Does quantum computing have an essential advantage in analyzing/controlling chaotic systems?

Question

1. The enthusiast-level, inaccurate knowledge about quantum computers is that they can solve many exponentially solvable problems in polynomial time.
2. The enthusiast-level, inaccurate knowledge about chaotic systems is that being highly sensitive to initial conditions, their prediction and control is very hard above a - typically, not enough - accuracy.

Today, one of the most famous practical usage of chaotic systems is the problem of modeling the weather of the Earth.

Putting (1) and (2) together, I think using quantum computers, we may have a significant (polynomial to exponential) step to handle them. Is it correct?

Do we have any essential advantage to handle chaos even more than this?

Answer 1

Not always. Some problems are non-deterministic (their solution). Apart from that, some problems are, as you say, so sensitive to changes in initial conditions, that most solutions are too localized.

But there are cases where quantum computers can provide insightful results, that might shed light on different approaches to solutions.

Another point to consider is the use of Numerical methods in chaotic systems. Some methods are more optimal than others, at the cost of accuracy. With quantum computers, computation time decreases by a lot (acc. to theory), which may allow more accurate calculations, leading to a better understanding of the more difficult chaotic systems.

To clarify: Quantum computers might not be able to give an analytical solution (even to problems that might have such solutions), but a more accurate approximation can often lead to a new understanding of the problem, which is a way to handle problems.

Answer 2

No.

Chaos (as described in chaotic systems) is deterministic, and the evolution of such a system can be calculated using classical deterministic equations. The problem is the strong divergence of the different trajectories that even small differences in initial values can lead to large differences in the final values.

Quantum computing does not help in this situation.