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Shor's Algorithm

Richard J. Lipton, Kenneth W. Regan

Published Jan 1, 2014• 1 citation
OpenAlex

Abstract

This chapter contains sections titled: 11.1 Strategy, 11.2 Good Numbers, 11.3 Quantum Part of the Algorithm, 11.4 Analysis of the Quantum Part, 11.5 Probability of a Good Number, 11.6 Using a Good Number, 11.7 Continued Fractions, 11.8 Problems, 11.9 Summary and Notes

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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.
Shor’s algorithm 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 Shor’s algorithm. Similarly, we provide a detailed computation of the estimation of the probability that convergents will provide the period required for determining a prime factor.
Shors algorithm is a quantum algorithm that finds with high probability the order of an element x∈ZN∗x \in Z_{N}^{*}x∈ZN∗​. One of its applications is the construction of an algorithm that finds the factors of N. In the initial chapters we approach necessary tools for the comprehension of Shors algorithm such as: modular arithmetic, algorithms, continued fractions, basic concepts of quantum computing and Fourier quantum transform. In the following chapters we present Shors algorithm an its application in factorization.