Quantum computer can not speed up iterated applications of a black box

Abstract

Let a classical algorithm be determined by sequential applications of a black box performing one step of this algorithm. If we consider this black box as an oracle which gives a value f (a) for a query a, we can compute T sequential applications of f on a classical computer relative to this oracle in time T. It is proved that if T = O(2n/7), where n is the length of input, then the result of T sequential applications of f can not be computed on quantum computer with oracle for f for all possible f faster than in time Ω(T). This means that there is no general method of quantum speeding up of classical algorithms provided in such a general method a classical algorithm is regarded as iterated applications of a given black box.