The Query Complexity of Order--nding

We consider the problem where is an unknown permutation on f0; 1; : : :; 2 n ?1g, y 0 2 f0; 1; : : :; 2 n ? 1g, and the goal is to determine the minimum r > 0 such that r (y 0) = y 0. Information about is available only via queries that yield x (y) from any x 2 f0; 1; : : :; 2 m ? 1g and y 2 f0; 1; : : :; 2 n ? 1g (where m is polynomial in n). The main resource under consideration is the number of these queries. We show that the number of queries necessary to solve the problem in the classical probabilistic bounded-error model is exponential in n. This contrasts sharply with the quantum bounded-error model, where a constant number of queries suuces.