Artificial intelligence reduces a quantum physics problem of 100,000 equations to just four equations

Using artificial intelligence, physicists have compressed a daunting quantum problem that so far required 100,000 equations into a bite-sized task of just four equations, all without sacrificing precision. The book, published in the September 23 issue of Physical examination letters, could revolutionize the way scientists study systems containing many interacting electrons. Moreover, if adaptable to other problems, the approach could potentially aid in the design of materials with sought-after properties such as superconductivity or utility for clean energy generation.

“We start with this huge object of all these coupled differential equations, and then we use machine learning to turn it into something so small you can count it on your fingers,” says the study’s lead author, Domenico Di Sante, guest researcher. member of the Flatiron Institute’s Center for Computational Quantum Physics (CCQ) in New York and assistant professor at the University of Bologna in Italy.

The daunting problem concerns the behavior of electrons as they move across a grid-like lattice. When two electrons occupy the same lattice site, they interact. This configuration, known as the Hubbard model, is an idealization of several important classes of materials and allows scientists to learn how the behavior of electrons gives rise to desirable phases of matter, such as superconductivity, in which electrons flow through a material without resistance. The model also serves as a testing ground for new methods before they are launched on more complex quantum systems.

Hubbard’s model, however, is deceptively simple. Even for a modest number of electrons and state-of-the-art computational approaches, the problem requires significant computational power. This is because when electrons interact, their fates can become quantum mechanically entangled: even once they are far apart at different lattice sites, the two electrons cannot be individually processed, so physicists must process all electrons at a time rather than one at a time. a time. With more electrons, more tangles appear, making the computational challenge exponentially more difficult.

One way to study a quantum system is to use what is called a renormalization group. It is a mathematical device that physicists use to observe how the behavior of a system, such as the Hubbard model, changes when scientists change properties such as temperature or examine properties at different scales. Unfortunately, a renormalization group that keeps track of all possible couplings between electrons and sacrifices nothing may contain tens of thousands, hundreds of thousands, or even millions of individual equations that need to be solved. On top of that, the equations are tricky: each represents a pair of interacting electrons.

Di Sante and his colleagues wondered if they could use a machine learning tool known as a neural network to make the renormalization group more manageable. The neural network is like a cross between a frantic switchboard operator and the evolution of survival of the fittest. First, the machine learning program creates connections within the full renormalization group. The neural network then adjusts the strengths of these connections until it finds a small set of equations that generates the same solution as the original giant-sized renormalization group. The program output captured the physics of the Hubbard model even with only four equations.

“It’s basically a machine that has the power to uncover hidden patterns,” says Di Sante. “When we saw the result, we said, ‘Wow, that’s more than we expected.’ We were really able to capture the relevant physics.”

Training the machine learning program required a lot of computing power and the program took weeks to complete. The good news, says Di Sante, is that now that they have their program framed, they can adapt it to work on other issues without having to start from scratch. He and his collaborators are also studying what machine learning actually “learns” about the system, which could provide additional insights that might otherwise be difficult for physicists to decipher.

Ultimately, the biggest open question is how well the new approach works on more complex quantum systems such as materials in which electrons interact at long distances. Additionally, there are exciting possibilities for using the technique in other fields that deal with renormalization groups, Di Sante says, such as cosmology and neuroscience.

Provided by the Simons Foundation