Volume 7 (2011) Article 2 pp. 19-25 [Note]
Inverting a Permutation is as Hard as Unordered Search
The reduction we present helps us bypass the hybrid argument due to Bennett, Bernstein, Brassard, and Vazirani (1997) and the quantum adversary method due to Ambainis (2002) that were earlier used to derive lower bounds on the quantum query complexity of the problem of inverting permutations. It directly implies that the quantum query complexity of the problem is asymptotically the same as that for unordered search, namely in $\Theta(\sqrt{n}\,)$.