  We provide a systematic framework for constructing generic models of
nonequilibrium quantum dynamics with a target stationary (mixed) state. Our
framework identifies (almost) all combinations of Hamiltonian and dissipative
dynamics that relax to a steady state of interest, generalizing the Davies'
generator for dissipative relaxation at finite temperature to nonequilibrium
dynamics targeting arbitrary stationary states. We focus on Gibbs states of
stabilizer Hamiltonians, identifying local Lindbladians compatible therewith by
constraining the rates of dissipative and unitary processes. Moreover, given
terms in the Lindbladian not compatible with the target state, our formalism
identifies the operations -- including syndrome measurements and local feedback
-- one must apply to correct these errors. Our methods also reveal new models
of quantum dynamics: for example, we provide a "measurement-induced phase
transition" where measurable two-point functions exhibit critical (power-law)
scaling with distance at a critical ratio of the transverse field and rate of
measurement and feedback. Time-reversal symmetry -- defined naturally within
our formalism -- can be broken both in effectively classical, and intrinsically
quantum, ways. Our framework provides a systematic starting point for exploring
the landscape of quantum dynamical universality classes, as well as identifying
new protocols for quantum error correction.
