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2n N using barcode integrated for none control to generate, create none image in none applications. Microsoft Office Word Website xk ,. Shor s factorization algorithm This chapter descri none for none bes what is generally considered to be one of the most important and historical contributions to the eld 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 dif culty. With the previous chapters and tools readily assimilated, and some patience in going through the different preliminary steps required, such a dif culty 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- nding algorithms. Both represent the purely quantum part of the approach: it cannot be implemented classically. Basically, phase estimation allows one to nd the periodicity r of a modular function by means of a multi-qubit quantum-gate circuit (Hadamard, controlled-U gates, inverse-Fourier transform), followed by a probabilistic, quantum-mechanical measurement of the resulting qubit state, which yields a phase estimate .

Order nding, from which the period r is determined with high probability and (the measurement being successful) without ambiguity, represents a particular case of quantum phase estimation. Such a determination eventually rests on the implementation of continued fraction expansion, a classical algorithm that is straightforward to run with a computer. The requirement for r to be the period is that the phase estimation = s/r , with s an integer, is such that s, r are co-prime.

Since this does not happen systematically, there is a nite chance that the endeavor may fail. Any such event is not a failure of Shor s algorithm, but rather a call for another try in this speci c implementation step. Such conditions of trial and error leading to factorizing success may sound strange to engineers but they are really embedded in the algorithm game! The key feature to grasp is that the.

Shor s factorization algorithm probability of such none for none intermediate failures remains comparatively small, or innocuous for computing logistics, and that the chances of success, after only a few trials in the worst of all cases, are relatively high. As we shall see, all of the above steps are of polynomial-time complexity. A comparison is made with nonpolynomial algorithms, such as the general number eld sieve (GNFS) approach, which is shown to require decades of CPU computing time to factorize numbers of 100-bits long! We then establish the connection between order nding and factorization of composite (or nonprime) numbers by using two basic theorems.

The rst theorem yields the two factors N , N of any given composite N such that N = N N , given the knowledge of the period. The second theorem establishes that the probability of the period meeting certain eligibility criteria is at least 75% for any composite. These two theorems combined validate and conclude Shor s factorization algorithm.

The factorization of the composite N = 15 = 3 5 is found in textbooks as the only illustrative example of Shor s algorithm. Here, we shall investigate the whole space of nontrivial composites N 100, as an emulation of the quantum computer. It is possible to do this based on the fact that for such relatively small numbers, we can compute (with a basic home computer) all the possibilities associated with each step of Shor s algorithm, namely what the period- nding quantum circuit should yield, and the associated probabilities of success or failure.

The result of this investigation is an original plot showing the probability of successfully concluding the factorization of nontrivial composites N 100 in a single run. The exercise helps one to grasp how Shor s algorithm would work when taking greater composite numbers. The last section, which brie y describes public key cryptography (PKC), is not completely out of place in this chapter.

The purpose of this addition is to show how the PKC algorithm works, as based on the product N = pq of two prime numbers p, q, whose factorization is indeed considered intractable by classical means. Should a quantum computer of corresponding computing power be implemented someday, the whole eld of PKC-based cryptography, and Internet security for that matter, would be compromised overnight! Fortunately, this remains a distant perspective, while from this chapter, we know that the theory works mathematically..

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