Robustness of Floquet Topological Time Crystals against Two-Level System Defects in Circuit QED

Discrete time crystals (DTCs) represent a nonequilibrium phase of matter characterized by spontaneousbreaking of discrete time-translational symmetry under periodic driving. While Floquet topological phases offer apromising route to stabilizing DTC behavior, their persistence in realistic open quantum systems remains an openproblem. In this work, we numerically investigate a periodically driven one-dimensional Ising chain locally coupledto a dissipative two-level system (TLS) defect, motivated by microscopic defects commonly observed in superconductingcircuit-QED devices. Using Lindblad master-equation simulations implemented in QuTiP, we show that weaklydissipative TLS defects can dynamically synchronize with the Floquet edge mode and inherit its subharmonic oscillations.As the defect coupling and dissipation strength increase, the system undergoes a crossover toward decoherence andFloquet thermalization, accompanied by the suppression of edge-localized period doubling. By combining quasienergyspectroscopy, boundary-condition analysis, rigidity diagnostics, and entropy evolution, we characterize the stabilitylimits of Floquet topological time-crystalline phases in realistic noisy quantum hardware.