My lay understanding of the problem with classical algorithms is basically that a lack of resolution means you need to monte carlo the thing millions of times... which is why it's slow. If you could model it as a set of quantum states of similar inaccuracy, wouldn't that by definition be just as (in)accurate but faster?
[edit] this reminds me of something I read about how NASA doesn't predict solar eclipses by trying to keep an exact model of the solar system, but rather uses pattern matching algorithms.
I don’t think there’s many three body problems in the real world where compute is the limiting factor. If you’re projecting so far out into the future that simulation speed is a problem, you’re initial measurement error will have compounded to the point that your solution is meaningless.
We struggle to predict the exact path of asteroids because of measurement errors, not because computing is slow. Minuscule changes to the initial condition manifest as massive differences in the outcome.
The reason this worked is, that the back-action of the moon on the sun is very insignificant. For a "real" chaotic three-body system, you need three bodies that interact with each other on comparable scales.
>> In summary, it is clear ancient people could predict timings for lunar eclipses and partial solar eclipses, but there is no convincing evidence of people predicting the times and locations of total solar eclipses.
>> Today, we don’t rely on calculating the orbits of the whole Solar System to predict eclipses. For example, NASA uses a highly advanced form of an ancient technique – pattern recognition. Using some 38,000 repeating mathematical terms, NASA can predict both solar and lunar eclipses for 1,000 years into the future. Beyond that, the Moon’s wobble and Earth’s changing rotation make eclipse prediction less accurate.