  The influence of discreteness on the fluxon scattering on the dipole-like
impurity is studied. This kind of impurity is used to model the qubit
inductively coupled to the Josephson transmission line (JTL). The previously
proposed fluxon assisted readout process of the qubit state is based on
measuring the passage time through the dipole impurity. The aim of this work is
to clarify the role of the discreteness in this qubit readout process. It is
demonstrated that the fluxon delay time on the qubit impurity increases
significantly if the discreteness of the JTL increases. Also it is shown that
the difference between the fluxon threshold currents for the positive and
negative qubits decreases with the increase of discreteness.
