Unfortunately, we've proven that all physically known processes can be simulated on a Turing machine, so unless there is some crazy new discovery of physics, there is no kind of computation that can do something Turing machines can't. It's not the only kind, but it's equivalent to any other that is possible we've come up with, and to any we could physically implement with our knowledge of physics. It would be very cool if that wasn't the case, though.
Feynman wrote a paper about it in the 80s, called "Simulating Physics with Computers", in which he lays out how all known laws of physics can be simulated using a computer to sufficient accuracy, but that without a quantum computer it would take exponential time to do so.
Unfortunately, we've proven that all physically known processes can be simulated on a Turing machine, so unless there is some crazy new discovery of physics, there is no kind of computation that can do something Turing machines can't. It's not the only kind, but it's equivalent to any other that is possible we've come up with, and to any we could physically implement with our knowledge of physics. It would be very cool if that wasn't the case, though.
Have we proven that? If so, super cool.
Feynman wrote a paper about it in the 80s, called "Simulating Physics with Computers", in which he lays out how all known laws of physics can be simulated using a computer to sufficient accuracy, but that without a quantum computer it would take exponential time to do so.