Publication Date

7-22-2026

Document Type

Article

Publication Title

IEEE Access

Volume

14

DOI

10.1109/ACCESS.2026.3716184

First Page

112622

Last Page

112638

Abstract

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.

Funding Number

2125906

Funding Sponsor

National Science Foundation

Keywords

inverse quantum Fourier transform (IQFT), quantum education, quantum Fourier transform (QFT), quantum phase estimation (QPE), Shor s algorithm

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.

Department

Physics and Astronomy; Electrical Engineering

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