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.
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.
Integer factorization, a fundamental problem in computational mathematics, holds critical significance for modern cryptography, particularly in RSA encryption. Traditional approaches such as the number field sieve face exponential complexity limitations, while Shor’s quantum algorithm remains impractical due to hardware constraints. This study proposes a universal algorithm for integer factorization based on Ising machines by transforming the problem into a Quadratic Unconstrained Binary Optimization (QUBO) formulation. The algorithm introduces an optimal reduction formula to optimize the parameter ranges of local field coefficients ( h ) and coupling coefficients ( J ) in the Ising model. Additionally, a non-uniform column grouping method is employed to resolve the conflict between coefficient ranges and carry auxiliary quantum bits(qubits), minimizing the number of auxiliary qubits with minimal compromise on coefficient ranges. Using this approach, we successfully factorized the 22-bit integer 2,093,809 with only 118 qubits. Extrapolating to existing photonic Ising machines with 100,000 qubits, our method demonstrates the potential to factorize 631-bit integers, highlighting its promise for efficient large-scale integer factorization. All results presented in this paper are obtained from simulations on Fixstars Amplify and D-Wave simulators.
The difficulty in factoring large integers is a security basis of Rivest-Shamir-Adleman public key cryptography. Quantum annealing, a completely different method as compared with that of Shor’s algorithm, can transform an integer factorization problem into a combinatorial optimization problem using the quantum tunneling effects of D-wave quantum annealing to leap from local minima. In this paper, a new distributed integer factorization algorithm based on quantum annealing, which transforms an arbitrary integer into a stable Ising model executable using a D-wave quantum computer, is proposed. The stability and range of the local field coefficient h and the coupling term coefficient J of the Ising model are important factors that affect the success rate of integer factorization. Compared with the algorithm of Purdue University’s Jiang et al., the proposed method reduces the number of logical bits and the parameters of h and J by more than 60% and 40%, respectively, and the ranges of the coefficients of the Ising model are stable. Compared with the algorithm of Lockheed Martin Warren, the parameters of the proposed method are reduced in the orders from 106 to 102 of magnitude when the Ising model is guaranteed to be stable. In addition, to prove the correctness of the algorithm, Warren traversed factorization within 1000 integers. The proposed method traverses all integers within 10000 and all of them are successfully factored. The experimental results of this paper exceed the results of Shor’s algorithm, Jiang’s algorithm and the maximum published factorization scale of Lockheed Martin Warren’s algorithm.
This paper provides a new (second) way, which is completely different from Shor's algorithm, to show the optimistic potential of a D-Wave quantum computer for deciphering RSA and successfully factoring all integers within 10000. Our method significantly reduced the local field coefficient [Formula: see text] and coupling term coefficient [Formula: see text] by more than 33% and 26%, respectively, of those of Ising model, which can further improve the stability of qubit chains and improve the upper bound of integer factorization. In addition, our results obtained the best index (20-bit integer (1028171)) of quantum computing for deciphering RSA via the quantum computing software environment provided by D-Wave. Furthermore, Shor's algorithm requires approximately 40 qubits to factor the integer 1028171, which is far beyond the capacity of universal quantum computers. Thus, post quantum cryptography should further consider the potential of the D-Wave quantum computer for deciphering the RSA cryptosystem in future.
Objective: to study the transformational potential of quantum annealing in solving the problem of simple factorization. Research method(s): the approach includes a comprehensive review of recent experimental breakthroughs and theoretical innovations. In particular, we analyze techniques such as surface code memory, HUBO and QUBO formulations, range-dependent Hamiltonian algorithms, modular locally structured embedding methods, and a modified multiplication table method random number. Research Output(s): the study evaluates the application of quantum annealing to factorization of primes, showing that advanced problem mapping techniques, such as the HUBO and QUBO formulations, significantly improve the efficiency of representing complex factorization problems on quantum hardware. Notably, the inclusion of surface code memory increases the stability of qubit states during annealing, reducing errors and improving computational accuracy It also demonstrates that Hamiltonian's range-dependent algorithms and modular, locally structured embedding methods help optimize qubit interaction, enabling a more accurate factorization process. A modified multiplication table method is presented, providing an optimized computational strategy, especially effective for large composite numbers. Preliminary experiments with random numbers confirm the theoretical conclusions, indicating that these integrated methods allow for better performance than traditional approaches. Taken together, the results highlight the potential of quantum annealing as a solid foundation for solving complex cryptographic problems and lay the foundation for future research into scalable quantum algorithms and hardware implementations. Scientific novelty: the work combines several advanced quantum annealing techniques for factorization of prime numbers, combining experimental innovations with theoretical developments to propose a new framework that improves the efficiency of cryptographic computing.
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.
This paper investigates novel techniques to solve prime factorization by quantum annealing (QA). First, we present a very-compact modular encoding of a multiplier circuit into the architecture of current D-Wave QA devices. The key contribution is a compact encoding of a controlled full-adder into an 8-qubit module in the Pegasus topology, which we synthesized using Optimization Modulo Theories. This allows us to encode up to a 21 × 12-bit multiplier (and a 22 × 8-bit one) into the Pegasus 5760-qubit topology of current annealers. To the best of our knowledge, these are the largest factorization problems ever encoded into a quantum annealer. Second, we investigated the problem of actually solving encoded PF problems by running an extensive experimental evaluation on a D-Wave Advantage 4.1 quantum annealer. In the experiments we introduced different approaches to initialize the multiplier qubits and adopted several performance enhancement techniques. Overall, 8,219,999 = 32,749 × 251 was the highest prime product we were able to factorize within the limits of our QPU resources. To the best of our knowledge, this is the largest number which was ever factorized by means of a quantum annealer; also, this is the largest number which was ever factorized by means of any quantum device without relying on external search or preprocessing procedures run on classical computers.