Charged Complexity and the Thermofield Double State

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.