  Quantum computers are able to outperform classical algorithms. This was long
recognized by the visionary Richard Feynman who pointed out in the 1980s that
quantum mechanical problems were better solved with quantum machines. It was
only in 1994 that Peter Shor came up with an algorithm that is able to
calculate the prime factors of a large number vastly more efficiently than
known possible with a classical computer. This paradigmatic algorithm
stimulated the flourishing research in quantum information processing and the
quest for an actual implementation of a quantum computer. Over the last fifteen
years, using skillful optimizations, several instances of a Shor algorithm have
been implemented on various platforms and clearly proved the feasibility of
quantum factoring. For general scalability, though, a different approach has to
be pursued. Here, we report the realization of a fully scalable Shor algorithm
as proposed by Kitaev. For this, we demonstrate factoring the number fifteen by
effectively employing and controlling seven qubits and four "cache-qubits",
together with the implementation of generalized arithmetic operations, known as
modular multipliers. The scalable algorithm has been realized with an ion-trap
quantum computer exhibiting success probabilities in excess of 90%.
