About the quantum mechanical speeding up of classical algorithms

TL;DR

This paper introduces a quantum transformation called RDT that enables certain classical algorithms to be sped up significantly on quantum computers, specifically reducing computation time from T to approximately S√T.

Contribution

The paper presents the RDT, a novel quantum transformation, and proves it can accelerate classical 1D cellular automaton computations to a time complexity of O(S√T).

Findings

RDT enables polynomial-time quantum simulation of classical automata.

Quantum computation time is reduced from T to O(S√T).

Multiprocessor quantum systems with classical interaction achieve this speedup.

Abstract

This work introduces a relative diffusion transformation (RDT) - a simple unitary transformation which acts in a subspace, localized by an oracle. Such a transformation can not be fulfilled on quantum Turing machines with this oracle in polynomial time in general case. It is proved, that every function computable in time T and space S on classical 1-dimensional cellular automaton, can be computed with certainty in time O(S \sqrt T) on quantum computer with RDTs over the parts of intermediate products of classical computation. This requires multiprocessor, which consists of \sqrt T quantum devices each of O(S) size, working in parallel-serial mode and interacting by classical lows.