Ion-trap quantum computation (QC) is promising for the long coherence times and the scalability of the number of qubits. However, at the moment multi-qubit experiments are hard to implement, therefore simulations are important tools for studying decoherence and the efficiency of quantum algorithms.

First we present our proposed general method for realizing the conditional phase-shift gate in multi-qubit ion-trap QC. We simulated Grover's algorithm and the Quantum Fourier Transform within the Cirac-Zoller QC proposal taking into consideration the laser fluctuations (intensity and phase) and investigated the fidelity and the heating rate. Furthermore Quantum Random Walks (QRWs) have recently attracted much attention because of their applications in developing new quantum algorithms. Here we propose scalable quantum networks for discrete QRWs on lines, circles and in higher- dimensions. With our method QRW can be implemented in any physical system, and as an example we show the results of the simulation in an ion-trap system. At last we introduce the prospect for developing new ion-trap QC schemes.