  Mandal and Jarzynski have proposed a fully autonomous information heat
engine, consisting of a demon, a mass and a memory register interacting with a
thermal reservoir. This device converts thermal energy into mechanical work by
writing information to a memory register, or conversely, erasing information by
consuming mechanical work. Here, we derive a speed limit inequality between the
relaxation time of state transformation and the distance between the initial
and final distributions, where the combination of the dynamical activity and
entropy production plays an important role. Such inequality provides a hint
that a speed-performance trade-off relation exists between the relaxation time
to functional state and the average production. To obtain fast
functionalization while maintaining the performance, we show that the
relaxation dynamics of information heat engine can be accelerated significantly
by devising an optimal initial state of the demon. Our design principle is
inspired by the so-called Mpemba effect, where water freezes faster when
initially heated.
