Artificial intelligence reduces a 100,000-equation quantum physics problem to only four equations

Scientists have reduced the complexity of a challenging quantum issue that formerly required 100,000 equations to as few as four equations using artificial intelligence, all without compromising accuracy. The research, which was released in the Physical Review Letters on September 23, has the potential to completely alter how scientists study systems with lots of interacting electrons. Furthermore, the technique may help in the creation of materials with desired features like superconductivity or usefulness for the production of renewable energy if it is scalable to other issues.

"We utilize machine learning to reduce this massive object of interconnected differential equations to a size that is small enough to be counted on your fingers," explains Domenico Di Sante, the lead author of the study. Di Sante is an assistant professor at the University of Bologna in Italy and a visiting research fellow at the Center for Computational Quantum Physics (CCQ) at the Flatiron Institute in New York City.

The difficult issue is how electrons behave when they go through a lattice that resembles a grid. Two electrons interact when they are located at the same lattice location. This configuration, referred to as the Hubbard model, idealizes a number of significant material classes and helps scientists understand how electron behavior leads to desired phases of matter, such superconductivity, in which electrons pass through a material without encountering resistance. Additionally, the model is used to test new techniques before releasing them onto more intricate quantum systems.

Still, the Hubbard model is very straightforward. However, the issue takes significant computer power, even for moderate numbers of electrons and state-of-the-art computational methods. This is due to the possibility of quantum mechanical entanglement between electrons when they interact: Physicists have to deal with all the electrons at once instead of treating each one separately since the two electrons cannot be dealt separately even though they are separated by a great distance on separate lattice sites. As the number of electrons increases, more entanglements arise, increasing the computational difficulty dramatically.

Using a renormalization group is one method of analyzing a quantum system. Scientists may examine how a system's behavior, like the Hubbard model, changes as they alter its attributes, like its temperature, or examine it at different scales using this mathematical tool. Unfortunately, there can be tens of thousands, hundreds of thousands, or even millions of unique equations in a renormalization group that maintains track of all conceivable couplings between electrons while sacrificing as little as feasible. Furthermore, the equations are challenging: Each one depicts the interaction of two electron pairs.

Dinte and his associates pondered if they might utilize a neural network—a machine learning instrument—to help manage the renormalization group. The neural network can be compared to a hybrid of the scrambling switchboard operator and evolution's survival-of-the-fittest theory. First, inside the full-size renormalization group, the machine learning algorithm establishes connections. Afterwards, the neural network adjusts the connections' strengths until it discovers a tiny set of equations that produce the same answer as the initial, jumbo-size renormalization group. With just four equations, the program's output accurately represented the physics of the Hubbard model.

According to Di Sante, "it's basically a machine that has the ability to find hidden patterns." We exclaimed, 'Wow, this is more than we imagined,' as soon as we saw the outcome.'" We truly succeeded in capturing the pertinent physics."

It took weeks for the machine learning algorithm to run, and a lot of processing power was needed for training. The good news, according to Di Sante, is that they can modify their program to focus on different difficulties without having to start from scratch now that they have a coach for it. Additionally, he and his colleagues are looking into what the machine learning is really "learning" about the system, which might provide physicists additional insights that they might not otherwise be able to understand.

The main unanswered question is ultimately how well the new method performs on more intricate quantum systems, including materials where electrons interact across large distances. Furthermore, Di Sante notes that there are intriguing prospects for using the method in other domains that deal with renormalization groups, such cosmology and neurology.