Shor’s integer factoring algorithm served as a significant breakthrough in the field of quantum computing owing to its potential applications in modern cryptography. Simulating quantum algorithms in a way that offers a close imitation of quantum computers is required to provide some insight viably. This paper proposes an FPGA implementation of the Shor’s algorithm by employing High Level Synthesis (HLS). A detailed method of implementation including design of data-flow graphs, pruning matrices using Python scripts, writing a HLS compatible C++ code, and subsequent hardware synthesis is presented for an 8-qubit system. An estimated maximum frequency of 180 MHz has been achieved in this design.
Shor’s integer factoring algorithm served as a significant breakthrough in the field of quantum computing owing to its potential applications in modern cryptography. Simulating quantum algorithms in a way that offers a close imitation of quantum computers is required to provide some insight viably. This paper proposes an FPGA implementation of the Shor’s algorithm by employing High Level Synthesis (HLS). A detailed method of implementation including design of data-flow graphs, pruning matrices using Python scripts, writing a HLS compatible C++ code, and subsequent hardware synthesis is presented for an 8-qubit system. An estimated maximum frequency of 180 MHz has been achieved in this design.
This article presents a detailed implementation of Shor’s quantum factoring algorithm, which served as the cornerstone for a pilot project at Jamk University of Applied Sciences. The primary goal was to leverage this significant algorithm to establish and validate a reusable quantum simulation environment on a high-performance computing (HPC) platform. We report the successful factorization of the integer 15, executed on the Finnish CSC’s MAHTI supercomputer using a 9-qubit quantum circuit developed in the Qiskit framework. This work provides a comprehensive overview of the principles behind Shor’s algorithm and a methodical breakdown of its simulation. Furthermore, it documents the practical software challenges faced during the environment’s configuration, focusing on library versioning and dependencies. The purpose of this work is to provide a systematic introduction to implement Shor’s algorithm at computer science or applied mathematics classes in institutions of higher education such as universities of applied sciences. The successful factorization provides an initial validation of the simulation workflow for this benchmark case, while the detailed account of implementing Shor’s algorithm offers a practical guide and valuable insights for future quantum computing initiatives at educational institutions with a focus on computer science or scientific computing.
Prime number factorization is a problem in computer science where the solution to that problem takes super-polynomial time classically. Shor’s quantum factoring algorithm is able to solve the problem in polynomial time by harnessing the power of quantum computing. The implementation of the quantum algorithm itself is not detailed by Shor in his paper. In this paper, an approach and experiment to implement Shor’s quantum factoring algorithm are proposed. The implementation is done using Python and a quantum computer simulator from ProjectQ. The testing and evaluation are completed in two computers with different hardware specifications. User time of the implementation is measured in comparison with other quantum computer simulators: ProjectQ and Quantum Computing Playground. This comparison was done to show the performance of Shor’s algorithm when simulated using different hardware. There is a 33% improvement in the execution time (user time) between the two computers with the accuracy of prime factorization in this implementation is inversely proportional to the number of qubits used. Further improvements upon the program that has been developed for this paper is its accuracy in terms of finding the factors of a number and the number of qubits used, as previously mentioned.
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.
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.
In recent years, advancements in quantum chip technology, such as Willow, have contributed to reducing quantum computation error rates, potentially accelerating the practical adoption of quantum computing. As a result, the design of quantum algorithms suitable for real-world applications has become a crucial research direction. This study focuses on the implementation of Shor’s algorithm, aiming to improve modular computation efficiency and demonstrate the factorization of a 4096-bit integer under specific constraints. Experimental results, when compared with state-of-the-art (SOTA) methods, indicate a significant improvement in efficiency—by an order of magnitude—while enabling the factorization of longer integers.
Quantum computing is inherently abstract and often counterintuitive from the perspective of classical physics. For learners to develop a deeper understanding, it is useful to study a practical example thoroughly, from its mathematical formalism to quantum circuit implementation. Shor's algorithm for period finding, which can then be used for prime factorization, is demonstrated in this paper due to its practical significance. While numerous publications on Shor's algorithm exist, they tend to focus on advanced theoretical analysis, specific implementation details, or narrowly defined research aspects; consequently, they are often not well-suited for educational purposes for beginning and intermediate learners. This paper presents a rigorous step-by-step walkthrough of Shor's algorithm, utilizing the conventional approach, and discusses the construction of the quantum oracle with illustrative numerical examples. This is followed by classical simulations and implementations on actual quantum hardware. The aim is to allow learners with minimal mathematical background to follow and understand the algorithm. Through this process, the paper also attempts to interpret and connect the key underlying quantum computing concepts of superposition, entanglement, interference, quantum oracle, and phase kick-back that are often assumed to be trivial and omitted in research papers, yet are particularly challenging for novice learners to internalize. In the numerical examples, reduced-qubit cases are included. By contrasting the results from the reduced-qubit configurations with the standard ones, learners can better appreciate the key enablers in Shor's algorithm and how the accuracy decreases with a smaller number of qubits. Moreover, reduced-qubit examples (e.g., 5 qubits) are small enough to be implemented in a hardware emulator, such as a low-cost and low-capacity Field Programmable Gate Array (FPGA), which can be used in experimental hardware classes for students to understand how to use an FPGA to simulate quantum circuits.