 Permutation-preimage quantum search

To find an input of a permutation whose output lies in a known $K$-element subset, compute the output, apply its marked-state phase oracle, and use uncomputation. A modular-addition quantum oracle costs two forward queries per marked reflection, because modular-oracle inversion by negation supplies its inverse. Amplitude amplification then uses $O(\sqrt{N/K})$ queries for fixed success probability. Positive-square outputs have $K=\lfloor \sqrt{N-1}\rfloor$ and hence query cost $O(N^{1/4})$.