 Quantum technologies require pure states which are often generated by extreme refrigeration. Heat bath algorithmic cooling is the theoretically optimal refrigeration technique. It shuttles entropy from a multi-particle system to a thermal bath, thereby generating a quantum state with a high degree of purity. Here we show how to surpass this hitherto optimal technique by taking advantage of a single binary outcome measurement. Our protocols can create arbitrary numbers of pure quantum states without any residual mixtness by using a recently discovered device known as a quantum switch to put two operations in superposition, with post-selection certifying the complete purification. This article was authored by Aaron Z. Goldberg and Kabat Hashami.