(ORDO NEWS) — Forecasting the weather, describing turbulent fluid flows, and many other “feedback” systems, such as those that affect themselves, require solving nonlinear differential equations. But we cannot solve them with the existing methods and will never be able to, regardless of the power of computers. Two teams of researchers from the University of Maryland and from the Massachusetts Institute of Technology have proposed efficient approaches to solving nonlinear problems using quantum computers. True, we do not yet have sufficiently powerful quantum computers.
Even the most powerful classical computers cannot and will never be able to provide long-term weather forecasts. But there is a chance that quantum computers can cope with this task.
Non-linear problems are, in particular, problems with “feedback”. This includes the weather forecast. What the weather will be like outside our window in three days depends on what the weather is today. What the weather will be in Moscow in three days depends on the weather in Moscow today. A six-day forecast – from the weather today and from the weather in three days. The longer the forecast is made, the less accurate it is, until 10-15 days it becomes just random. Then we forget about the calculations and move on to climate predictions based on observation statistics: it is colder in winter than in summer.
This is the feedback: the system influences its behavior and depends on its state, and not only on the input parameters. Such problems are not described by systems of ordinary differential equations, which we have already learned to solve well and quickly. Many problems that we need to solve are just nonlinear.
On the other hand, the very creation of algorithms for quantum computers is a non-trivial task. Today, there are very few quantum algorithms that solve problems that cannot be solved in real time on classical computers. And then two more appeared.
A team from the University of Maryland adapted a quantum computer to solve nonlinear problems using the Carleman Linearization algorithm. This method allows you to reduce a nonlinear problem to a set of linear equations. The problem is that there are infinitely many such equations. Although the method was developed in the 1930s, it has been reliably forgotten. But in Maryland, they learned to cut off the “extra” infinity in time and solve a nonlinear problem with a given accuracy on a quantum machine.
At the Massachusetts Institute of Technology, they found an even more radical way. Scientists have proposed to model any nonlinear problem like a Bose-Einstein condensate. In this state of matter, all particles are interconnected, and the behavior of each of them affects the rest, and all the others affect it, that is, the interaction returns along the loop.
The mathematics describing the Bose-Einstein condensate is well developed and it helped a lot in building the model. Tobias Osborne, a quantum information scientist at Leibniz University of Hanover, commented on the MIT solution: “Give me your favorite nonlinear differential equation and I will build you a Bose-Einstein condensate that will model it.”
In fact, MIT proposed to build a kind of “chip” on a quantum computer that works like a model of a Bose-Einstein condensate.
In both cases, scientists used “old”, deeply researched mathematical theories for new applications.
The matter is small. We need to build a quantum computer with several thousand qubits and use it to test the models proposed in Maryland and MIT.
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