  We study some discrete symmetries of unbiased (Hadamard) and biased quantum
walk on a line, which are shown to hold even when the quantum walker is
subjected to environmental effects. The noise models considered in order to
account for these effects are the phase flip, bit flip and generalized
amplitude damping channels. The numerical solutions are obtained by evolving
the density matrix, but the persistence of the symmetries in the presence of
noise is proved using the quantum trajectories approach. We also briefly extend
these studies to quantum walk on a cycle. These investigations can be relevant
to the implementation of quantum walks in various known physical systems. We
discuss the implementation in the case of NMR quantum information processor and
ultra cold atoms.
