Unlikely. Check out "Can NP-complete problems be solved efficiently in the physical universe?" (http://www.scottaaronson.com/papers/npcomplete.pdf) and "BQP and the Polynomial Hierarchy" (http://www.scottaaronson.com/papers/bqpph.pdf).
There's, of course, also Wikipedia ( https://en.wikipedia.org/wiki/BQP).
As for the assumptions, I agree. Hence it's wild guessing.
If you're not assuming the universe is simulated and is instead "fundamental" (whatever that means), then who are you to put limits on its processing power?'
I think you have a level-confusion here - the universe isn't packing computational machinery into itself. Remember that space-time is a part of this universe. If something is computing our universe, it has to hold the representation of space-time, too. The computation machinery isn't in here with us.
Except black holes are a thing.
1. Simulating QM-level events is NP-complete.
2.Assuming P=/=NP, a machine that simulates a volume of size X in QM-correct terms must be exponentially larger than X.
3. This includes X itself.
4. Therefore, the universe is taking sweeping shortcuts that look like classical mechanics at large scale. These are not 'our' computational shortcuts, they're actually how things work at a fundamental level.
5. ...maybe if we build a big enough quantum computer we can overflow the universe's buffer.