The researchers trained a machine learning tool to model the physics of electrons moving on a lattice with far fewer equations than would normally be required, without sacrificing accuracy.
A difficult quantum problem that previously required 100.000 equations has been condensed by physicists using artificial intelligence into a manageable task that requires as few as four equations. Accuracy has been maintained throughout this entire process. The research could completely change the way scientists study systems with large numbers of interacting electrons. The method could also help design materials with exceptionally valuable properties, such as superconductivity or usefulness for clean energy generation, if transferable to other subjects.
According to the study's lead author, Domenico Di Sante, “we start with this huge object with all these combined differential equations and then we use machine learning to transform it into something small enough that you can count with your fingers.” Di Sante is a visiting research fellow at the Center for Computational Quantum Physics (CCQ) at the Flatiron Institute in New York and an assistant professor at the University of Bologna in Italy.
The hard quantum problem has to do with the movement of electrons on a grid-like lattice. Interaction occurs when two electrons are in the same lattice position. Known as the Hubbard model, this configuration idealizes many important types of materials and enables researchers to understand how electron behavior leads to highly desirable phases of matter, such as superconductivity, where electrons move through a material without resistance. New techniques can be tested on the model before being applied to more complex quantum systems.
Still, the Hubbard model is quite simple. This task requires a small number of electrons and enormous processing power, even for state-of-the-art computational methods. This is because interactions between electrons can cause their fates to become entangled quantum mechanically. This indicates that two electrons cannot be considered separately, even if they are far apart and in different lattice positions. As a result, physicists have to deal with each electron all at once, not individually. This enormous computational difficulty becomes increasingly difficult when there are more electrons as there is more entanglement.
Renormalization groups are a tool that can be used to study a quantum system. The Hubbard model is an example of a system where physicists use this mathematical tool to study how the behavior of a system changes when researchers change parameters such as temperature or when considering properties at various scales. Unfortunately, there may be tens of thousands, hundreds of thousands, or even millions of unique equations in an uncompromising renormalization group that keeps track of all potential couplings between electrons. Also, the equations are pretty compelling: Each symbolizes the interaction of two electrons.
Di Sante and his colleagues questioned whether they could use a machine learning technology, a neural network, to simplify the renormalization set. The neural network resembles a cross between an anxious switchboard operator and evolution according to its strongest principle. The full-size renormalization group is first connected to the machine learning algorithm. The neural network adjusts the strengths of these connections to find a smaller set of equations that give the same result as the original, jumbo-sized renormalization set. Even with just four equations, the output of the program was able to reproduce the physics of the Hubbard model.
Di Sante describes it as "basically a machine capable of finding hidden patterns." Wow, this is more than we expected, we thought when we saw the result. We have successfully captured the relevant physics.
The machine learning algorithm took weeks to train because it required a lot of computing power. The good news, according to Di Sante, is that they can change their curriculum to address additional issues without having to start from scratch because it goes through coaching. Together with his colleagues, he studies what the machine learning algorithm "learns" about the system. This can offer extra information that would otherwise be difficult for physicists to understand.
The main unanswered question is how well the new method applies to more complex quantum systems, such as materials with long-range electron interactions. According to Di Sante, there is also interesting potential for application of the method to other disciplines working with renormalization groups, such as cosmology and neurology.