**Of the 100,000 equations to solve, there are only 4 left! This computational feat was achieved using artificial intelligence and now makes it more accessible to solve a famous quantum physics problem describing the behavior of electrons moving in a network. This breakthrough could help design materials with desirable properties, such as superconductivity.**

Hubbard’s model is a model studied in condensed matter theory, describing the movement of electrons in a network of atoms. It is used to determine how the behavior of electrons gives rise to sought-after phases of matter, such as superconductivity. But when the electrons are on the same site, they interact and can become entangled; therefore, simulating the behavior of an electron means simultaneously following the range of possibilities of all the electrons in the system, their quantum state depending on each other.

Thus, even considering a small number of electrons, the problem requires immense computing power, as physicists have to process all the electrons at once. With a larger number of electrons, the rate of entanglement is even higher and the calculation becomes exponentially more difficult. To study this type of quantum system, scientists use a renormalization group — a mathematical approach that observes how a system behaves when certain properties (such as temperature) are changed. However, this requires solving hundreds of thousands or even millions of individual equations.

## A machine “capable of discovering hidden patterns”

To offer the most accurate possible capture of the system, a renormalization group must indeed take into account all the possible couplings between the electrons. Also, the equations are particularly complex, as each represents an interacting pair of electrons. Researchers at the Flatiron Institute in New York therefore set out to use a machine learning tool, an artificial neural network, to simplify the task without sacrificing the precision of the calculation.

” *We start with this huge object made up of all these differential equations coupled together, and then we use machine learning to turn it into something so small you can count it on your fingers.* », explains Domenico Di Santevisiting scholar at the Center for Computational Quantum Physics at the Flatiron Institute and co-author of the study describing this approach.

Concretely, the machine learning program creates connections within the renormalization group of normal size. The neural network then adjusts the strength of these connections until it finds a small set of equations that generates the same solution as the original renormalization group. For Di Sante, it is essentially a machine “capable of discovering hidden patterns”.

The result of the program exceeded all the team’s expectations: the physics of Hubbard’s model was reduced to just four equations. This means that a visualization of the problem (as shown in the header of this article) would only require four pixels. In other words, studying the emergent properties of complex quantum materials is now much more manageable.

## Potential applications in cosmology and neuroscience

As a reminder, superconductivity is a phenomenon characterized by the total absence of electrical resistance within certain materials; the latter conduct electricity without loss of energy and their potential applications are therefore crucial for the energy sector. But obtaining this superconductivity generally requires extremely low temperatures, close to absolute zero.

Harnessing superconductivity at more reasonable temperatures could lead to the development of much more efficient electrical networks and devices. That’s why physicists try to predict, using various models — including Hubbard’s model — how electrons might behave under different circumstances.

Like any machine learning algorithm, the one that was used in this study had to be trained beforehand on a set of data; the training took several weeks. The simulations only capture a relatively small number of variables in the grid network, but now that the program is trained it can be adapted to work on other problems in condensed matter physics, the researchers believe. According to Di Sante, this technique could advantageously be used in other fields that deal with renormalization groups, such as cosmology and neuroscience.

The real test, the team points out, will be whether this new approach works well on more complex quantum systems, such as materials in which electrons interact at long distances. Di Sante and his collaborators are also looking to find out what their algorithm actually “learns” about the system, which could provide additional insights that are otherwise hard for physicists to decipher.

Meanwhile, this work demonstrates the possibility of using artificial intelligence to extract compact representations of correlated electrons, “a goal of the utmost importance for the success of state-of-the-art quantum field theory methods to tackle the problem.” many electrons,” the team concludes in their paper.

##### Source : D. Di Sante et al., Physical Review Letters

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A 100,000 equation quantum physics problem simplified to 4 equations thanks to AI

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