  Scar eigenstates in a many-body system refers to a small subset of
non-thermal finite energy density eigenstates embedded into an otherwise
thermal spectrum. This novel non-thermal behaviour has been seen in recent
experiments simulating a one-dimensional PXP model with a
kinetically-constrained local Hilbert space realized by a chain of Rydberg
atoms. We probe these small sets of special eigenstates starting from
particular initial states by computing the spread complexity associated to time
evolution of the PXP hamiltonian. Since the scar subspace in this model is
embedded only loosely, the scar states form a weakly broken representation of
the Lie Algebra. We demonstrate why a careful usage of the Forward Scattering
Approximation (or similar strategies thereof) is required to extract an
appropriate set of Lanczos coefficients in this case as the consequence of this
approximate symmetry. This leads to a well defined notion of a closed Krylov
subspace and consequently, that of spread complexity. We show how the spread
complexity shows approximate revivals starting from both $|\mathbb{Z}_2\rangle$
and $|\mathbb{Z}_3\rangle$ states and how these revivals can be made more
accurate by adding optimal perturbations to the bare Hamiltonian. We also
investigate the case of the vacuum as the initial state, where revivals can be
stabilized using an iterative process of adding few-body terms.
