  The quantum Zeno effect is well-known for fixing a system to an eigenstate by
frequent measurements. It is also known that applying frequent unitary pulses
induces a Zeno subspace that can also pin the system to an eigenspace. Both
approaches have been studied as means to maintain a system in a certain
subspace. Extending the two concepts, we consider making the
measurements/pulses dynamical so that the state can move with the motion of the
measurement axis/pulse basis. We show that the system stays in the dynamical
eigenbasis when the measurements/pulses are slowly changing. Explicit bounds
for the apply rate that guarantees a success probability are provided. In
addition, both methods are inherently resilient against non-Markovian noise.
Finally, we discuss the similarities and differences between the two methods
and their connection to adiabatic quantum computation.
