  The Index-Erasure problem is a quantum state generation problem that asks a
quantum computer to prepare a uniform superposition over the image of an
injective function given by an oracle. We prove a tight $\Omega(\sqrt{n})$
lower bound on the quantum query complexity of the non-coherent case of the
problem, where, in addition to preparing the required superposition, the
algorithm is allowed to leave the ancillary memory in an arbitrary
function-dependent state. This resolves an open question of Ambainis et al.,
who gave a tight bound for the coherent case, the case where the ancillary
memory must return to its initial state.
  To prove our main result, we first extend the automorphism principle of
H{\o}yer et al. to the general adversary method of Lee et al. for state
generation problems, which allows one to exploit the symmetries of these
problems to lower bound their quantum query complexity. Using this method, we
establish a strong connection between the quantum query complexity of
non-coherent symmetric state generation problems and the Krein parameters of an
association scheme defined on injective functions. In particular, we use the
spherical harmonics a finite symmetric Gelfand pair associated with the space
of injective functions to obtain asymptotic bounds on certain Krein parameters,
from which the main result follows.
