  We develop a microscopic theory for biasing the quantum trajectories of an
open quantum system, which renders rare trajectories typical. To this end we
consider a discrete-time quantum dynamics, where the open system collides
sequentially with qubit probes which are then measured. A theoretical framework
is built in terms of thermodynamic functionals in order to characterize its
quantum trajectories (each embodied by a sequence of measurement outcomes). We
show that the desired biasing is achieved by suitably modifying the Kraus
operators describing the discrete open dynamics. From a microscopical viewpoint
and for short collision times, this corresponds to adding extra collisions
which enforce the system to follow a desired rare trajectory. The above extends
the theory of biased quantum trajectories from Lindblad-like dynamics to
sequences of arbitrary dynamical maps, providing at once a transparent physical
interpretation.
