[quanto]

The challenges you mentioned, and techniques to address them, are not unique to quantum physics. I am still not understanding how quantum physics require "new" kind of numerical analysis. And what are these new kinds of techniques you hint at? Could you give me some examples of unique techniques that arose from quantum physics and are not used elsewhere?

[sigmoid10]

They are pretty unique. Many of them were actually developed for it. But I wouldn't even know where to begin to explain this to a layperson without knowing their level of education first. Like, how much do you know about Dyson series? Path integrals? Wick rotations? You're basically asking for a graduate level lecture on numerical methods in quantum field theories. If you really want to know the details, a quick google will probably net you some books or lecture notes.