A Tight Lower Bound for Non-coherent Index Erasure

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. As an application, we show that the quantum query complexity of non-coherently generating the uniform superposition of the symmetries of an $n$-vertex graph is $\Theta(\sqrt{n!})$. This improves the known $\Omega(\sqrt[3]{n!})$ lower bound obtained through previous results on the Set-Equality problem. 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.