Using artificial intelligence, physicists have compressed a
daunting quantum problem that until now required 100,000 equations into a
bite-size task of as few as four equations—all without sacrificing
accuracy. The work, published in the September 23 issue of Physical Review Letters,
could revolutionize how 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.
"We start with this huge object of all these coupled-together differential equations; then we're using machine learning to turn it into something so small you can count it on your fingers,"
says study lead author Domenico Di Sante, a visiting research fellow at
the Flatiron Institute's Center for Computational Quantum Physics (CCQ)
in New York City and an assistant professor at the University of Bologna
in Italy.
The formidable problem concerns 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 model also serves as a testing ground for new methods
before they're unleashed on more complex quantum systems.
The Hubbard model is deceptively simple, however. For even a modest
number of electrons and cutting-edge computational approaches, 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 and doesn't
sacrifice anything 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. "When we saw the result, we said, 'Wow, this
is more than what we expected.' We were really able to capture the
relevant physics."
Training the machine learning program required a lot of computational
muscle, and the program ran for entire weeks. The good news, Di Sante
says, is that now that they have their program coached, they can adapt
it to work on other problems without having to start from scratch. He
and his collaborators are also investigating just what the machine
learning is actually "learning" about the system, which could provide
additional insights that might otherwise be hard for physicists to
decipher.
Ultimately, the biggest open question is how well the new approach
works on more complex quantum systems such as materials in which
electrons interact at long distances. In addition, there are exciting
possibilities for using the technique in other fields that deal with
renormalization groups, Di Sante says, such as cosmology and
neuroscience.
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