Global unitary transformations (OPTSWAPS) that optimally increase the bias of any mixed computation qubit in a quantum system -- represented by a diagonal density matrix -- towards a particular state of the computational basis which, in effect, increases its purity are presented. Quantum circuits that achieve this by implementing the above data compression technique -- a generalization of the 3B-Comp used before -- are described. These circuits enable purity increment in the computation qubit by maximally transferring part of its von Neumann or Shannon entropy to any number of surrounding qubits and are valid for the complete range of initial biases. Using the optswaps, a practicable new method that algorithmically achieves hierarchy-dependent cooling of qubits to their respective limits in an engineered quantum register opened to the heat-bath is delineated. In addition to multi-qubit purification and satisfying two of DiVincenzo's criteria for quantum computation in some architectures, the implications of this work for quantum data compression and quantum thermodynamics are discussed.