Turing machines and their computational power
Your question is closely related to the Church&ndash;Turing thesis, which says that Turing machines can compute anything that can reasonably be described as an algorithm (i.e., a sequence of discrete steps). Variants, such as the physical Church&ndash;Turing thesis state that Turing machines can compute anything that can be computed by any physical mechanism. Note that these are described as "theses", rather than theorems. They're not formal claims about mathematics so they're not possible to prove.

We don't know how to build any physical device that could compute a non-Turing-computable function. This includes quantum computers, which can be simulated (though inefficiently) to arbitrary accuracy by non-quantum devices such as Turing machines.

We can certainly consider more powerful devices, though, even though we don't know of any way to build them. A typical construction is to take something that Turing machines can't do, and then imagine that we have a subroutine that instantly solves that problem. These are known as oracle machines, and they are provably more powerful than Turing machines. Indeed, one can define a whole infinite hierarchy of successively more powerful machines using this idea, with the caveat that we have no clue how to physically build such a thing.