Quantum Factorization Algorithm Research Articles

Digitized adiabatic quantum factorization

Quantum integer factorization is a potential quantum computing solution that may revolutionize cryptography. Nevertheless, a scalable and efficient quantum algorithm for noisy intermediate-scale quantum computers looks far-fetched. We propose an alternative factorization method, within the digitized-adiabatic quantum computing paradigm, by digitizing an adiabatic quantum factorization algorithm enhanced by shortcuts to adiabaticity techniques. We find that this fast factorization algorithm is suitable for available gate-based quantum computers. We test our quantum algorithm in an IBM quantum computer with up to six qubits, surpassing the performance of the more commonly used factorization algorithms on the long way towards quantum advantage.

Speed improvement of the quantum factorization algorithm of P. Shor by upgrade its classical part

This report discusses Shor’s quantum factorization algorithm and ρ–Pollard’s factorization algorithm. Shor’s quantum factorization algorithm consists of classical and quantum parts. In the classical part, it is proposed to use Euclidean algorithm, to find the greatest common divisor (GCD), but now exist large number of modern algorithms for finding GCD. Results of calculations of 8 algorithms were considered, among which algorithm with lowest execution rate of task was identified, which allowed the quantum algorithm as whole to work faster, which in turn provides greater potential for practical application of Shor’s quantum algorithm. Standard quantum Shor’s algorithm was upgraded by replacing the binary algorithm with iterative shift algorithm, canceling random number generation operation, using additive chain algorithm for raising to power. Both Shor’s algorithms (standard and upgraded) are distinguished by their high performance, which proves much faster and insignificant increase in time in implementation of data processing. In addition, it was possible to modernize Shor’s quantum algorithm in such way that its efficiency turned out to be higher than standard algorithm because classical part received an improvement, which allows an increase in speed by 12%.

Integer Factorization – Cryptology Meets Number Theory

Integer factorization is one of the oldest mathematical problems. Initially, the interest in factorization was motivated by curiosity about be­haviour of prime numbers, which are the basic building blocks of all other integers. Early factorization algorithms were not very efficient. However, this dramatically has changed after the invention of the well-known RSA public-key cryptosystem. The reason for this was simple. Finding an efficient fac­toring algorithm is equivalent to breaking RSA. The work overviews development of integer factoring algorithms. It starts from the classical sieve of Eratosthenes, covers the Fermat algorithm and explains the quadratic sieve, which is a good representative of modern fac­toring algorithms. The progress in factoring is illustrated by examples of RSA challenge moduli, which have been factorized by groups of mathemati­cians and cryptographers. Shor's quantum factorization algorithm with poly­nomial complexity is described and the impact on public-key encryption is discussed.

Fast quantum modular exponentiation architecture for Shor's factoring alogrithm

We present a novel and efficient, in terms of circuit depth, design for Shor's quantum factorization algorithm. The circuit effectively utilizes a diverse set of adders based on the Quantum Fourier transform (QFT) Draper's adders to build more complex arithmetic blocks: quantum multiplier/accumulators by constants and quantum dividers by constants. These arithmetic blocks are effectively architected into a quantum modular multiplier which is the fundamental block for the modular exponentiation circuit, the most computational intensive part of Shor's algorithm. The proposed modular exponentiation circuit has a depth of about $2000n^2$ and requires $9n+2$ qubits, where $n$ is the number of bits of the classic number to be factored. The total quantum cost of the proposed design is $1600n^3$. The circuit depth can be further decreased by more than three times if the approximate QFT implementation of each adder unit is exploited.

T-bit semiclassical quantum Fourier transform

Because of the difficulty of building a high-dimensional quantum register, this paper presents an implementation of the high-dimensional quantum Fourier transform (QFT) based on a low-dimensional quantum register. First, we define the t-bit semiclassical quantum Fourier transform. In terms of probability amplitude, we prove that the transform can realize quantum Fourier transformation, illustrate that the requirement for the two-qubit gate reduces obviously, and further design a quantum circuit of the transform. Combining the classical fixed-window method and the implementation of Shor’s quantum factorization algorithm, we then redesign a circuit for Shor’s algorithm, whose required computation resource is approximately equal to that of Parker’s. The requirement for elementary quantum gates for Parker’s algorithm is O(⌈ log N⌉3, and the quantum register for our circuit requires t−1 more dimensions than Parker’s. However, our circuit is t 2 times as fast as Parker’s, where t is the width of the window.

Analysis of Quantum Algorithm to Find the Solution of Integer factoriza-tion Problem

Quantum computing has emerged as an important interdisciplinary field, merging theories in mathematics, physics and computer science. So far, a significant portion of research in quantum computing has focused on the design of quantum algorithms. Quantum Com-puters requires very different algorithms for factorization, which plays an important role in the field of quantum computing and shor’s algo-rithm is used to address integer factorization problem. Shor's algorithm is important because it breaks a widely used public-key cryptography scheme known as RSA. This paper discusses the refinement of quantum factorization algorithm. Keywords: - quantum algorithm, factorization, gcd, prime decomposition, shor’s algorithm.

Numerical Estimation of Efficiency of the Algorithms Assisting Quantum Factorization

One of the most significant achievements of quantum information processing is the construction of the probabilistic algorithm providing factoring of numbers in polynomial time. However, properties of the classic algorithms assisting an operation of the quantum device have not been well investigated and rough estimates of their effectiveness have been put forth. This paper presents results of the numerical simulation of the quantum factorization algorithm for composite numbers, which have been formed as a product of exactly two factors. It was shown that average factorization efficiency significantly exceeds the commonly used bound. The presented results also provided strong evidence that there exists an internal structure in probability distribution of successful factorization.

Group IV solid state proposals for quantum computation

The discovery of the quantum factorization algorithm more than a decade ago triggeredintense interest in exploring possible physical realizations of quantum computers. Amongthe many solid state proposals, electron and nuclear spins in Si and group IV relatedmaterials have long coherence times and the capability of state preparation, gating andread-out using electric, magnetic and optical fields. Proposals involving silicon seek to takeadvantage of an existing mature technology and the implicit promise of scalability fromsolid state materials. Nevertheless, building such quantum systems depends in many caseson the development of fabrication techniques with nearly atomic precision. Managingdecoherence, initialization and read-out in any quantum computer remains a daunting task.In this review we summarize proposals and recent developments relevant to the possiblerealization of a quantum computer constructed out of Si (or group IV) materials.

Quantum factorization algorithm by NMR ensemble computers

In this paper, we design our quantum factorization algorithm by using NMR ensemble computers. Our algorithm can find the prime factor p of the RSA-modulus N = pq by using NMR ensemble search scheme, where p < N , p and q are primes. We prepare massive molecules with the same states and use an auxiliary spin to record the result of computing. In each computation, we input states with one fixed spin and determine one bit value of the prime factor p. After ⌈ln p⌉computations, we can discover the factor p. Thus, our algorithm requires at most ln N oracle queries for factoring the RSA-modulus N.

Quantum computers and dissipation

We analyse dissipation in quantum computation and its destructive impact on efficiency of quantum algorithms. Using a general model of decoherence, we study the time evolution of a quantum register of arbitrary length coupled with an environment of arbitrary coherence length. We discuss relations between decoherence and computational complexity and show that the quantum factorization algorithm must be modified in order to be regarded as efficient and realistic.