  Almost all of the most successful quantum algorithms discovered to date
exploit the ability of the Fourier transform to recover subgroup structure of
functions, especially periodicity. The fact that Fourier transforms can also be
used to capture shift structure has received far less attention in the context
of quantum computation.
  In this paper, we present three examples of ``unknown shift'' problems that
can be solved efficiently on a quantum computer using the quantum Fourier
transform. We also define the hidden coset problem, which generalizes the
hidden shift problem and the hidden subgroup problem. This framework provides a
unified way of viewing the ability of the Fourier transform to capture subgroup
and shift structure.
