  Motivated by a connection, described here for the first time, between the
hidden normal subgroup problem (HNSP) and abelian hypergroups (algebraic
objects that model collisions of physical particles), we develop a stabilizer
formalism using abelian hypergroups and an associated classical simulation
theorem (a la Gottesman-Knill). Using these tools, we develop the first
provably efficient quantum algorithm for finding hidden subhypergroups of
nilpotent abelian hypergroups and, via the aforementioned connection, a new,
hypergroup-based algorithm for the HNSP on nilpotent groups. We also give
efficient methods for manipulating non-unitary, non-monomial stabilizers and an
adaptive Fourier sampling technique of general interest.
