  We predict that continuously monitored quantum dynamics can be chaotic. The
optimal paths between past and future boundary conditions can diverge
exponentially in time when there is time-dependent evolution and continuous
weak monitoring. Optimal paths are defined by extremizing the global
probability density to move between two boundary conditions. We investigate the
onset of chaos in pure-state qubit systems with optimal paths generated by a
periodic Hamiltonian. Specifically, chaotic quantum dynamics are demonstrated
in a scheme where two non-commuting observables of a qubit are continuously
monitored, and one measurement strength is periodically modulated. The optimal
quantum paths in this example bear similarities to the trajectories of the
kicked rotor, or standard map, which is a paradigmatic example of classical
chaos. We emphasize connections with the concept of resonance between
integrable optimal paths and weak periodic perturbations, as well as our
previous work on "multipaths", and connect the optimal path chaos to
instabilities in the underlying quantum trajectories.
