Shor's algorithm proved a significant milestone in quantum computing since it promises an exponential speedup over conventional algorithms for integer factorization, a problem critical to modern cryptography systems. This work covers implementation steps of Shor’s algorithm and analyzes Quantum Fourier Transform (QFT) usage for extracting periodicity from quantum superpositions. Firstly, theoretical foundations of algorithm are illustrated concentrating on QFT construction and operation within the setting of the quantum circuit. Thereafter, each implementation step such as qubit optimization, modular exponentiation, and circuit design is discussed. Our findings validate the theoretical effectiveness of the QFT in solving the period-finding subroutine, while also highlighting the practical difficulties and scaling constraints presented by existing quantum hardware. The paper ends with a summary of possible advancements and future paths for error mitigation strategies and quantum algorithm design.
Shor's algorithm proved a significant milestone in quantum computing since it promises an exponential speedup over conventional algorithms for integer factorization, a problem critical to modern cryptography systems. This work covers implementation steps of Shor’s algorithm and analyzes Quantum Fourier Transform (QFT) usage for extracting periodicity from quantum superpositions. Firstly, theoretical foundations of algorithm are illustrated concentrating on QFT construction and operation within the setting of the quantum circuit. Thereafter, each implementation step such as qubit optimization, modular exponentiation, and circuit design is discussed. Our findings validate the theoretical effectiveness of the QFT in solving the period-finding subroutine, while also highlighting the practical difficulties and scaling constraints presented by existing quantum hardware. The paper ends with a summary of possible advancements and future paths for error mitigation strategies and quantum algorithm design.
Quantum Shor’s algorithm offers an exponential speedup over classical methods for integer factorization, a central problem to modern cryptography, and the implica- tions of Shor’s algorithm in factoring large numbers. The algorithm leverages on quantum parallelism to compute the period of a modular exponentiation function, employ- ing the Inverse Quantum Fourier Transform (IQFT) to extract periodicity, and classical post-processing to derive the factors. This final report of the Quantum Engineer- ing post-graduate course of the Fundacio Polit ́ enica de ` Catalunya, presents the main findings encountered in the implementation and benchmarking of Schor’s algorithm in real IBM quantum computers. To accomplish this, we created a command line tool made in Python, that allows the integration of the circuits implementing the Shor’s algorithm with the overall set of configuration parameters needed to perform the benchmark. In this report, we first provide an overview of the quantum factoring problem, followed by a detailed description of Shor’s algorithm, and its implementation using the Qiskit Python library. The circuit depth and noise effects are analyzed across different backend types, also different values for the optimization level and approximation degrees are configured when performing the circuit transpilation. As mentioned earlier different backend types were utilized for performance comparison, namely backend class with or without a noise model (AerSim), a fake backend class provided by the Qiskit library (fake provider), and the real quantum hardware (IBM Quantum Processing Units in January- June 2025).The study follows by analyzing the different error mitigation techniques: dynamic decoupling, Pauli twirling, the SABRE Optimization (Stochastic Algorithm for Quantum Boolean Optimization and Routing), and any combination of them. Once the optimum configuration is found, the maximum number that can be reasonably factorized with success is explored. Finally, the main conclusions, findings, and possible future research lines are presented, including the factorization of a N ≈ 1000 if appropriate parameters of the algorithm are selected.
Factoring large numbers poses a significant chal-lenge in contemporary cryptography and security practices. Shor's algorithm presents a remarkable advancement over clas- sical methods by providing exponential acceleration in factoring large numbers, presenting a possible risk to traditional cryptography methods. The algorithm relies on the Quantum Fourier Transform (QFT) and modular exponentiation principles to efficiently ascertain the period (x) of a given large number (N). Upon determining the period using classical methods, the prime factors of the large number can be easily computed. To eval- uate the algorithm's efficiency and scalability, both simulation- based experiments and theoretical analyses are conducted. The algorithm is simulated utilizing the Qiskit back end to estimate resources, including Clifford gates, T gates, T depth, Full depth, and qubits, across a range of large numbers.
This chapter analyzes the applicability of Fourier transformations to determine the periodic structure of finite-length sequences. The classical factorization problem is reformulated to solve it on modern quantum computers with 16 or more qubits. The chapter presents Shor's algorithm for factorization of natural numbers. Performing two basic quantum operations is required to implement the described Shor algorithm, namely exponentiation and quantum discrete Fourier transformation. The chapter estimates the amount of quantum resources needed to solve some asymmetric cryptographic problems using derivatives of the Shor algorithm, with different parameters of these problems, and compares them with the complexity of solving the iterative problem when searching for the key of a symmetric cryptosystem. Using the example of solving the problem of finding the exponent (multiplicative order) of a simple integer modulo by another simple integer, the chapter shows the main features of modeling and engineering implementation of the quantum Shor algorithm.
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.
A fundamental problem in factorization by Shor's algorithm [1] is the implementation of the modular exponentiation function. The period of this function is important for determining the individual factors, so its accurate implementation is for successful factorization critical.The imprecision of current quantum computers requires that any quantum circuits should be as simple as possible. Both in terms of the number of qubits used, the number of quantum gates and the depth of the quantum circuit. Therefore, it is necessary to search for an optimal variant for the implementation of the above-mentioned modular exponentiation function.The standard method is to implement individual arithmetic functions [2] which can be combined to achieve the desired modular exponentiation function. This process, however, creates a very complex quantum circuit, so that its measured result (probability distribution) will be so skewed that it will not be possible to reliably obtain the period of sought.Another way is to create a special quantum circuit for one particular modular exponentiation function [3]. Such a circuit is of course simpler than the circuit created by the above universal implementation, but in practice it only makes sense for testing the functionality of the surrounding quantum circuit for period detection.A very interesting possibility for the implementation of arithmetic functions is to use the direct and inverse quantum Fourier transform [4]. For a quantum computer, these transformations are very easy operations, and this interesting feature can then be used to advantage in these various implementations.This contribution will explore these possible approaches and show their advantages and disadvantages through concrete examples.
The article explores the fundamental role of quantum period determination algorithms, particularly in the context of cryptography. The work focuses on quantum algorithms that exploit periodicity, such as Shor's algorithm, which is central to efficient integer factorization. It highlights the challenges quantum algorithms face when applied to non-Abelian groups such as the Suzuki, Hermitage, and Rie groups, which exhibit complex periodic structures that are difficult to solve with existing quantum methods. The study delves into the structure and properties of these groups, explaining the complexity of their representations and the challenges posed by the quantum Fourier transform (QFT) in these cases. This contrasts the relative ease with which Abelian groups can be handled by quantum algorithms with the exponential complexity encountered with non-Abelian groups. The study provides a comparative analysis of the computational complexity between classical and quantum approaches to find the period in different types of groups, highlighting that while quantum algorithms offer exponential speedup for Abelian cases, non-Abelian structures remain a frontier for further research. The conclusion calls for continued research in quantum representation theory and cryptanalysis, especially for non-Abelian groups where current quantum methods have not yet provided efficient solutions. The problem of finding the period has been identified as critical to the advancement of both quantum computing and cryptographic applications.