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The cost of fault-tolerant magic state preparations have been going down rapidly over the past few years. Motivated by this development, I will discuss some surprising examples of quantum circuits that can be realized in constant T-depth. Some of these constructions, such as single-qubit rotation and its programmable variants, as well as quantum part of Shor's factoring algorithm, require a catalyst state. But there are also other constructions that do not, such as reversible encoded addition.
The discovery of the random purification channel has surprised many of us who work on quantum learning, as it is so natural a statement, so powerful a technique, and spent so long under our noses. In this talk, I will survey the developments it led to over this past year. I will also give a rough theory for why it took decades to find, including why it was found now.