We will discuss the quantum Fourier transform for an arbitrary
finite abelian group, and Hallgren's adaptation of Shor's algorithm to
uncountable abelian groups—namely, to $\mathbb{R}$. Both pieces are
essential ingredients in the quantum algorithm of
Eisentr\"ager-Hallgren-Kitaev-Song to compute the unit group of a number
field. Suggested readings are Hallgren's Pell equation paper and Jozsa's
exposition on the quantum Fourier transform; as usual, both are
available at

http://public.csusm.edu/ssharif/crypto