  We establish a systematic framework for studying quantum computational
complexity of Gaussian states of charged systems based on Nielsen's geometric
approach. We use this framework to examine the effect of a chemical potential
on the dynamics of complexity. As an example, we consider the complexity of a
charged thermofield double state constructed from two free massive complex
scalar fields in the presence of a chemical potential. We show that this state
factorizes between positively and negatively charged modes and demonstrate that
this fact can be used to relate it, for each momentum mode separately, to two
uncharged thermofield double states with shifted temperatures and times. We
evaluate the complexity of formation for the charged thermofield double state,
both numerically and in certain analytic expansions. We further present
numerical results for the time dependence of complexity. We compare various
aspects of these results to those obtained in holography for charged black
holes.
