I was thinking about the roots-of-unity algorithm for proposed by Jason, Anshu, Aram, and others. As you all convinced me, the expectation works out very nicely. The variance is a bit large: naively it seems to grow like instead of and controlling that requires some ideas. Here is a paper by Sumit Ganguly that pursues this direction; maybe you all have ideas of how to do things differently.
The notes for today’s lecture will be only soon.