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.
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