  Landauer's universal limit on heat dissipation during information erasure
becomes increasingly crucial as computing devices shrink: minimising
heat-induced errors demands optimal pure-state preparation. For this, however,
Nernst's third law posits an infinite-resource requirement: either energy,
time, or control complexity must diverge. Here, we address the practical
challenge of efficiently cooling quantum systems using finite resources. We
investigate the ensuing resource trade-offs and present efficient protocols for
finite distinct energy gaps in settings pertaining to coherent or incoherent
control, corresponding to quantum batteries and heat engines, respectively.
Expressing energy bounds through thermodynamic length, our findings illuminate
the optimal distribution of energy gaps, detailing the resource limitations of
preparing pure states in practical settings.
