Turing machine and quantum computers

[bifolco84]

Can anyone help me...from Nielsen-Chuang "quantum computation and quantum information":how might we recognize that a process in nature computes a function not computable by a turing machine?

[haael]

how might we recognize that a process in nature computes a function not computable by a turing machine
This can not be proven by observations alone. The strong hint would be that the system could not be described by any simple equation.

For any function (computable or not) there exists a Turing machine that can reproduce an arbitrary large finite subset of it. So if you have a black box producing some output, then you never know whether there is an uncomputable system inside or just a very smart Turing machine.