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

Quantum Shor’s algorithm offers an exponential speedup over classical methods for integer factorization, a central problem to modern cryptography, and the implications of Shor’s algorithm in factoring large numbers. The algorithm leverages on quantum parallelism to compute the period of a modular exponentiation function, employing the Inverse Quantum Fourier Transform (IQFT) to extract periodicity, and classical post-processing to derive the factors. This final report of the Quantum Engineering
post-graduate course of the Fundació Politènica 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.