  We consider how the energy cost of bit reset scales with the time duration of
the protocol. Bit reset necessarily takes place in finite time, where there is
an extra penalty on top of the quasistatic work cost derived by Landauer. This
extra energy is dissipated as heat in the computer, inducing a fundamental
limit on the speed of irreversible computers. We formulate a
hardware-independent expression for this limit in the framework of stochastic
processes. We derive a closed-form lower bound on the work penalty as a
function of the time taken for the protocol and bit reset error. It holds for
discrete as well as continuous systems, assuming only that the master equation
respects detailed balance.
