Factorization of a large semiprime is a hard problem on classical computers, and quantum annealing is one way to solve the prime factorization problem by converting to a combinatorial optimization problem. A conversion method using the block-wise sum of partial products has been proposed but that works well up to numbers with 21-bit. To enhance the input bit width of the prime factorization, the paper introduces three proposals for prime factorization based on quantum annealing. First one is to divide the original problem into several sub-problems and solve them using multiple annealings. Second and third ones are analytical variable reduction of LSB and MSB parts. The proposals have been implemented on Python for Fixstars Amplify Annealing Engine (AE) and can factorize numbers with 46-bit or less. The proposals can also factorize numbers up to 57-bit but with a lot of trials for each number.
Factorization of a large semiprime is a hard problem on classical computers, and quantum annealing is one way to solve the prime factorization problem by converting to a combinatorial optimization problem. A conversion method using the block-wise sum of partial products has been proposed but that works well up to numbers with 21-bit. To enhance the input bit width of the prime factorization, the paper introduces three proposals for prime factorization based on quantum annealing. First one is to divide the original problem into several sub-problems and solve them using multiple annealings. Second and third ones are analytical variable reduction of LSB and MSB parts. The proposals have been implemented on Python for Fixstars Amplify Annealing Engine (AE) and can factorize numbers with 46-bit or less. The proposals can also factorize numbers up to 57-bit but with a lot of trials for each number.
Factorization of large semiprimes remains one of the most challenging problems for classical computers. Shor’s algorithm offers a quantum approach that reduces computational complexity, but its practical application is currently limited by hardware constraints. Meanwhile, as a provisional approach, quantum annealing (QA) has been explored through formulations of the quadratic unconstrained binary optimization (QUBO) problem. Among existing methods, the blockwise partial-product approach effectively reduced the QUBO variable count but was limited to semiprimes up to 21 bits. To extend factorization to larger semiprimes, this paper addresses key engineering challenges in constructing efficient QUBO formulations for prime factorization. We propose five techniques to reduce variable counts and improve scalability with current QA hardware: (1) dividing the problem into subproblems with partial constraints; (2) applying analytical reductions near the LSB; (3) applying analytical reductions near the MSB; (4) exploiting special patterns in semiprimes, with odd bit widths and long MSB-side zero sequences; and (5) balancing variable usage across both sides of the subproblem. Integrated into a QUBO converter, these methods enable stable factorization of semiprimes up to 47-bits within 20 seconds and can extend to special 2049-bit instances with 1001 consecutive MSB-side zeros.
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.
The RSA cryptosystem, which relies on the computational difficulty of prime factorization, faces growing challenges with the advancement of quantum computing. In this study, we propose a quantum annealing based approach to integer factorization using both high order unconstrained binary optimization (HUBO) and constrained quadratic model (CQM) formulations. We begin by modeling binary multiplication with explicit carry propagation, translating this into a HUBO representation and subsequently reducing it to a quadratic unconstrained binary optimization form compatible with current quantum solvers. To address scalability limitations, we implement a CQM approach with constraint relaxation and global product consistency. While the HUBO model successfully factors small semiprimes, it exhibits exponential memory growth, making it impractical for inputs larger than 10 bits. In contrast, the CQM model achieves accurate factorization of semiprimes up to 60 bits including N = 1152921423002469787 demonstrating significantly improved scalability. Experimental results further show that applying global product constraints enhances factorization accuracy and consistency across all tested instances. This work highlights both the promise and current limitations of quantum-assisted factorization and establishes a foundation for evaluating RSA security in the emerging quantum era.
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.
Abstract Probabilistic computing has been introduced to operate functional networks using a probabilistic bit (p-bit), broadening the computational abilities in non-deterministic polynomial searching operations. However, previous developments have focused on emulating the operation of quantum computers similarly, implementing every p-bit with large weight-sum matrix multiplication blocks and requiring tens of times more p-bits than semiprime bits. In addition, operations based on a conventional simulated annealing scheme required a large number of sampling operations, which deteriorated the performance of the Ising machines. Here we introduce a prime factorization machine with a virtually connected Boltzmann machine and probabilistic annealing method, which are designed to reduce the hardware complexity and number of sampling operations. From 10-bit to 64-bit prime factorizations were performed, and the machine offers up to 1.2 × 108 times improvement in the number of sampling operations compared with previous factorization machines, with a 22-fold smaller hardware resource.
Factoring large integers using a quantum computer is an outstanding research problem that can illustrate true quantum advantage over classical computers. Exponential time order is required in order to find the prime factors of an integer by means of classical computation. However, the order can be drastically reduced by converting the factorization problem to an optimization one and solving it using a quantum computer. Recent works involving both theoretical and experimental approaches use Shor's algorithm, adiabatic quantum computation and quantum annealing principles to factorize integers. However, our work makes use of the generalized Grover's algorithm as proposed by Liu, with an optimal version of classical algorithm/analytic algebra. We utilize the phase-matching property of the above algorithm for only amplitude amplification purposes to avoid an inherent phase factor that prevents perfect implementation of the algorithm. Here we experimentally demonstrate the factorization of two bi-primes, 4088459 and 966887 using IBM's 5- and 16-qubit quantum processors, hence making those the largest numbers that have been factorized on a quantum device. Using the above 5-qubit processor, we also realize the factorization of a tri-prime integer 175, which had not been achieved to date. We observe good agreement between experimental and theoretical results with high fidelities. The difficulty of the factorization experiments has been analyzed and it has been concluded that the solution to this problem depends on the level of simplification chosen, not the size of the number factored. In principle, our results can be extended to factorize any multi-prime integer with minimum quantum resources.