Journal of Quantum Information Science, 2013, 3, 51-56 http://dx.doi.org/10.4236/jqis.2013.32010 Published Online June 2013 (http://www.scirp.org/journal/jqis) Survival Probability of the Quantum Walk with Phase Parameters on the Two-Dimensional Trapped Lattice Clement Ampadu 1 , Meltem Gönülol 2 , Ekrem Aydıner 2 1 31 Carrolton Road, Boston, MA, USA 2 İstanbul University Theoretical Physics Group, Department of Physics, İstanbul University, İstanbul, Turkey Email: drampadu@hotmail.com, ekrem.aydiner@istanbul.edu.tr Received March 27, 2013; revised May 5, 2013; accepted May 18, 2013 Copyright © 2013 Clement Ampadu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT We investigate the time dependence of the survival probability of quantum walks governed by Fibonacci walks with phase parameters on the trapped two-dimensional lattice. We have shown that the survival probability of the quantum walk decays with time obey to the stretched exponential law for all initial states of walkers. We have also shown that stretched exponential decay parameter β can be arranged by phase parameter combination. Obtained numerical results show that phase parameters can be used as a control parameter to determine the decay rate of the survival probability of the quantum walk. Keywords: Quantum Walk; Survival Probability 1. Introduction In recent years, the quantum walk on the trapped lattice has been intensively investigated due to importance ap- plications in quantum information and computing. Therefore, many theoretical and experimental studies have been carried out to understand the effect of the trapping states on the quantum walk. For example, Agliari [1] considered a continuous-time quantum walk propagating on Erdos-Renyi random graphs in the pres- ence of a random distribution of traps, and showed that the survival probability exhibits an exponential character which fluctuates depending on the trap concentration. Zahringer et al. [2] implemented a quantum walk using trapped ions. They used an experimental technique to determine the probability distribution along a line in phase space. It is shown that instabilities in the trapping frequency leads to decoherence and by change in the coupling strength due to high phonon numbers. Schimitz et al. [3] implemented the proof of principle for the quantum walk of one ion in a linear ion trap. It is shown that quantum interference enforces asymmetric, non- classical distributions in the highly entangled degrees of freedom (of coin and position states). Xue et al. [4] im- plemented a multi-step quantum walk for a single trapped ion with interpolation between quantum and random walk by randomizing the generalized Hadamard coin flip phase. It is shown that the distribution of the walker spreads over unbounded position space rather than being folded back on itself. Agliari et al. [5] studied the continuous time quantum walk focusing on trapping processes on a ring and show that when the traps are ar- ranged periodically the survival probability decays as- ymptotically, when the traps are arranged to form a clus- ter the survival probability decays exponentially, on the other hand for randomly distributed traps the survival probability decays algebraically. Eckert et al. [6] imple- mented the quantum walk with a neutral atom trapped in a ground state of optical potentials by using the concept of spatially delocalized quibits, that is, a coin defined through the presence of the atom in one out of two trap- ping potentials. It is shown that the shaking of the trap position on the quantum walk leads to a flat distribution in the transition from quantum to classical in the inter- mediate. Wojcik et al. [7] studied changing a phase at a single point in a discrete quantum walk. For certain val- ues of this phase change and the internal coin-state of the walker it is shown the distribution exhibits the localiza- tion phenomenon. Altmann et al. [8] studied chaotic scattering, it is shown that noise enhances the trapping of trajectories in scattering systems. In particular, they show that weak noise leads to a slower decay of the survival probability. Karksi et al. [9] implemented a quantum walk on the line with single neutral atoms by determinis- tically delocalizing them over the sites of a one-dimen- Copyright © 2013 SciRes. JQIS