  Motivated by holographic complexity proposals as novel probes of black hole
spacetimes, we explore circuit complexity for thermofield double (TFD) states
in free scalar quantum field theories using the Nielsen approach. For TFD
states at t = 0, we show that the complexity of formation is proportional to
the thermodynamic entropy, in qualitative agreement with holographic complexity
proposals. For TFD states at t > 0, we demonstrate that the complexity evolves
in time and saturates after a time of the order of the inverse temperature. The
latter feature, which is in contrast with the results of holographic proposals,
is due to the Gaussian nature of the TFD state of the free bosonic QFT. A novel
technical aspect of our work is framing complexity calculations in the language
of covariance matrices and the associated symplectic transformations, which
provide a natural language for dealing with Gaussian states. Furthermore, for
free QFTs in 1+1 dimension, we compare the dynamics of circuit complexity with
the time dependence of the entanglement entropy for simple bipartitions of
TFDs. We relate our results for the entanglement entropy to previous studies on
non-equilibrium entanglement evolution following quenches. We also present a
new analytic derivation of a logarithmic contribution due to the zero momentum
mode in the limit of vanishing mass for a subsystem containing a single degree
of freedom on each side of the TFD and argue why a similar logarithmic growth
should be present for larger subsystems.
