one for every pixel Scientists reduced the problem to four equations using machine learning. That means that a comparable visualisation for the compressed version would only require four pixels. Photographer: Domenico Di Sante/Flatiron Institute
Physicists have used artificial intelligence to reduce a daunting quantum problem that previously required 100,000 equations to a manageable task of as few as four equations—all without sacrificing accuracy. The findings, which were published in the September 23 issue of Physical Review Letters, have the potential to change how scientists investigate systems with many interacting electrons. Furthermore, if the approach is scalable to other problems, it could potentially aid in the design of materials with desirable properties such as superconductivity or utility for clean energy generation.
"We start with this massive object of all these coupled-together differential equations, and then we use machine learning to reduce it to something so small you can count it on your fingers," explains study lead author Domenico Di Sante, 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 problem is how electrons behave as they move across a gridlike lattice. Interaction occurs when two electrons occupy the same lattice site. This setup, known as the Hubbard model, idealises several important classes of materials and allows scientists to learn how electron behaviour gives rise to desired phases of matter, such as superconductivity, in which electrons flow without resistance through a material. The model is also used to test new methods before they are applied to more complex quantum systems.
However, the Hubbard model is deceptively simple. Even a small number of electrons and cutting-edge computational approaches are insufficient to solve the problem. Because electrons' fates can become quantum mechanically entangled when they interact: Even when separated by a large distance on different lattice sites, the two electrons cannot be treated separately, so physicists must deal with all of the electrons at once rather than one at a time. More electrons result in more entanglements, making the computational challenge exponentially more difficult.
A renormalization group is one technique for studying a quantum system. That is a mathematical apparatus used by physicists to examine how the behaviour of a system, such as the Hubbard model, changes when scientists change properties such as temperature or examine the properties on different scales. Unfortunately, a renormalization group that keeps track of all possible electron couplings and makes no sacrifices can have tens of thousands, hundreds of thousands, or even millions of individual equations that must be solved. Furthermore, the equations are difficult: Each one represents an electron pair interacting.
Di Sante and his colleagues wondered if they could use a neural network, a machine learning tool, to make the renormalization group more manageable. The neural network resembles a frantic switchboard operator crossed with survival-of-the-fittest evolution. The machine learning programme begins by making connections within the full-size renormalization group. The neural network then adjusts the strength of those connections until it finds 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.
"It's essentially a machine with the ability to discover hidden patterns," Di Sante explains. "When we saw the outcome, we thought, 'Wow, this is more than we expected.'"We were able to accurately capture the relevant physics."
The machine learning program's training required a lot of computational muscle, and the programme ran for weeks. The good news, according to Di Sante, is that now that their programme has been coached, they can adapt it to work on other problems without having to start from scratch. He and his colleagues are also looking into what the machine learning is "learning" about the system, which could provide additional insights that would otherwise be difficult for physicists to decipher.
The biggest unknown is how well the new method works on more complex quantum systems, such as materials in which electrons interact over long distances. Furthermore, Di Sante believes that there are exciting possibilities for using the technique in other fields that deal with renormalization groups, such as cosmology and neuroscience.