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

With the help of artificial intelligence, physicists have compressed a daunting quantum problem that previously required 100,000 equations into a bite-sized task of just four equations—all without sacrificing accuracy. The work, published in the September 23 issue of Physical Verification Letters, could revolutionize the way scientists study systems containing many interacting electrons. Additionally, if the approach is scalable to other problems, it could potentially aid in the design of materials with coveted properties such as superconductivity or utility for clean energy generation.

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

The daunting problem concerns the behavior of electrons when moving on a lattice-like lattice. When two electrons occupy the same lattice site, they interact. This setup, known as the Hubbard model, is an idealization of several important classes of materials and allows scientists to learn how electron behavior leads to desirable phases of matter, such as B. Superconductivity, where electrons flow through a material without resistance. The model also serves as a testing ground for new methods before they are unleashed on more complex quantum systems.

However, the Hubbard model is deceptively simple. Even for a modest number of electrons and the most modern computational approaches, the problem requires significant computational power. Because when electrons interact, their fates can become quantum mechanically entangled: even if they are far apart on different lattice sites, the two electrons cannot be treated individually, so physicists have to deal with all the electrons at once rather than just one at a time. With more electrons, more entanglement occurs, making the computational task exponentially more difficult.

One way to study a quantum system is to use what is called a renormalization group. This is a mathematical apparatus that physicists use to study how the behavior of a system – such as the Hubbard model – changes when scientists change properties such as temperature or look at properties at different scales. Unfortunately, a renormalization group that tracks all possible couplings between electrons and sacrifices nothing may contain tens, hundreds of thousands, or even millions of individual equations that need to be solved. In addition, the equations are tricky: each one represents an interacting pair of 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 telephone operator and the evolution of survival of the fittest. First, the machine tutorial creates connections within the full-size renormalization group. The neural network then optimizes the strength of these connections until it finds a small set of equations that produces the same solution as the original jumbo-sized renormalization set. The program’s output even captured the physics of the Hubbard model with only four equations.

“It’s essentially a machine that can discover 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 ran for whole weeks. The good news, Di Sante says, is that now that they’ve coached their program, they can adapt it to work on other issues without having to start from scratch. He and his collaborators are also investigating what machine learning actually “learns” about the system, which could provide additional insights that would 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 where electrons interact over long distances. Additionally, there are exciting opportunities to use the technique in other fields that look at renormalization groups, Di Sante says, such as in cosmology and neuroscience.

Provided by the Simons Foundation