  We present a systematic method to expand the quantum complexity of
interacting theory in series of coupling constant. The complexity is evaluated
by the operator approach in which the transformation matrix between the second
quantization operators of reference state and the target state defines the
quantum gate. We start with two coupled oscillators and perturbatively evaluate
the geodesic length of the associated group manifold of gate matrix. Next, we
generalize the analysis to $N$ coupled oscillators which describes the lattice
$\lambda\phi^4$ theory. Especially, we introduce simple diagrams to represent
the perturbative series and construct simple rules to efficiently calculate the
complexity. General formulae are obtained for the higher-order complexity of
excited states. We present several diagrams to illuminate the properties of
complexity and show that the interaction correction to complexity may be
positive or negative depending on the magnitude of reference-state frequency.
