  Almost strong edge-mode operators arising at the boundaries of certain
interacting 1D symmetry protected topological phases with \(Z_2\) symmetry have
infinite temperature lifetimes that are non-perturbatively long in the
integrability breaking terms, making them promising as bits for quantum
information processing. We extract the lifetime of these edge-mode operators
for small system sizes as well as in the thermodynamic limit. For the latter, a
Lanczos scheme is employed to map the operator dynamics to a one dimensional
tight-binding model of a single particle in Krylov space. We find this model to
be that of a spatially inhomogeneous Su-Schrieffer-Heeger model with a hopping
amplitude that increases away from the boundary, and a dimerization that
decreases away from the boundary. We associate this dimerized or staggered
structure with the existence of the almost strong mode. Thus the short time
dynamics of the almost strong mode is that of the edge-mode of the
Su-Schrieffer-Heeger model, while the long time dynamics involves decay due to
tunneling out of that mode, followed by chaotic operator spreading. We also
show that competing scattering processes can lead to interference effects that
can significantly enhance the lifetime.
