Grover's algorithm constitutes the optimal quantum solution to the search
problem and provides a quadratic speed-up over all possible classical search
algorithms. Quantum interference between computational paths has been posited
as a key resource behind this computational speed-up. However there is a limit
to this interference, at most pairs of paths can ever interact in a fundamental
way. Could more interference imply more computational power? Sorkin has defined
a hierarchy of possible interference behaviours---currently under experimental
investigation---where classical theory is at the first level of the hierarchy
and quantum theory belongs to the second. Informally, the order in the
hierarchy corresponds to the number of paths that have an irreducible
interaction in a multi-slit experiment. In this work, we consider how Grover's
speed-up depends on the order of interference in a theory. Surprisingly, we
show that the quadratic lower bound holds regardless of the order of
interference. Thus, at least from the point of view of the search problem,
post-quantum interference does not imply a computational speed-up over quantum
theory.Comment: Updated title and exposition in response to referee comments. 6+2
pages, 5 figure
