  In this paper, the Nielsen geometric method is used to study the position
dependence of the Nielsen complexity for the thermofield double state of a
harmonic oscillator. We present the state shift under the influence of an
external electric field and demonstrate its importance for the construction of
the corresponding circuit. By numerical analysis, we investigate the effect of
the frequency and the external field on the dynamics of complexity. Our
observation reveals that the system's complexity diminishes considerably with
the rise of the frequency. Furthermore, our findings indicate that the
complexity exhibits a distinct behavior under a feeble external electric field,
as it grows more intricate with the escalation of the frequency. However, with
higher magnitudes of the electric field, the system reverts to its prior
behavior. We also remark on the influence of the reference state's frequency on
the complexity.
