  We consider the dynamics of fluctuations in the quantum asymmetric simple
exclusion process (Q-ASEP) with periodic boundary conditions. The Q-ASEP
describes a chain of spinless fermions with random hoppings that are induced by
a Markovian environment. We show that fluctuations of the fermionic degrees of
freedom obey evolution equations of Lindblad type, and derive the corresponding
Lindbladians. We identify the underlying algebraic structure by mapping them to
non-Hermitian spin chains and demonstrate that the operator space fragments
into exponentially many (in system size) sectors that are invariant under time
evolution. At the level of quadratic fluctuations we consider the Lindbladian
on the sectors that determine the late time dynamics for the particular case of
the quantum symmetric simple exclusion process (Q-SSEP). We show that the
corresponding blocks in some cases correspond to known Yang-Baxter integrable
models and investigate the level-spacing statistics in others. We carry out a
detailed analysis of the steady states and slow modes that govern the late time
behaviour and show that the dynamics of fluctuations of observables is
described in terms of closed sets of coupled linear differential-difference
equations. The behaviour of the solutions to these equations is essentially
diffusive but with relevant deviations, that at sufficiently late times and
large distances can be described in terms of a continuum scaling limit which we
construct. We numerically check the validity of this scaling limit over a
significant range of time and space scales. These results are then applied to
the study of operator spreading at large scales, focusing on out-of-time
ordered correlators and operator entanglement.
