TITLE: Quantum algorithms based on transforms other than Fourier transforms
QUESTION [19 upvotes]: In Quantum Computation and Quantum Information by Nielsen and Chuang they say that many of the algorithms based on quantum Fourier transforms rely on the Coset Invariance property of Fourier transforms and suggests that invariance properties of other transforms might yield new algorithms.
Has there been any fruitful research on other transforms?

REPLY [4 votes]: Not sure if this is directly linked to your question, but reading it made me think about an article by Peter Høyer I read some years ago. In it, he shows how the most popular quantum algorithms like Grover's or Shor's follow the same pattern of applying what he calls "conjugated operators" and he builds new algorithms also based on that same pattern. 
As I said, it's been a few years since I've read it so my description is a bit sloppy, but here's the link in case you want to check it out. 
http://journals.aps.org/pra/abstract/10.1103/PhysRevA.59.3280
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