# How classical computers can beat quantum computers at their own game

Earlier this year, researchers at the Flatiron Institute’s Center for Computational Quantum Physics (CCQ) used classical computers and sophisticated mathematical models to solve what some say only quantum computers can solve. They announced that they had succeeded in demonstrating performance that completely outperformed quantum computers at the task they were considering.

Now, those researchers have discovered why they were able to beat quantum computers at their own game. Their answer, published in Physical Review Letters, reveals that the quantum problem they worked on, including a particular two-dimensional quantum system that flips a magnet, exhibits a behavior known as confinement. This behavior has so far only been seen in one-dimensional systems in quantum condensed matter physics.

This unexpected discovery helps scientists better understand the line that separates the power of quantum and classical computers, and provides a framework for testing new quantum simulations, said lead researcher at CCQ. Author Joseph Tyndale says:

“There are some boundaries between what you can do with quantum computing and what you can do with classical computers,” he says. “Right now, that line is incredibly blurred. I think our work will help make that line a little clearer.”

By harnessing the principles of quantum mechanics, quantum computers promise significant advantages in processing power and speed over classical computers. While classical computation is limited by binary operations of 1 and 0, quantum computers can process information in a fundamentally different way, using qubits that can represent both 0 and 1 simultaneously.

However, quantum technology is still in its infancy and has yet to convincingly demonstrate its superiority over classical computers. As scientists work to understand where quantum computers have an advantage, they come up with complex problems that test the limits of classical and quantum computers.

The results of recent tests of quantum computers will be published in June 2023, with IBM researchers publishing a paper in the journal Nature. Their paper detailed an experiment that simulated a system with an array of small reversing magnets that evolve over time. The researchers argued that this simulation could only be achieved with quantum computers, not classical ones. Tyndall learned about this new paper in the news and decided to give it a try.

Tyndall has spent the last few years working with colleagues to develop better algorithms and code for solving complex quantum problems on classical computers. He applied these techniques to IBM’s simulations and demonstrated in just two weeks that the problem could be solved with very little computing power, even on a smartphone.

“We haven’t really implemented cutting-edge technology,” Tyndall said. “We brought together many ideas in a concise and elegant way that allowed us to solve problems. I couldn’t.’

Tyndall and his colleagues published their findings in the journal PRX Quantum in January 2024, but Tyndall didn’t stop there. Inspired by the simplicity of their results, he and co-author Dries Sells of the Flatiron Institute and New York University wondered why this system, which on the surface looks like I tried to figure out how easy it is to solve. A very complex problem.

“We started thinking about this problem and realized there were a lot of similarities in the way this system worked to what people had seen in one dimension: confinement,” Tyndall said.

Confinement is a phenomenon that can occur under special circumstances in closed quantum systems, and is similar to the quark confinement known from particle physics. To understand confinement, let’s start with quantum basics. At the quantum scale, individual magnets can point up or down, or they can be in a “superposition,” a quantum state in which the magnet points both up and down at the same time. The top and bottom height of a magnet affects the amount of energy it has when in a magnetic field.

In the initial setup of the system, all magnets were oriented in the same direction. The system was then perturbed by a small magnetic field, causing some magnets to try to flip, which encouraged neighboring magnets to flip as well. This action, where the magnets affect each other’s reversals, can cause entanglement, which is a combination of superpositions of magnets. Over time, the entanglement of the system increases, making traditional computer simulations difficult.

However, in a closed system, there is only so much energy that can circulate. Tyndall and Sells showed that in a closed system, only enough energy exists to flip small, sparsely separated clusters of orientation, directly limiting tangle growth. This energy-based restriction on entanglement is known as confinement, and it arose as a completely natural consequence of the two-dimensional shape of the system.

“In this system, the magnet doesn’t just suddenly go wild, it actually just oscillates around its initial state, even over very long time scales,” Tyndall says. “This is very interesting from a physics perspective, because it means that the system is not only completely disordered, but remains in a state with a very specific structure.”

Coincidentally, in their first test, IBM set up a problem in which the configuration of magnets in a closed two-dimensional array caused confinement. Tindall and Sels realized that confining the system reduces the amount of entanglement, thus keeping the problem simple enough to be written using classical techniques. Using simulations and mathematical calculations, Tyndall and Sells devised a simple and accurate mathematical model to explain this behavior.

“One of the big unsolved problems in quantum physics is understanding when entanglement increases rapidly and when it does not,” Tyndall says. “This experiment provides a good understanding of an example where large-scale entanglement did not occur due to the model used and the two-dimensional structure of the quantum processor.”

This result suggests that confinement itself can appear in a variety of two-dimensional quantum systems. If so, the mathematical models developed by Tyndall and Sells provide a valuable tool for understanding the physics occurring in those systems. Additionally, the code used in the paper could provide a benchmarking tool that experimental scientists can use when developing new computer simulations for other quantum problems.

Further information: Joseph Tindall et al, “Confinement in a transverse magnetic Ising model on a heavy hex lattice,” Physical Review Letters (2024). DOI: 10.1103/PhysRevLett.133.180402

Provided by Simons Foundation

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