  Integrable models are characterized by the existence of stable excitations
that can propagate indefinitely without decaying. This includes multi-magnon
bound states in the celebrated XXZ spin chain model and its integrable Floquet
counterpart. A recent Google Quantum AI experiment [A. Morvan et al., Nature
612, 240 (2022)] realizing the Floquet model demonstrated the persistence of
such collective excitations even when the integrability is broken: this
observation is at odds with the expectation of ergodic dynamics in generic
non-integrable systems. We here study the spectrum of the model realized in the
experiment using exact diagonalization and physical arguments. We find that
isolated bands corresponding to the descendants of the exact bound states of
the integrable model are clearly observable in the spectrum for a large range
of system sizes. However, our numerical analysis of the localization properties
of the eigenstates suggests that the bound states become unstable in the
thermodynamic limit. A perturbative estimate of the decay rate agrees with the
prediction of an eventual instability for large system sizes.
