  A Josephson quantum filter (JQF) protects a data qubit (DQ) from the
radiative decay into transmission lines in superconducting quantum computing
architectures. A transmon, which is a weakly nonlinear harmonic oscillator
rather than a pure two-level system, can play a role of a JQF or a DQ. However,
in the previous study, a JQF and a DQ were modeled as two-level systems
neglecting the effects of higher levels. We theoretically examine the effects
of the higher levels of the JQF and the DQ on the control of the DQ. It is
shown that the higher levels of the DQ cause the shift of the resonance
frequency and the decrease of the maximum population of the first excited state
of the DQ in the controls with a continuous wave (cw) field and a pulsed field,
while the higher levels of the JQF do not. Moreover, we present optimal
parameters of the pulsed field, which maximize the control efficiency.
