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Quantum Algorithm for Prime Factorization with High Probability

Zhong Pu-Cha

Published Jan 1, 2011• 1 citation
OpenAlex

Abstract

In this paper,based on the quantum Fourier transform,we give a new quantum algorithm for prime factorization.The algorithm turns r to be phase factor under repeated application of Fourier transform and variate transform,where r is the order of a selective element in the ring of integers modulo N.Furthermore the amplitude of the non-target states except zero state is modified to be 0.Our algorithm′s success probability,which is more than 3/4 and higher than Shor′s algorithm,doesn′t depend on the size of r other than Shor′s algorithm.Meanwhile,we present a comparison of the required resource between the new algorithm and Shor′s algorithm.

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Wuhan University Journal of Natural Sciences•Jan 9, 2017•4 citations
In this paper,based on the quantum Fourier transform,we give a new quantum algorithm for prime factorization.The algorithm turns r to be phase factor under repeated application of Fourier transform and variate transform,where r is the order of a selective element in the ring of integers modulo N.Furthermore the amplitude of the non-target states except zero state is modified to be 0.Our algorithm′s success probability,which is more than 3/4 and higher than Shor′s algorithm,doesn′t depend on the size of r other than Shor′s algorithm.Meanwhile,we present a comparison of the required resource between the new algorithm and Shor′s algorithm.
The intent of this thesis is to elucidate the quantum computing algorithm developed by Peter Shor called Shor's algorithm. We will provide a detailed description and simulation of the algorithm using MATLAB. Precursory information regarding quantum phenomena such as superposition, entanglement, and Dirac notation, will be described in great detail so that the reader may have a better understanding of the operations in Shor's algorithm. Quantum computers store and transport information quite differently than their classical counterparts. We will provide a quick overview of these differences to high- light the benefit of utilizing quantum phenomena in a computer in order to create massive parallel computations. Thus, reducing the complexity time for classical algorithms used to solve problems such as the prime factorization problem and the period finding problem. The Quantum Fourier Transform is a principle component in Shor's algorithm. We will explicitly define the Quantum Fourier Transform and show that it is a unitary transformation. We will also show how the Quantum Fourier Transform, as well as another unitary transformation called the Hadamard transform, functions in Shor's algorithm. One of the initial parameters in Shor's algorithm is to select a random variable. We will examine the erratic effects of this random variable as well as how it effects the probability of us successfully reducing an integer into a product of two primes. We will provide a thorough analysis of the randomness in Shor's algorithm. We will also show how measuring the state of our quantum system as well as selecting a suitable random variable impacts finding the period of the Quantum Fourier Transform which in turn will either give a high or low probability of obtaining a factor of some integer.
Abstract Expanding on the factoring and periodicity in Chapter 14, the Shor algorithm is explained in detail, with a warning that this is one of the toughest chapters in the book. While the steps of Shor algorithm may look just like the RSA in reverse, the quantum computer is needed to find the periodicity of the factors and thereby the prime factors themselves. Fourier analysis is the tool for this. The quantum Fourier transform is described, including the quantum circuit, after first showing what the classical Fourier transform accomplishes. Extensive examples of all the steps are given.
We consider a version of Shor's quantum factoring algorithm such that the quantum Fourier transform is replaced by an extremely simple one where decomposition coefficients take only the values of 1,i,−1,−i1,i,-1,-i1,i,−1,−i. In numerous calculations which have been carried out so far, our algorithm has been surprisingly stable and never failed. There are numerical indications that the probability of period finding given by the algorithm is a slowly decreasing function of the number to be factorized and is typically less than in Shor's algorithm. On the other hand, quantum computer (QC), capable of implementing our algorithm, will require a much less amount of resources and will be much less error-sensitive than standard QC. We also propose a modification of Coppersmith' Approximate Fast Fourier Transform. The numerical results show that the probability is signifacantly amplified even in the first post integral approximation. Our algorithm can be very useful at early stages of development of quantum computer.
Quantum Fourier transform (QFT) plays an eminent role in quantum computation.It creates a superposition of different quantum states, allowing simultaneous calculation, which would take many steps were the same program implemented on a classical computer.The computational speed of a quantum computer is thus boosted dramatically.In this article, we explain the application of QFT in Shor's algorithm, which was proposed by Peter Shor to factor large integers on quantum computers.Specifically, the principle and design of quantum Fourier transform are explained.We stress the subtle distinction between QFT and its inverse.Since former articles did not emphasize it, we hope it could be a supplement to former articles.Our next work is to verify the existing articles on executing Shor's algorithm by conducting two experiments on IBMQ of factoring N = 15 in two ways (a = 7 and a = 11).We find that the effect of quantum entanglement might be crucial to the speed boost of factoring large integers in Shor's algorithm.
The Shor quantum factorization algorithm allows the factorization or large integers in logarithmic squared time whereas classical algorithms require an exponential time increase with the bit length of the number to be factored. The hardware implementation of the Shor algorithm would thus allow the factorization of the very large integers employed by commercial encryption methods. We propose some modifications of the algorithm by employing some simplification to the stage employing the quantum Fourier transform. The quantum Hadamard transform may be used to replace the quantum Fourier transform in certain cases. This would reduce the hardware complexity of implementation since phase rotation gates with only two states of 0 and π would be required.