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Insights and Limitations of Shor’s Factorization Algorithm on IBM Real Quantum Computers

Camps Carmona, Adriano José, Manresa, Gerard De, Villarreal D’Angelo, Juan Manuel

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

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.

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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.
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 stands as a breakthrough in quantum computing due to its potential to factor large integers exponentially quicker than classical algorithms. However, implementing and evaluating this algorithm on real quantum computer hardware remains exciting due to qubit limitations, gate noise, and hardware constraints. This research presents a comprehensive performance evaluation of Shor’s algorithm using simulated quantum backends provided by Qiskit. A flexible and generic implementation is proposed, allowing dynamic input of integers to be factored, with randomized co-prime selection and automated circuit generation. The algorithm is tested on various semiprime numbers, such as 15, 21, and 35, using IBM’s Aer simulator. A major contribution of this work is the circuit-level analysis conducted both before and after transpilation. Metrics such as gate counts, circuit depth, and simulator runtime are extracted to assess scalability and resource requirements. High-resolution plots of the pre-transpiled circuits are saved to visualize algorithmic complexity, while post-transpilation metrics inform future quantum hardware feasibility. The output measurement distributions are analyzed to estimate periodicity and derive correct factors. The proposed implementation is compared with existing fixed-instance Shor demonstrations to highlight its flexibility and extensibility. Experimental results show consistent success in factor retrieval and provide valuable insight into circuit growth and complexity under realistic constraints. This analysis lays the groundwork for future adaptation to NISQ hardware and contributes to understanding Shor’s algorithm from both computational and architectural perspectives.
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.
“Shor's Algorithm: Factoring and Cryptanalysis” focuses on Shor's Algorithm, particularly its application to factoring and cryptanalysis. It begins with an overview of Shor's algorithm, detailing the classical extraction of periods from measured values and the quantum gates required for the Quantum Fourier Transform (QFT). Further, this chapter explores the applications and limitations of Shor's algorithm, including its impact on RSA encryption. A literature review follows, discussing Shor's algorithm in the context of quantum computation, its effectiveness on RSA, and the effects of imperfections on its factorization process. The section also includes an experimental study of Shor's algorithm using IBM Q, followed by a practical implementation of factoring numbers N=21 and N=35. Further, it covers resource estimation, quantum simulations, and different types of simulators in IBM Qiskit, including state vector, QASM, Matrix Product State (MPS), and stabilizer simulators, providing insights into their use cases and the limits of classical simulation.
Quantum algorithms for integer factorization represent one of the most prominent applications of quantum computation, with far-reaching implications for modern cryptography. While Shor's algorithm provides a polynomial-time solution in the ideal quantum model, its practical implementation is severely constrained by the limitations of current noisy intermediate-scale quantum (NISQ) hardware. These constraints have motivated the exploration of alternative factoring algorithms with different structural and resource trade-offs. In this work, we present an experimental study of Regev's quantum factoring algorithm, implemented on real quantum hardware, and compare its behavior with that of Shor's algorithm under analogous conditions. Focusing on the case N = 15, we execute both algorithms on the QMIO quantum computer at the Centro de Supercomputacion de Galicia (CESGA) and contrast the results with one of IBM's open-access quantum computers and ideal simulations. This parallel execution enables a low-level comparison of the two algorithms, highlighting how their respective quantum implementations interact with hardware noise, limited circuit depth, and finite sampling. Our analysis emphasizes the different ways in which Shor's and Regev's algorithms encode arithmetic structure into quantum states through Fourier sampling in one and higher dimensions, respectively, and how these differences manifest in experimental outcomes. Although neither algorithm demonstrates a practical advantage in the small N regime, the results provide insight into their relative robustness and failure modes on contemporary quantum devices. This study illustrates the value of experimental benchmarking of alternative quantum factoring algorithms as a means of understanding the practical implications of algorithmic design choices in the NISQ era.
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.