Continued Fractions and Probability Estimations in the Shor Algorithm -- A Detailed and Self-Contained Treatise

TL;DR

This paper provides a comprehensive, detailed explanation of the classical continued fraction analysis and probability estimation steps in Shor's quantum algorithm for prime factorization, filling gaps in existing literature.

Contribution

It offers an in-depth, self-contained presentation of continued fractions and probability estimations specifically tailored for understanding Shor's algorithm.

Findings

Detailed proofs of continued fraction theory relevant to Shor's algorithm

Explicit computation of probability estimates for period detection

Enhanced clarity for complete comprehension of the classical part of Shor's algorithm

Abstract

The algorithm of Shor for prime factorization is a hybrid algorithm consisting of a quantum part and a classical part. The main focus of the classical part is a continued fraction analysis. The presentation of this is often short, pointing to text books on number theory. In this contribution, we present the relevant results and proofs from the theory of continued fractions in detail (even in more detail than in text books) filling the gap to allow a complete comprehension of the algorithm of Shor. Similarly, we provide a detailed computation of the estimation of the probability that convergents will provide the period required for determining a prime factor.