  We evaluate the complexity of the free scalar field by the operator approach
in which the transformation matrix between the second quantization operators of
reference state and target state is regarded as the quantum gate. We first
examine the system in which the reference state is two non-interacting
oscillators with same frequency $\omega_0$ while the target state is two
interacting oscillators with frequency $\tilde \omega_1$ and $\tilde \omega_2$.
We calculate the geodesic length on the associated group manifold of gate
matrix and reproduce the known value of ground-state complexity. Next, we study
the complexity in the excited states. Although the gate matrix is very large we
can transform it to a diagonal matrix and obtain the associated complexity. We
explicitly calculate the complexity in several excited states and prove that
the square of geodesic length in the general state $|{\rm n,m}\rangle$ is
$D_{\rm (n,m)}^2={\rm (n+1)}\left(\ln {\sqrt{\tilde \omega_1\over
\omega_0}}\,\right)^2 +{\rm (m+1)}\left(\ln {\sqrt{\tilde \omega_2\over
\omega_0}}\,\right)^2$. The results are extended to the N couple harmonic
oscillators which correspond to the lattice version of free scalar field.
