  Quantum computers have recently become available as noisy intermediate-scale
quantum devices. Already these machines yield a useful environment for research
on quantum systems and dynamics. Building on this opportunity, we investigate
open-system dynamics that are simulated on a quantum computer by coupling a
system of interest to an ancilla. After each interaction the ancilla is
measured and the sequence of measurements defines a quantum trajectory. Using a
thermodynamic analogy, which identifies trajectories as microstates, we show
how to control the dynamics of the open system in order to enhance the
probability of quantum trajectories with desired properties, e.g., particular
patterns or temporal correlations. We discuss how such biased -- generally
non-Markovian -- dynamics can be implemented on a unitary, gate-based quantum
computer and show proof-of-principle results on the publicly accessible
\texttt{ibm\_jakarta} machine. While our study is solely conducted on small
systems, it highlights the challenges in controlling complex aspects of
open-system dynamics on digital quantum computers.
