  The advantages of quantum information processing are in many cases obtained
as consequences of quantum interactions, especially for computational tasks
where two-qubit interactions are essential. In this work, we establish the
framework of analyzing and quantifying loss or gain of information on a quantum
system when the system interacts with its environment. We show that the
information flow, the theoretical method of characterizing (non-)Markovianity
of quantum dynamics, corresponds to the rate of the minimum uncertainty about
the system given quantum side information. Thereafter, we analyze the
information exchange among subsystems that are under the performance of quantum
algorithms, in particular, the amplitude amplification algorithms where the
computational process relies fully on quantum evolution. Different realizations
of the algorithm are considered, such as i)quantum circuits, ii) analog
computation, and iii) adiabatic computation. It is shown that, in all the
cases, our formalism provides insights about the process of amplifying the
amplitude from the information flow or leakage on the subsystems.
