Motivated by holographic complexity proposals as novel probes of black hole spacetimes, the talk will explore circuit complexity for thermofield double states (TFDs) in free quantum field theories (QFTs). The talk is based on an upcoming paper with Chapman, Eisert, Hackl, Jefferson, Marrochio, Myers and Pastawski. To set the stage, I will also discuss elements of earlier works 1707.08582 and 1707.08570 by various members of the present collaboration.