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

TL;DR

This paper proves that quantum computers cannot universally speed up classical algorithms that are based on sequential black box applications, establishing fundamental lower bounds on quantum simulation efficiency.

Contribution

It demonstrates that there is no general quantum speedup method for classical algorithms modeled as iterative black box applications, providing lower bounds on quantum simulation time.

Findings

Quantum speedup is impossible for certain black box algorithms.

Lower bound of (T^{1/2}) for quantum simulation time.

No universal quantum acceleration for classical iterative algorithms.

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 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}).