The Query Complexity of Order-Finding

We consider the problem where is an unknown permutation on f0; 1; : : : ; 2n 1g, y0 2 f0; 1; : : : ; 2n 1g, and the goal is to determine the minimum r > 0 such that r(y0) = y0. Information about is available only via queries that yield x(y) from any x 2 f0; 1; : : : ; 2m 1g and y 2 f0; 1; : : : ; 2n 1g (where m is polynomial in n). The resource under consideration is the number of these queries (hence our model of computation is the decision tree). We show that the number of queries necessary to solve the problem in the classical probabilistic boundederror model is exponential in n. This contrasts sharply with the quantum bounded-error model, where a constant number of queries suffices.