It is as yet unclear as to how powerful the quantum computing paradigm is. The
fact that an NP problem such as factoring can be solved in exponential time is
an extremely encouraging indication. Quantum Algorithms are difficult to design
due to the difficulty of engineering the system to collapse into a state that
leads to the solution of a problem. It is known that the complexity space of
the quantum computer is a subset of PSPACE, ie the class of problems
that can be solved on a Turing machine with a bound on memory, but with
unlimited time. It is probable that Quantum Computing can solve a few more
NP problems, and can speedup the solution to many more.
A useful Quantum Computer has never been built, due to the engineering
difficulties in preventing the quantum state decohering, and in isolating the
quantum state from the outside world. It is probable however that a prototype
will be constructed some time in the near future. Shor's algorithm was
demonstrated in 2001 by a group at IBM, which factored 15 into 3 and 5, using a
quantum computer with 7 qubits.