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Communications in computer and information science

Solving Prime Factorization Using Quantum Ising Model

Wen-Li Wang, Mei-Huei Tang, Shahid Hussain, Kevin Wang

Published Jan 1, 2026• 1 citation
Publisher
DOI
10.1007/978-3-032-08977-9_37

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The security of the RSA cryptosystem is based on the difficulty of factoring a large number N into prime numbers [Formula: see text] and [Formula: see text] satisfying [Formula: see text]. This paper presents a prime factorization method using a D-Wave quantum computer that could threaten the RSA cryptosystem in the future. The starting point for this method is very simple, representing two prime numbers as qubits. Then, we set the difference between the product of the two prime numbers expressed in qubits and N as a cost function, and we find the solution when the cost function is minimized. D-Wave's quantum annealer can find the minimum value of any quadratic problem. However, the cost function must be a higher-order unconstrained optimization (HUBO) model because it contains second- or higher-order terms. We used a hybrid solver accessible via Leap, D-Wave's real-time quantum cloud service, and the dimod package provided by the D-Wave Ocean software development kit (SDK) to solve the HUBO problem. We also successfully factorized 102,454,763 with 26 logical qubits. In addition, we factorized 1,000,070,001,221 using the range-dependent Hamiltonian algorithm.
We have developed a framework to convert an arbitrary integer factorization problem to an executable Ising model by first writing it as an optimization function then transforming the k-bit coupling (k ≥ 3) terms to quadratic terms using ancillary variables. Our resource-efficient method uses [Formula: see text] binary variables (qubits) for finding the factors of an integer N. We present how to factorize 15, 143, 59989, and 376289 using 4, 12, 59, and 94 logical qubits, respectively. This method was tested using the D-Wave 2000Q for finding an embedding and determining the prime factors for a given composite number. The method is general and could be used to factor larger integers as the number of available qubits increases, or combined with other ad hoc methods to achieve better performances for specific numbers.