A Tight Lower Bound for Index Erasure
Abstract
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 noncoherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary functiondependent 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2019
 arXiv:
 arXiv:1902.07336
 Bibcode:
 2019arXiv190207336L
 Keywords:

 Quantum Physics;
 Computer Science  Computational Complexity;
 Mathematics  Combinatorics;
 Mathematics  Representation Theory