  Developing protocols for preserving information in quantum systems is a
central quest for implementing realistic quantum computation. In this regard,
the quantum Zeno effect has emerged as a widely utilized technique to safeguard
classical information stored in quantum systems. However, existing results
pertaining to this method often assume operations performed infinitely fast on
the system of interest, which only serves as an approximation to real-world
scenarios where the temporal resolution of any experimental apparatus is
inherently finite. In this study, we go beyond this conventional assumption and
derive the effective Zeno dynamics for any time interval between operations.
Our analysis considers a qubit undergoing thermalization, as described by a
generalized amplitude damping channel, while the operations performed consist
of projections onto an orthonormal basis that may or may not coincide with the
pointer basis to which the system is thermalizing. By obtaining the probability
of successfully storing a bit of information after a given time, we investigate
the performance of the protocol in two important scenarios: the limit of many
interventions, with a first-order correction to the Zeno limit, and the limit
of very few interventions. In doing so, we provide valuable insights into the
protocol's performance by establishing bounds on its efficacy. These findings
enhance our understanding of the practical applicability of the quantum Zeno
effect in preserving classical information stored in quantum systems, allowing
for better design and optimization of quantum information processing protocols.
