces the same solution as the original, jumbo-sized renormalization group. Even with only four equations, the program’s output captured the physics of the Hubbard model.

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“It’s simply a machine with the ability to find hidden patterns,” Di Sante explains. “When we saw the outcome, we said, ‘Wow, this is more than we imagined.'” We were able to accurately represent the key physics.” Training the machine learning software took a lot of processing power, and the program ran for weeks. The good news, according to Di Sante, is that now that their program has been coached, they can adjust it to work on other problems without having to start from zero. He and his colleagues are also looking into what the machine learning is “learning” about the system. This could reveal new information that would be difficult for physicists to comprehend otherwise.

The largest unknown is how well the new method works on more complex quantum systems, such as materials in which electrons interact across long distances. Furthermore, Di Sante believes that there are intriguing potential for employing the technique in other domains that deal with renormalization groups, such as cosmology and neurology.

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