Liouvillian Gap in Dissipative Haar-Doped Clifford Circuits

Quantum chaos is commonly assessed through probe-dependent signatures that need not coincide. Recently, a dissipative signature was proposed for chaotic Floquet systems, where infinitesimal bulk dissipation induces a non-zero constant intrinsic relaxation rate quantified by the Liouvillian gap. This raises a question: what minimal departure from Clifford dynamics is required to generate such intrinsic relaxation? To address this, we study a Floquet two-qubit Clifford circuit doped with Haar-random single-qubit gates and subject to local dissipation of strength $\gamma$. We find a structure-dependent crossover. The undoped iSWAP-class circuit exhibits a weak-dissipation singularity, with a gap that grows with $N$ for any $\gamma>0$. Haar doping preserves this undoped-like growth for any subextensive doping pattern. At finite doping density, there exist patterns that yield an $\mathcal{O}(1)$ gap for any fixed $\gamma$ as $N\to\infty$, yet remain singular as $\gamma\to0^+$. Because our bounds depend only on the spatial doping pattern, they remain valid even when the Haar rotations are independently redrawn each Floquet period. Overall, our findings provide a circuit-level perspective on intrinsic relaxation, and thus irreversibility, in open many-body systems.