A tight lower bound for non-coherent index erasure

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