# Quantum Showdown: Competing Methods of Solving Quantum Systems Put to the Test

A many-method study of the Hubbard model demonstrates how the field of computational quantum physics is evolving to be more collaborative.

Unlocking the complexities of quantum systems containing gobs of interacting particles remains one of the most daunting computational challenges ever faced. Scientists have developed many different methods of tackling this problem, but those methods have primarily been developed separately and not directly compared.

A new study in *Physical Review X* shows how well almost all available methods stack up against each other. The testing ground was an infinite grid of interacting electrons called the Hubbard model. The researchers ran each method on the same setup of the model to see how well they captured the system’s physics.

The friendly competition showed that “surprisingly, there was no one method that rules them all,” says study lead author Thomas Schäfer, head of the Theory of Strongly Correlated Quantum Matter research group at the Max Planck Institute for Solid State Research Stuttgart. “Some of the methods are better suited for certain properties we want to analyze about the system.”

The results show the increasing need to use many methods in concert, says study co-author and director of the Center for Computational Quantum Physics (CCQ) at the Flatiron Institute in New York City, Antoine Georges. “Over the last few years, there’s been a culture change in the field,” he says. “We’re trying to incorporate more than one method at a time and establish a dialogue between the methods. CCQ and the many electron collaboration are committed to promoting this culture.”

In the new paper, the researchers dub this the “multi-method, multi-messenger” approach — using several methods to investigate many different physical properties. This terminology is borrowed from astrophysicists, who ushered in a new multi-messenger era of astronomy in 2017 by observing both light and gravitational waves from the merger of two neutron stars.

Bringing so many computational methods together is tricky because they are often targeted for different problems. The setup of the Hubbard model used in this work ultimately proved a fair, neutral ground for the tests. “In our community, the Hubbard model is what the fruit fly is to biology,” says Schäfer.

The Hubbard model is deceptively simple. The two-dimensional variant used in the study is essentially an infinitely large checkerboard. Each space on the board is a site that electrons can inhabit. Electrons can hop from their current site to an adjacent one. Each electron can have an up or down spin, and two electrons can share a site if they have opposite spins. In the researchers’ setup, there were as many electrons as there were spaces on the board.

In the model, electrons only interact when sharing a site. In quantum mechanics, though, those short-range interactions can have far-reaching, long-lasting effects. When two particles interact, they become quantum mechanically entangled with one another. Even when they venture off to different sites, they can’t be treated separately.

The system’s temperature acts like a knob that controls just how ‘quantum’ the system is. Heat fluctuations break quantum connections. At high temperatures, the system behaves as expected in classical physics (an easier regime to calculate). As the temperature drops, the system becomes more quantum mechanically ensnared and therefore harder to compute. At temperatures near absolute zero, even the best computational methods struggle.

The researchers needed a definitive baseline to determine how accurately the various methods reproduced the system’s physics. They set up the model such that two approaches based on the Monte Carlo method could yield exact solutions. The Monte Carlo method, named for the Mediterranean casino, uses random sampling to compute an answer to a problem. The two Monte Carlo methods approached the problem in very different ways but yielded the same answers. That agreement gave the researchers confidence that they had a reliable benchmark for the other methods.

The methods evaluated included the two Monte Carlo approaches, static mean-field theory, dynamical mean-field theory, cluster extensions of dynamical mean-field theory, vertex-based extensions of dynamical mean-field theory, and many more.

"The dynamical mean-field theory has been a great step forward in the description of the Hubbard model. However, it does not include all types of its correlations and, hence, extensions are needed”, says Marcel Klett, expert of cluster extensions of dynamical mean-field theory, postdoc in the research group of Thomas Schäfer and coauthor of the study. “Cluster extensions of dynamical mean-field theory lead to insights of how spatial correlations affect the electrons. Calculations of large clusters are numerically very challenging, but with increasing cluster size they will eventually converge to the exact solution of the model.”

One of the most significant tests for the various methods involved capturing the Hubbard model’s behavior as the temperature dropped. At high temperatures, heat fluctuations break down quantum entanglements, and electrons move every which way. In the new study, the researchers describe this phase as an “incoherent soup.” As the temperature drops, quantum effects grow, and the system becomes a metal. As the temperature drops lower still, quasi-ordered patches appear in the system. To a passing electron, these patches seem like one giant particle. Electrons scatter off these obstacles such that, when the temperature approaches absolute zero, the patches inhibit electron movement so severely that the system becomes an insulator.

In addition to examining performance in these different regimes, the researchers also evaluated how well the methods predicted how many sites housed two electrons, how susceptible the system would be to an external magnetic field, and the size of the quasi-ordered patches.

The study did reveal some new information about the Hubbard model, including the nature of the regime changes and the behavior of the quasi-ordered patches. Despite the model’s simplicity, it provides useful information about real-world systems, such as certain superconductors and lattices of ultra-cold gas.

The more significant takeaway, Schäfer says, was that the various methods could be used in tandem for better results. Each of the tested methods had strengths and weaknesses, and there was no standout winner. Some methods did better with a given property or in a different temperature regime. Two complementary methods could make up for each other’s weaknesses or confirm a given result.

“We can say what are the methods that are accurate in a given regime,” Georges says. “Our paper offers a clear road map of which methods to pick for which regimes.”

The study itself is a testament to the field’s shift toward larger collaborations combining many methods, Georges says. The paper has 26 authors, including experts in many of the methods tested. That’s unusual for a computational quantum physics study, he says, and he credits Schäfer for bringing the project together.

“There’s a human adventure behind this paper,” Georges says. “Large collaborative teams will help push our knowledge of the field and make quantum condensed matter less individualistic.”

*Thomas Sumner/Simons Foundation*