  Discrete-time quantum walks are known to exhibit exotic topological states
and phases. Physical realization of quantum walks in a noisy environment may
destroy these phases. We investigate the behavior of topological states in
quantum walks in the presence of a lossy environment. The environmental effects
in the quantum walk dynamics are addressed using the non-Hermitian Hamiltonian
approach. We show that the topological phases of the quantum walks are robust
against moderate losses. The topological order in one-dimensional split-step
quantum walk persists as long as the Hamiltonian is $\mathcal{PT}$-symmetric.
Although the topological nature persists in two-dimensional quantum walks as
well, the $\mathcal{PT}$-symmetry has no role to play there. Furthermore, we
observe the noise-induced topological phase transition in two-dimensional
quantum walks.

Quantum Physics