  Using a recent proposal of circuit complexity in quantum field theories
introduced by Jefferson and Myers, we compute the time evolution of the
complexity following a smooth mass quench characterized by a time scale $\delta
t$ in a free scalar field theory. We show that the dynamics has two distinct
phases, namely an early regime of approximately linear evolution followed by a
saturation phase characterized by oscillations around a mean value. The
behavior is similar to previous conjectures for the complexity growth in
chaotic and holographic systems, although here we have found that the
complexity may grow or decrease depending on whether the quench increases or
decreases the mass, and also that the time scale for saturation of the
complexity is of order $\delta t$ (not parametrically larger).
