quantum random walk Mozhgan Torabifard Department of Physics, Sharif University of Technology Introduction Result Many classical algorithms are based on random walks, so it is natural to ask whether quantum random walks might be useful for quantum computation. There also, an exponential separation was found between the classical and quantum times to propagate (Y. Aharonov, L. Davidovich, and N.Zagury, 1993). Two models of quantum random walks have been suggested. Discrete Time(D. Aharonov, A. Ambainis, J. Kempe, and U. Vazirani, 2001) Continuous Time(A. Childs, E. Farhi, and S. Gutmann, 2002) Discrete Time We follow this model on a circle. The states of quantum random walk are in the space : Position Coin The quantum random walk of is defined as the transformation: Where is Hadamard. Initial state is .We get the eigen-vectors of U are: Where: With this analytical method we could reach the same results like previous studies. Decoherense  Decoherense is usually modeled as a non-unitary evolution. is a projection that represent decoherense. On the position (V.Kenden and O.Tregenna, 2003)  On the coin (C.C.Lopez and J.P.Paz,2003)   Complete depolarization   We study decoherense on both of them. · In each step, the coin and the position lost their memory with probability p.   We follow the analytic solution for this case. acknowledgment Partially supported by CeCsCm Great thanks to my supervisor Dr V.Karimipour for his worthwhile guidances International Iran Conference on Quantum Information, September 2007 , Kish Island, Iran