A note on quantum one-way permutations

We discuss the question of the existence of quantum one-way permutations. First, we prove the equivalence between inverting a permutation and that of constructing a polynomial size network for reflecting about a given quantum state. Next, we consider the question: if a state is difficult to prepare, is the operator reflecting about that state difficult to construct? By revisiting Grover's algorithm, we present the relationship between this question and the existence of one-way permutations. Moreover, we compare our method to Grover's algorithm and discuss possible applications of our results.