  Practical implementations of quantum technologies require preparation of
states with a high degree of purity---or, in thermodynamic terms, very low
temperatures. Given finite resources, the Third Law of thermodynamics prohibits
perfect cooling; nonetheless, attainable upper bounds for the asymptotic ground
state population of a system repeatedly interacting with quantum thermal
machines have been derived. These bounds apply within a memoryless (Markovian)
setting, in which each refrigeration step proceeds independently of those
previous. Here, we expand this framework to study the effects of memory on
quantum cooling. By introducing a memory mechanism through a generalized
collision model that permits a Markovian embedding, we derive achievable bounds
that provide an exponential advantage over the memoryless case. For qubits, our
bound coincides with that of heat-bath algorithmic cooling, which our framework
generalizes to arbitrary dimensions. We lastly describe the adaptive step-wise
optimal protocol that outperforms all standard procedures.
