Perturbative complexity of an interacting theory

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