  Dissipation is a ubiquitous phenomenon in nature that affects the fate of
chaotic quantum dynamics. To characterize the interplay between quantum chaos
and dissipation in generic quantum many-body systems, we consider a minimal
dissipative Floquet many-body circuit. In particular, we study the dissipative
form factor (DFF), an extension of the spectral form factor to open quantum
systems, of Floquet systems in the presence of arbitrary on-site dissipation
modeled by quantum channels. For a solvable model, in the limit of large local
Hilbert space dimension, we obtain an exact expression for the DFF averaged
over the random unitary gates, with simple, closed-form expressions in the
limit of large times. We find that, for long enough times, the system always
relaxes (i.e., the DFF decays) with two distinctive regimes characterized by
the presence or absence of gap-closing. While the system can sustain a robust
ramp for a long (but finite) time interval in the gap-closing regime,
relaxation is "assisted" by quantum chaos in the regime where the gap remains
nonzero. In the latter regime, we find that, if the thermodynamic limit is
taken first, the gap does not close even in the dissipationless limit, a
recently uncovered phenomenon dubbed anomalous relaxation. We complement our
findings with numerical results for quantum circuits with a small Hilbert space
dimension.
