Topological edge states in non-Hermitian systems are generally believed to be robust based on non-Bloch topology. Here, we show that in the Floquet non-Hermitian Su-Schrieffer-Heeger model, infinitesimal symmetry-preserving perturbations can completely destroy edge states, revealing the breakdown of the non-Bloch bulk-boundary correspondence in finite systems. We trace this fragility to the instability in the quasienergy spectrum and establish a correspondence between stable zero-mode singular states and midgap states of quasienergies in the thermodynamic limit. Then we develop a unified theoretical framework to characterize the number of zero-mode and $\pi /T$-mode edge states by the singular-value-based diagnostic scheme. This supplies a useful way to explore non-Hermitian Floquet topological phases.

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• Topological phase transition
• Floquet systems
• Non-Hermitian systems
• Bloch-Floquet theorem