  Landauer's principle makes a strong connection between information theory and
thermodynamics by stating that erasing a one-bit memory at temperature $T_0$
requires an average energy larger than $W_{LB}=k_BT_0 \ln2$, with $k_B$
Boltzmann's constant. This tiny limit has been saturated in model experiments
using quasi-static processes. For faster operations, an overhead proportional
to the processing speed and to the memory damping appears. In this article, we
show that underdamped systems are a winning strategy to reduce this extra
energetic cost. We prove both experimentally and theoretically that, in the
limit of vanishing dissipation mechanisms in the memory, the physical system is
thermally insulated from its environment during fast erasures, i.e. fast
protocols are adiabatic as no heat is exchanged with the bath. Using a fast
optimal erasure protocol we also show that these adiabatic processes produce a
maximum adiabatic temperature $T_a=2T_0$, and that Landauer's bound for fast
erasures in underdamped systems becomes the adiabatic bound: $W_a = k_B T_0$.
