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arXiv:1505.00642

Simulating Factorization with a Quantum Computer

José Luis Rosales

Published May 4, 2015• 0 citations
arXivHTMLPDF

Abstract

Modern cryptography is largely based on complexity assumptions, for example, the ubiquitous RSA is based on the supposed complexity of the prime factorization problem. Thus, it is of fundamental importance to understand how a quantum computer would eventually weaken these algorithms. In this paper, one follows Feynman's prescription for a computer to simulate the physics corresponding to the algorithm of factoring a large number $N$ into primes. Using Dirac-Jordan transformation theory one translates factorization into the language of quantum hermitical operators, acting on the vectors of the Hilbert space. This leads to obtaining the ensemble of factorization of $N$ in terms of the Euler function $φ(N)$, that is quantized. On the other hand, considering $N$ as a parameter of the computer, a Quantum Mechanical Prime Counting Function $π_{QM}(x)$, where $x$ factorizes $N$, is derived. This function converges to $π(x)$ when $N\gg x$. It has no counterpart in analytic number theory and its derivation relies on semiclassical quantization alone.

DOI
10.48550/arxiv.1505.00642

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Modern cryptography is largely based on complexity assumptions, for example, the ubiquitous RSA is based on the supposed complexity of the prime factorization problem. Thus, it is of fundamental importance to understand how a quantum computer would eventually weaken these algorithms. In this paper, one follows Feynman's prescription for a computer to simulate the physics corresponding to the algorithm of factoring a large number NNN into primes. Using Dirac-Jordan transformation theory one translates factorization into the language of quantum hermitical operators, acting on the vectors of the Hilbert space. This leads to obtaining the ensemble of factorization of NNN in terms of the Euler function φ(N)φ(N)φ(N), that is quantized. On the other hand, considering NNN as a parameter of the computer, a Quantum Mechanical Prime Counting Function πQM(x)π_{QM}(x)πQM​(x), where xxx factorizes NNN, is derived. This function converges to π(x)π(x)π(x) when N≫xN\gg xN≫x. It has no counterpart in analytic number theory and its derivation relies on semiclassical quantization alone.
Feynman's prescription for a quantum simulator was to find a Hamitonian for a system that could serve as a computer. The Pólya-Hilbert conjecture proposed the demonstration of Riemann's hypothesis through the spectral decomposition of Hermitian operators. Here we study the problem of decomposing a number into its prime factors, N=xy, using such a simulator. First, we derive the Hamiltonian of the physical system that simulates a new arithmetic function formulated for the factorization problem that represents the energy of the computer. This function rests alone on the primes below sqrt[N]. We exactly solve the spectrum of the quantum system without resorting to any external ad hoc conditions, also showing that it obtains, for x≪sqrt[N], a prediction of the prime counting function that is almost identical to Riemann's R(x) function. It has no counterpart in analytic number theory, and its derivation is a consequence of the quantum theory of the simulator alone.
Many cryptographic algorithms depend on computational complexity assumptions. Notorious cases are the RSA algorithm for public key criptography or the Diffie-Hellman key exchange protocol, to publicly agree on a common secret key. Both algorithms are known to be broken by quantum computing as well as those that can be reduced to a discrete logarithm problem. These are key algorithms in our digital society and are at the basis of everyday tasks, specially those that rely on digital signatures. The RSA algorithm, in particular, is probably the most used algorithm and is its assumed security the one that guards most of the e-commerce nowadays. In this case, it is the time complexity of finding the prime factors of a large number, that grows worse than polinomially with the size of the number, the main guardian of our cyberinfrastructure. The fact that a quantum computer can solve this problem in polynomial time using Shor's algorithm is seen as a potentially major disruption and has prompted security agencies to advice the progressive deprecation of these algorithms.
The arithmetic problem of factoring an integer NNN can be translated into the physics of a quantum device, a result that supports P\'olya's and Hilbert's conjecture to demonstrate Riemann's hypothesis. The energies of this system, being univocally related to the factors of NNN, are the eigenvalues of a bounded Hamiltonian. Here we solve the quantum conditions and show that the histogram of the discrete energies, provided by the spectrum of the system, should be interpreted in number theory as the relative probability for a prime to be a factor candidate of NNN. This is equivalent to a quantum sieve that is shown to require only o(lnN)3o{(ln\sqrt{N})}^{3}o(lnN​)3 energy measurements to solve the problem, recovering Shor's complexity result. Hence the outcome can be seen as a probability map that a pair of primes solve the given factorization problem. Furthermore, we show that a possible embodiment of this quantum simulator corresponds to two entangled particles in a Penning trap. The possibility to build the simulator experimentally is studied in detail. The results show that factoring numbers, many orders of magnitude larger than those computed with experimentally available quantum computers, is achievable using typical parameters in Penning traps.
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.
The security of messages encoded via the widely used RSA public key encryption system rests on the enormous computational effort required to find the prime factors of a large number N using classical (i.e., conventional) computers. In 1994, however, Peter Shor showed that for sufficiently large N a quantum computer would be expected to perform the factoring with much less computational effort. This paper endeavors to explain, in a fashion comprehensible to the non-expert readers of this journal: (i) the RSA encryption protocol; (ii) the various quantum computer manipulations constituting the Shor algorithm; (iii) how the Shor algorithm performs the factoring; and (iv) the precise sense in which a quantum computer employing Shor's algorithm can be said to accomplish the factoring of very large numbers with less computational effort than a classical computer can. It is made apparent that factoring NNN generally requires many successive runs of the algorithm. The careful analysis herein reveals, however, that the probability of achieving a successful factorization on a single run is about twice as large as commonly quoted in the literature.
One of the most well known mathematically hard problems in number theory is the integer factorization problem, roughly stated that decomposition of a composite number into its prime factors. In modern cryptography, RSA encryption algorithm whose security is based on integer factorization problem is highly practical, widespread and up to date no classical algorithm having polynomial running time for the factorization of large numbers is known. In 1994, Peter Shor proposed an efficient algorithm on quantum computer. In this paper, we  mention about the fundamentals of  Shor's quantum algorithm illustrating a concrete example.