  One dimensional topological insulators are characterized by edge states with
exponentially small energies. According to one generalization of topological
phases to non-Hermitian systems, a finite system in a non-trivial topological
phase displays surface states with exponentially long life times. In this work
we explore the possibility of exploiting such non-Hermitian topological phases
to enhance the quantum coherence of a fiducial qubit embedded in a dissipative
environment. We first show that a network of qubits interacting with lossy
cavities can be represented, in a suitable super-one-particle sector, by a
non-Hermitian "Hamiltonian" of the desired form. We then study, both
analytically and numerically, one-dimensional geometries with up to three sites
per unit cell, and up to a topological winding number $W=2$. For finite-size
systems the number of edge modes is a complicated function of $W$ and the
system size $N$. However we find that there are precisely $W$ modes localized
at one end of the chain. In such topological phases the quibt's coherence
lifetime is exponentially large in the system size. We verify that, for $W>1$,
at large times, the Lindbladian evolution is approximately a non-trivial
unitary. For $W=2$ this results in Rabi-like oscillations of the qubit's
coherence measure.
