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Variational Quantum Computation Integer Factorization Algorithm

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Abstract

The integer factorization problem is a major challenge in the field of computer science, and Shor’s algorithm provides a promising solution for this problem. However, Shor’s algorithm involves complex modular exponentiation computation, which leads to the construction of complicated quantum circuits. Moreover, the precision of continued fraction computations in Shor’s algorithm is influenced by the number of qubits, making it difficult to implement the algorithm on Noisy Intermediate-Scale Quantum (NISQ) computers. To address these issues, this paper proposes variational quantum computation integer factorization (VQCIF) algorithm based on variational quantum algorithm (VQA). Inspired by classical computing, this algorithm utilizes the parallelism of quantum computing to calculate the product of parameterized quantum states. Subsequently, the quantum multi-control gate is used to map the product satisfying \(pq=N\) onto an auxiliary qubit. Then the variational quantum circuit is adjusted by the optimizer, and it is possible to obtain a prime factor of the integer N with a high probability. While maintaining generality, VQCIF has a simple quantum circuit structure and requires only \(2n+1\) qubits. Furthermore, the time complexity is exponentially accelerated. VQCIF algorithm is implemented using the Qiskit framework, and tests are conducted on factorization instances to demonstrate its feasibility.

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The datasets used or analyzed during the current study are available from the corresponding author upon reasonable request.

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Funding

This work is supported by the Beijing Natural Science Foundation No.4212015.

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Authors and Affiliations

  1. Faculty of Information Technology, Beijing University of Technology, Chaoyang District, Beijing, 100124, China

    Xinglan Zhang & Feng Zhang

  2. Beijing Key Laboratory of Trusted Computing, Chaoyang District, Beijing, 100124, China

    Xinglan Zhang & Feng Zhang

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Contributions

All authors contributed to the study conception and design. Dr. Xinglan Zhang provided guidance for the experimental work and manuscript writing, as well as offered assistance in securing funding. Experimental preparation and analysis were performed by Feng Zhang. Xinglan Zhang and Feng Zhang wrote the main manuscript text. All authors read and approved the final manuscript.

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Correspondence to Feng Zhang.

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Zhang, X., Zhang, F. Variational Quantum Computation Integer Factorization Algorithm. Int J Theor Phys 62, 245 (2023). https://doi.org/10.1007/s10773-023-05473-y

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