Shor's factorization algorithm

Abstract

This chapter describes what is generally considered to be one of the most important and historical contributions to the field of quantum computing, namely Shor's factorization algorithm . As its name indicates, this algorithm makes it possible to factorize numbers, which consists in their decomposition into a unique product of prime numbers. Other classical factorization algorithms previously developed have a complexity or computing time that increases exponentially with the number size, making the task intractable if not hopeless for large numbers. In contrast, Shor's algorithm is able to factor a number of any size in polynomial time , making the factorization problem tractable should a quantum computer ever be realized in the future. Since Shor's algorithm is based on several nonintuitive properties and other mathematical subtleties, this chapter presents a certain level of difficulty. With the previous chapters and tools readily assimilated, and some patience in going through the different preliminary steps required, such a difficulty is, however, quite surmountable. I have sought to make this description of Shor's algorithm as mathematically complete as possible and crack-free, while avoiding some academic considerations that may not be deemed necessary from any engineering perspective. Eventually, Shor's algorithm is described in only a few basic instructions. What is conceptually challenging is to grasp why it works so well, and also to feel comfortable with the fact that its implementation actually takes a fair amount of trial and error. The two preliminaries of Shor's algorithm are the phase estimation and the related order-finding algorithms.