  In this paper, we investigate the circuit complexity of a quantum harmonic
oscillator subjected to an external magnetic field. Utilizing the Nielsen
approach within the thermofield dynamics (TFD) framework, we determine the
complexity of thermofield double states as functions of time, temperature, and
the external magnetic field. Our subsequent analysis reveals various features
of this complexity. For instance, as temperature increases, the amplitude of
complexity oscillations also rises, while at low temperatures, complexity
stabilizes at a constant positive value. Furthermore, the magnetic field
creates two distinct sectors: strong magnetic fields exhibit periodic
complexity oscillations, whereas weak magnetic fields induce a beating effect.
Finally, we confirm that the rate of complexity obeys the Lloyd bound.
