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Wednesday, January 06, 2021

New Quantum Algorithms Finally Crack Nonlinear Equations | Mathematics - Quanta Magazine

Two teams found different ways for quantum computers to process nonlinear systems by first disguising them as linear ones, argues Max G. Levy, science journalist based in Los Angeles.

Sometimes, it’s easy for a computer to predict the future. Simple phenomena, such as how sap flows down a tree trunk, are straightforward and can be captured in a few lines of code using what mathematicians call linear differential equations. But in nonlinear systems, interactions can affect themselves: When air streams past a jet’s wings, the air flow alters molecular interactions, which alter the air flow, and so on. This feedback loop breeds chaos, where small changes in initial conditions lead to wildly different behavior later, making predictions nearly impossible — no matter how powerful the computer.

“This is part of why it’s difficult to predict the weather or understand complicated fluid flow,” said Andrew Childs, a quantum information researcher at the University of Maryland. “There are hard computational problems that you could solve, if you could [figure out] these nonlinear dynamics.”

That may soon be possible. In separate studies posted in November, two teams — one led by Childs, the other based at the Massachusetts Institute of Technology — described powerful tools that would allow quantum computers to better model nonlinear dynamics...

Knowing One’s Limits

While these are significant steps, they are still among the first in cracking nonlinear systems. More researchers will likely analyze and refine each method — even before the hardware needed to implement them becomes a reality. “With both of these algorithms, we are really looking in the future,” Kieferová said. Using them to solve practical nonlinear problems requires quantum computers with thousands of qubits to minimize error and noise — far beyond what’s possible today.

And both algorithms can realistically handle only mildly nonlinear problems. The Maryland study quantifies exactly how much nonlinearity it can handle with a new parameter, R, which represents the ratio of a problem’s nonlinearity to its linearity — its tendency toward chaos versus the friction keeping the system on the rails.

“[Childs’ study is] mathematically rigorous. He gives very clear statements of when it will work and when it won’t work,” Osborne said. “I think that’s really, really interesting. That’s the core contribution.”

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Source: Quanta Magazine