AI reduces a quantum problem of 100,000 equations to just 4 equations

Hubbard’s model is a model studied in condensed matter theory and a formidable quantum problem. A team of physicists used the deep learning to condense this problem that previously required 100,000 equations, into just four equations, without sacrificing accuracy. The study entitled “Deep Learning the Functional Renormalization Group” was published on September 21 in Physical examination letters.

Dominique Di Sante is the main author of this study. Since 2021, he holds the position of Assistant Professor (tenure track) in the Department of Physics and Astronomy, at the University of Bologna. At the same time, he is Visiting Professor at the Center for Computational Quantum Physics (CCQ) at the Flatiron Institute in New York as part of a “Marie Sklodowska-Curie Actions” (MSCA) grant which, among other things, encourages the mobility of researchers.

With colleagues from the Flatiron Institute and other international researchers, he conducted this study which has the potential to revolutionize the way scientists study systems containing many interacting electrons. In addition, if they succeed in adapting this method to other problems, the approach could contribute to the design of materials with desired properties, for example superconductivity, or contributing to the production of clean energy.

Hubbard’s model

This model studied in condensed matter theory was introduced by John Hubbard in 1963. It describes fermions (generally electrons) on a lattice (generally the atoms that form a solid), which only interact when they are on the same site (i.e. on the same atom).

With the use of this configuration, scientists can discover how the behavior of electrons gives rise to desired phases of matter, such as superconductivity, in which electrons move through a material without resistance. The model is also used to test new methods before applying them to more complex quantum systems.

Hubbard’s model, however, is deceptively simple. If it is possible to solve it for a single dimension, beyond that there is no exact solution.

On the other hand, even for a small number of electrons and advanced computational approaches, the problem requires serious computational power. Indeed, when electrons interact, their fates can become quantum mechanically entangled: although far apart at different lattice sites, electrons cannot be individually processed, so physicists must process all electrons at once rather than ‘one by one. With more electrons, more entanglements appear, and the computational challenge becomes exponentially more difficult…

Renormalization group and deep learning

One way to study a quantum system is to use what is called a renormalization group. Introduced in 1963, the latter was developed primarily by Kenneth Wilson, who was awarded the Nobel Prize in 1982 for his contributions. It is a complex set of transformations that demonstrates how the collective behavior observed in critical systems can result from microscopic interactions and has played a crucial role in the study of phase transitions in particular.

Physicists use it to examine how the behavior of a system, such as Hubbard’s model, changes when scientists change properties such as temperature or examine properties at different scales. Unfortunately, a renormalization group keeps track of all possible couplings between electrons and may contain hundreds of thousands or even millions of individual equations that need to be solved.

Di Sante and his colleagues decided to use the deep learning to better manage the renormalization group. Initially, the neural network creates connections within the full-size renormalization group, then adjusts the strengths of those connections until it finds a small set of equations that generates the same solution as the renormalization group. original renormalization. The program output captured the physics of the hubbard model with only four equations.

Algorithm training deep learning took many weeks. The good news, according to Di Sante, is that now that they have their program coached, 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.

Sources of the article:

“Deep Learning Functional Renormalization Group”

Physical Examination Letters 129, 136402

Dominique Di Sante co-wrote this study with:

  • Matija Medvidović, Visiting Researcher at CCQ (Columbia University graduate student);
  • Alessandro Toschi from TU Wien in Vienna;
  • Giorgio Sangiovanni from the University of Würzburg in Germany;
  • Cesare Franchini from the University of Bologna in Italy;
  • Anirvan M. Sengupta, senior researcher at CCQ and the Center for Computational Mathematics;
  • Andy Millis, co-director of the CCQ.

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