Just to point it out (if only for your edification), this is clearly false. Specifically:
If something is computable, a Turing machine is capable of computing it. By any definition that allows your statement to make sense to my reading, a “neural net computer” is performing (Turing) computation. The “current computers” – particularly when removing any and all constraints on their power – would be able to perform the same computation.
Thus, “neural net computers” are not the “only” way to do it (you’ve not defined exactly what “it” is, so I’m using the pronoun to retain the ambiguity).