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 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.
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.
With the advancement of quantum technologies, there is a potential threat to traditional encryption systems based on integer factorization. Therefore, developing techniques for accurately measuring the performance of associated quantum algorithms is crucial, as it can provide insights into their practical feasibility from the current perspective. In this chapter, we aim to analyze the time required for integer factorization tasks using Shor's algorithm within a gate-based quantum circuit simulator of the matrix product state type. Additionally, we observe the impact of parameter pre-selection in Shor's algorithm. Specifically, this pre-selection is expected to increase the success rate of integer factorization by reducing the number of iterations and facilitating performance measurement under fixed conditions, thus enabling scalable performance evaluation even on real quantum hardware.
With the advancement of quantum technologies, there is a potential threat to traditional encryption systems based on integer factorization. Therefore, developing techniques for accurately measuring the performance of associated quantum algorithms is crucial, as it can provide insights into the practical feasibility from the current perspective. In this chapter, we aim to analyze the time required for integer factorization tasks using Shor's algorithm within a gate-based quantum circuit simulator of the matrix product state type. Additionally, we observe the impact of parameter pre-selection in Shor's algorithm. Specifically, this pre-selection is expected to increase the success rate of integer factorization by reducing the number of iterations and facilitating performance measurement under fixed conditions, thus enabling scalable performance evaluation even on real quantum hardware.
Quantum circuit simulation is crucial for algorithm development and cryptanalysis given the limited qubit counts and high noise in today’s quantum hardware. We present a Qiskit MPI framework that accelerates classical simulation of quantum circuits focusing on Shor’s integer factoring algorithm by distributing workloads across high performance computing (HPC) resources. A novel dynamic shot allocation strategy balances simulation tasks across MPI processes, preventing idle CPUs even when the total number of circuit "shots" (independent runs) is not evenly divisible. This load balancing approach, combined with support for hybrid CPU/GPU execution on each node, achieves near linear speedups (e.g. 7.5× on 8 processes and 14× on 16 processes) in simulating Shor’s algorithm circuits. We validate the correctness of our parallel simulator by showing that distributed execution produces identical quantum output distributions as a serial run, thus maintaining fidelity. All experiments are performed with fixed random seeds and provided in open source form for reproducibility. We also discuss compatibility with the emerging Quantum Message Passing Interface (QMPI), which extends MPI to quantum computing contexts. By enabling fast, scalable simulation of cryptographically relevant algorithms like Shor’s on classical clusters, our framework allows researchers to prototype quantum attacks and assess quantum algorithm performance ahead of large scale quantum hardware. We report extensive experiments, including factoring small composite numbers, that demonstrate significant wall clock speedups without loss of accuracy. These results position our approach as a practical tool for scalable quantum algorithm prototyping and cryptanalysis research in the pre-quantum era.
Quantum computing promises revolutionary advances in cryptography, particularly through Shor's algorithm, which can factor large integers in polynomial time and threaten traditional RSA encryption. Through simulation experiments, our research compares Shor's algorithm implementation in ideal versus noisy quantum environments. Under ideal conditions, we factored semiprimes up to N=299 with reasonable resources, but with noise introduction, this ceiling dropped to approximately N=95, beyond which computational costs increased exponentially. The performance gap is striking: ideal simulations factored N=91 in 4 seconds, while noisy simulations required 2,600 seconds—a 650-fold increase. Our hybrid quantum-classical approach showed that noise simulation required modified circuit architectures, reducing gate count growth from 50 to 42.5 gates for comparable test numbers. Qubit requirements followed a step-function pattern in both environments, scaling from 12 qubits for N=15 to 21 qubits for N=91-95, while circuit depth increased from 11 to 17-19 units, with slightly shallower circuits in noisy conditions. In our paper, we propose a hybrid implementation strategy that addresses hardware limitations, present empirical findings on the impact of quantum noise on factoring efficiency, and evaluate the cryptographic security implications under realistic conditions. Our findings underscore the need for fault-tolerant quantum computing and error correction, especially as optimizations suggest factoring 2048-bit RSA keys would require approximately 20 million qubits running for 8 hours. This highlights the urgency in transitioning to post-quantum cryptographic algorithms like CRYSTALS-Dilithium and CRYSTALS-Kyber being standardized by NIST.
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.