Simultaneous near‐certain preparation of qubits (quantum bits) in their ground states is a key hurdle in quantum computing proposals as varied as liquid‐state NMR and ion traps. “Closed‐system” cooling mechanisms are of limited applicability due to the need for a continual supply of ancillas for fault tolerance and to the high initial temperatures of some systems. “Open‐system” mechanisms are therefore required. We describe a new, efficient initialization procedure for such open systems. With this procedure, an n‐qubit device that is originally maximally mixed, but is in contact with a heat bath of bias $\varepsilon \gg 2^{-n}$, can be almost perfectly initialized. This performance is optimal due to a newly discovered threshold effect: For bias $\varepsilon \ll 2^{-n}$ no cooling procedure can, even in principle (running indefinitely without any decoherence), significantly initialize even a single qubit.