As far as we can tell, all the laws of the universe are completely deterministic. (And that includes quantum mechanics.) As far as we can tell, human beings obey the laws of physics.

(To explain: quantum mechanics as a theory is completely deterministic and even linear. Some outdated interpretations of quantum mechanics, like Copenhagen, use randomisation. But interpretations don't make a difference to what the underlying theory actually is. And more widely accepted interpretations like 'Many Worlds' preserve the determinism of the underlying theory.)

Btw, neural nets are typically sampled from, and you can use as good a random number generator (even a physical random number generator) as there is, if you want to. I don't think it'll change what we think neural nets are capable of.