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The stability and dynamics of almost strong zero and $\pi$ modes in weaklynon-integrable Floquet spin chains are investigated. Such modes can also beviewed as localized Majorana modes at the edge of a topological superconductor.Perturbation theory in the strength of integrability-breaking interaction $J_z$is employed to estimate the decay rates of these modes, and compared to decayrates obtained from exact diagonalization. The structure of the perturbationtheory and thus the lifetime of the modes is governed by the conservation ofquasi-energy modulo $2 \pi/T$, where $T$ is the period of the Floquet system.If the quasi-energies of minimally $4 n-1$ quasi-particles adds up to zero (or$\pi/T$ for a $\pi$ mode), the lifetime is proportional to $1/J_z^{2 n}$. Thusthe lifetime is sensitively controlled by the width of the single-particleFloquet bands. For regimes where the decay rates are quadratic in $J_z$, ananalytic expression for the decay rate in terms of an infinite temperatureautocorrelation function of the integrable model is derived, and shown to agreewell with exact diagonalization.

Decay rates of almost strong modes in Floquet spin chains beyond Fermi's Golden Rule