Quantum Computer Can Not Speed Up Iterated Applications of a Black Box

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 any 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(2^{n/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 \Omega (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. For an arbitrary time complexity T a lower bound for the time of quantum simulation was found to be \Omega (T^{1/2}).