The numbers you are describing are very very large. To the point, it appears that they are numbers whose representation in decimal (or binary) are large enough that it there is little to no prospect of there being enough matter in the entire universe to store those numbers, in a place-value representation such as those. This being the case, no technology — quantum or otherwise — will be able to produce a representation (or anything which can be described as a conventional 'estimate') of those numbers.
Furthermore, it appears that these functions are faster growing than any provably total function (e.g., faster than any function which we know to be computable even in exponential time) relative to some more-or-less sensible model of set theory. If you are interested in things which you can compute in a reasonable time-bound with quantum computers — e.g. in polynomial time, which can be simulated in at worst exponential time on a conventional computer — it follows that on mathematical grounds as well as physical grounds, you should expect these functions not to be practically computable even on an idealised quantum computer.