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Implementation and Analysis of Scalability of Quantum Algorithms Based on Circuits and Subroutines

María Luisa Rodríguez Castillo

Published Jan 21, 2026• 0 citations
OpenAlex

Abstract

Quantum computing has given rise to significant challenges that make its wide availability difficult, especially when it comes to the accurate calculation of execution costs of Quantum Runtime Services, which in turn complicates the ability of a programmer to take well-informed decisions given an expectation of both cost and success rates. Quantum algorithms are intertwined in a complex web of dependencies, in which modern algorithms commonly incorporate other algorithms as subroutines. This paper studies whether the characterization of success metrics in scaling scenarios of specific quantum subroutines can be extrapolated to the quantum algorithms that integrate them as subroutines. It focuses on Shor’s algorithm, and the Quantum Fourier Transform algorithm as a subroutine, analyzing possible correlations in different scaling scenarios created by modifying the input parameters ’N’ and ’a’, running three jobs with different noise models for each one. The results showed a clear correlation that closely resembles a 2:1 ratio for the circuit where N = 15. For N = 15 circuits, success rates varied significantly, only reflecting ratios close to 2:1 when the period was equal to 2, an ideal, small and even number that reduced the required operations of the modular exponentiation stage. Larger periods such as r = 6 showed lesser success rates due to the necessity of performing more operations during the modular exponentiation stage and the noise accumulation it entails. Results achieved in high noise conditions had similar Shor and QFT success rates, evidencing how the dispersion of shots can alter the expected results. There is a meaningful connection between the success rates of an algorithm and its subroutine, which can be shadowed or decreased when error rates and qubit counts are increased.

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Quantum computing has given rise to significant challenges that make its wide availability difficult, especially when it comes to the accurate calculation of execution costs of Quantum Runtime Services, which in turn complicates the ability of a programmer to take well-informed decisions given an expectation of both cost and success rates. Quantum algorithms are intertwined in a complex web of dependencies, in which modern algorithms commonly incorporate other algorithms as subroutines. This paper studies whether the characterization of success metrics in scaling scenarios of specific quantum subroutines can be extrapolated to the quantum algorithms that integrate them as subroutines. It focuses on Shor’s algorithm, and the Quantum Fourier Transform algorithm as a subroutine, analyzing possible correlations in different scaling scenarios created by modifying the input parameters ’N’ and ’a’, running three jobs with different noise models for each one. The results showed a clear correlation that closely resembles a 2:1 ratio for the circuit where N = 15. For N = 15 circuits, success rates varied significantly, only reflecting ratios close to 2:1 when the period was equal to 2, an ideal, small and even number that reduced the required operations of the modular exponentiation stage. Larger periods such as r = 6 showed lesser success rates due to the necessity of performing more operations during the modular exponentiation stage and the noise accumulation it entails. Results achieved in high noise conditions had similar Shor and QFT success rates, evidencing how the dispersion of shots can alter the expected results. There is a meaningful connection between the success rates of an algorithm and its subroutine, which can be shadowed or decreased when error rates and qubit counts are increased.
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.
Quantum algorithms face significant challenges due to qubit susceptibility to environmental noise, and quantum error correction typically requires prohibitive resource overhead. This paper proposes that quantum algorithms may possess inherent noise resilience characteristics that could reduce implementation barriers. We investigate Shor's algorithm by applying circuit-level noise models directly to the original algorithm circuit. Our findings reveal that Shor's algorithm demonstrates superior fault tolerance under Z noise compared to X and Y noise. Focusing on the modular exponentiation circuit which is the core component of the algorithm, we conduct fault-tolerant position statistics on circuits with bit lengths from 4 to 9. The results show that under Z noise, fault-tolerant positions grow with the same quartic polynomial order as potential error positions as the problem scale increases. In contrast, fault tolerance under X and Y noise exhibits a strong dependence on the composite number N and the parameter a. Based on these findings, we develop an extrapolation method predicting that the minimum probability of a correct output of the modular exponentiation circuit to factor 2048 bit integers under biased noise is approximately 1.417*{10}^{-17}.
The publication in 1994 of Shor's algorithm, which allows factorisation of composite number N in a time polynomial in its binary length L has been the primary catalyst for the race to construct a functional quantum computer. However, it seems clear that any practical system that may be developed will not be able to perform completely error free quantum gate operations or shield even idle qubits from inevitable error effects. Hence, the practicality of quantum algorithms needs to be investigated to not only determine limitations on such algorithms in a noisy quantum computer, but also to estimate what demands must be made of quantum error correction (QEC). Shor's algorithm is a combination of both classical pre and post-processing, and also a quantum period finding subroutine (QPF) which allows for the exponential speed up of this algorithm on a quantum device. This paper will look at the stability of this quantum subroutine under the effects of several error models. Direct simulation of the entire QPF subroutine required to factorise a given composite number N in the presence of errors shows that the circuit required to implement Shor's algorithm is very sensitive to a small number of errors within the entire calculation. Well designed and efficient error correction codes, quick gate times and very low gate error rates will be essential for any physical realisation of Shor's factoring algorithm.
Quantum computation is becoming an increasingly interesting field, especially with the rise of real quantum computers. However, current quantum processors contain a few tens of error-prone qubits and the realization of large-scale quantum computers is still very challenging. Therefore, quantum computer simulators are particularly suitable for testing and analysing quantum algorithms without having a real quantum computer at one's disposal. In this thesis, different quantum algorithms such as Grover's and Shor's algorithm as well as key quantum routines such as the Quantum Fourier Transform (QFT) and a quantum adder/subtractor are described and analysed (optimal number of iterations, time complexity). Some of them have been implemented for an arbitrary number of qubits and have been simulated using two different quantum simulators, the QX simulator developed at QuTech and the Liquid simulator from Microsoft. In addition, how errors affect the success rate of the algorithms has been investigated.
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.
The Quantum Fourier Transform (QFT) grants competitive advantages, especially in resource usage and circuit approximation, for performing arithmetic operations on quantum computers, and offers a potential route towards a numerical quantum-computational paradigm. In this paper, we utilize efficient techniques to implement QFT-based integer addition and multiplications. These operations are fundamental to various quantum applications including Shor's algorithm, weighted sum optimization problems in data processing and machine learning, and quantum algorithms requiring inner products. We carry out performance evaluations of these implementations based on IBM's superconducting qubit architecture using different compatible noise models. We isolate the sensitivity of the component quantum circuits on both one-/two-qubit gate error rates, and the number of the arithmetic operands' superposed integer states. We analyze performance, and identify the most effective approximation depths for quantum add and quantum multiply within the given context. We observe significant dependency of the optimal approximation depth on the degree of machine noise and the number of superposed states in certain performance regimes. Finally, we elaborate on the algorithmic challenges - relevant to signed, unsigned, modular and non-modular versions - that could also be applied to current implementations of QFT-based subtraction, division, exponentiation, and their potential tensor extensions. We analyze performance trends in our results and speculate on possible future development within this computational paradigm.