The Index Erasure problem 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, Magnin, Roetteler, and Roland (CCC 2011), who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. The proof is based on evaluating certain Krein parameters of a symmetric association scheme defined over partial permutations. The study of this association scheme may be of independent interest.
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