  We use the wave-function method developed in area of quantum information to
investigate the quantum circuit complexity of the small quantum fluctuations
around the probe $D_3$ brane moving in $AdS_5\times S^5$ bulk. In our
consideration, the reference and target states are chosen as the vacuum state
and the squeezed quantum state respectively. The evolution of parameters
characterizing the squeezed quantum state are governed by the time-dependent
$Schr\ddot{o}dinger$ equation, in which the Hamiltonian operator is derived
from the perturbative action of $D_3$ brane. For a quantum chaotic system, some
recent works indicate that the evolution of quantum circuit complexity could
provide equivalent information like the out-of-time-order correlators. Basing
on this inference, our results show that the quantum fluctuations around the
non-BPS brane manifestly evolve into the chaotic regime at the late time, while
the chaotic behavior is not easy to observe in case of BPS brane. In
holographic viewpoint, it implies that the thermodynamic system consist of the
$N=4$ supersymmetric particles in non-BPS states evolve into a chaotic system
more easily than the one in BPS state.
