Physicists recently use a neural net to compressed a daunting quantum problem that required 100,000 equations into a solution that requires as few as four equations—all without sacrificing accuracy.
The problem consists of how electrons behave as they move on a gridlike 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 enables scientists to learn how electron behavior gives rise to sought-after phases of matter, such as superconductivity, in which electrons flow through a material without resistance.
The Hubbard model is deceptively simple, however. For even a modest number of electrons the problem requires serious computing power. That’s because when electrons interact, their fates can become quantum mechanically entangled: Even once they’re far apart on different lattice sites, the two electrons can’t be treated individually, so physicists must deal with all the electrons at once rather than one at a time. With more electrons, more entanglements crop up, making the computational challenge exponentially harder.
One way of studying a quantum system is by using what’s called a renormalization group. That’s a mathematical apparatus physicists use to look at how the behavior of a system—such as the Hubbard model—changes when scientists modify properties such as temperature or look at the properties on different scales. Unfortunately, a renormalization group that keeps track of all possible couplings between electrons can contain tens of thousands, hundreds of thousands or even millions of individual equations that need to be solved. On top of that, the equations are tricky: Each represents a pair of electrons interacting.
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 switchboard operator and survival-of-the-fittest evolution. First, the machine learning program creates connections within the full-size renormalization group. The neural network then tweaks the strengths of those connections until it finds a small set of equations that generates the same solution as the original, jumbo-size renormalization group. The program’s output captured the Hubbard model’s physics even with just four equations.
“It’s essentially a machine that has the power to discover hidden patterns,” Di Sante says.
The work, published in the September 23 issue of Physical Review Letters, could revolutionize how quantum scientists investigate systems containing many interacting electrons. Moreover, if scalable to other problems, the approach could potentially aid in the design of materials with sought-after properties such as superconductivity or utility for clean energy generation.
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