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Quantum computers hold the promise to solve certain problems exponentially faster than their classical counterparts. Trapped atomic ions are among the physical systems in which building such a computing device seems viable. In this work we present a small-scale quantum information processor based on a string of 40Ca+ ions confined in a macroscopic linear Paul trap. We review our set of operations which includes non-coherent operations allowing us to realize arbitrary Markovian processes. In order to build a larger quantum information processor it is mandatory to reduce the error rate of the available operations which is only possible if the physics of the noise processes is well understood. We identify the dominant noise sources in our system and discuss their effects on different algorithms. Finally we demonstrate how our entire set of operations can be used to facilitate the implementation of algorithms by examples of the quantum Fourier transform and the quantum order finding algorithm. [ABSTRACT FROM AUTHOR]

We study the effect of an environment consisting of noninteracting two level systems on Landau-Zener transitions with an interest on the performance of an adiabatic quantum computer. We show that if the environment is initially at zero temperature, it does not affect the transition probability. An excited environment, however, will always increase the probability of making a transition out of the ground state. For the case of equal intermediate gaps, we find an analytical upper bound for the transition probability in the limit of large number of environmental spins. We show that such an environment will only suppress the probability of success for adiabatic quantum computation by at most a factor close to 1/2. [ABSTRACT FROM AUTHOR]

Certain algorithms for quantumcomputers are able to outperform their classical counterparts. In 1994, Peter Shor came up with a quantum algorithm that calculates the prime factors of a large number vastly more efficiently than a classical computer. For general scalability of such algorithms, hardware, quantum error correction, and the algorithmic realization itself need to be extensible. Here we present the realization of a scalable Shor algorithm, as proposed by Kitaev. We factor the number 15 by effectively employing and controlling seven qubits and four "cache qubits" and by implementing generalized arithmetic operations, known as modular multipliers. This algorithm has been realized scalably within an ion-trap quantum computer and returns the correct factors with a confidence level exceeding 99%. [ABSTRACT FROM AUTHOR]