[erikpukinskis]

> The wavefunction, which is the core of a quantum calculation, "runs" all the time whether any part of it is observed or not.

The wave function doesn't really have to "run" to exist though... a wave doesn't actually have any influence on anything until it is collapsed. If nothing observes the wave, it won't have any affect at all. It will just be there, in some cosmic register, waiting for some dust cloud to inquire about it.

Consider a polygon in a game engine, which started at 0,0 and has a known velocity. You are at tick 4762, and that polygon is represented by a position function, but it doesn't actually "run" until you declare a tick, and do the math.

[comicjk]

I dispute whether a wavefunction can influence things without collapsing, but putting that aside...

Your example of a polygon with a constant velocity is carefully chosen: that equation has an analytical solution, so you can calculate x(t) and "skip" forward in time. This is not possible in general, even in classical mechanics. If it were a system of more than two interacting particles, you wouldn't be able to fast-forward; you would have to calculate all the intermediate timesteps even if you only wanted the last one.

[EGreg]

That's only because as a human being you deal with closed form solutions.

Who is to say that the "universe simulator" is only doing 21st century math?

[comicjk]

If the universe simulator can solve iterative problems in O(1), then I question whether our concept of optimization is meaningful enough to it for this discussion to make sense.

[bicubic]

This is a very interesting question. What if the universe is stateless and any state at time T can be calculated in O(1)

What if we just lack the expressiveness or the initial conditions to model a stateless universe?

A fun consequence of stateless universe would be that you can rewind, fast forward, loop, speed up, slow down time at no extra cost.

[comicjk]

There are some physical systems (mathematically, "Hamiltonians") which have this "stateless" property. However, if the time/energy uncertainty principle is true, simulating time T in O(1) cannot be possible in general unless BQP=PSPACE (the unlikely idea that quantum computers can efficiently solve any problem that can be stored in polynomial amounts of memory). See this paper: https://arxiv.org/abs/1610.09619