Can Quantum Computing solve Problems not even a Turing Machine can solve? In his book "The Fabric of Reality", Penguin Books 1998, p. 218, David Deutsch says that the first quantum computer (built 1989 in the office of Charles Bennet, IBM Reasearch) "became the first machine ever to perform non-trivial computations that no Turing machine could perform".

My question: Do we really know that quantum computing is powerful enough to solve problems that not even Turing machines can solve? (Or is this just a personal belief of David Deutsch?)
While it is true that the computation of a quantum Turing machine is vastly different from that of a classical one, nevertheless quantum Turing machines can be simulated on a classical Turing machine, albeit with exponential slowdown. In particular, everything that can be computed on a quantum Turing machine can also be computed on a classical Turing machine.

The main advantage of quantum Turing machines is that they appear to solve some problems much faster than classical Turing machines. As Sasho comments, at the moment we can't quite prove this, yet this advantage is believed to hold by most researchers.

For more, check out this talk by Ashley Montanaro.