Introduction. Modern information security systems use methods of asymmetric cryptography to transfer encryption keys, which are based on the high computational complexity of factorization of large numbers. Quantum computers (QCs) theoretically make it possible to accelerate the solution of the problem of factorization of numbers in comparison with classical computers and pose a potential threat to information security systems. However, real QCs have a limited number of connections between them and problems with preserving a stable low temperature, which reduces the probability of detecting a global minimum. The joint use of QCs with classical computers based on hybrid cloud services is advisable when the search for the optimal solution by direct methods is a complex problem both in the theoretical sense and in the sense of the required amount of calculations for tasks with specific data. The article proposes a method for improving the accuracy of solving the factorization problem based on multiple minimum search by the method of hardware reverse quantum annealing with a variation of its parameters. The results of numerical experiments for two different QC processors and a hybrid quantum-classical computer by D-Wave are presented, it is shown that the maximum number that can be factorized exclusively by direct annealing is 143, and with a combination of direct and reverse annealing 255. The purpose. Examination of the influence of the parameters of quantum annealing and the corresponding solutions for the adiabatic CC, developed by D-Wave, on the quality of the solution of the factorization problem. To give recommendations for improving the accuracy of solving the factorization problem and increasing the statistical frequency of the appearance of correct pairs of multipliers. Results. Numerical experiments have shown that for the problem of factorization of numbers, the successive application of direct and reverse annealing makes it possible to improve the probability of obtaining the correct pair of multipliers and to more than double the statistical frequency of its occurrence. Quantum annealing modes: pause and quenching reduce the probability of obtaining the correct solution and worsen the statistical frequency of the appearance of correct pairs of multipliers. Conclusions. The use of direct and reverse annealing makes it possible to increase the probability of obtaining the correct solution of the factorization problem for the adiabatic QC of D-Wave. Increasing the calculation time of the problem is justified, since it allows increasing the probability of a correct solution. The use of hybrid quantum-classical computing and cloud services allows factorization for numbers with a bit depth of up to twenty-two bits. Keywords: quantum annealing, factorization of natural numbers, asymmetric shifts, hardening, reverse annealing, combinatorial optimization.
Introduction. Modern information security systems use methods of asymmetric cryptography to transfer encryption keys, which are based on the high computational complexity of factorization of large numbers. Quantum computers (QCs) theoretically make it possible to accelerate the solution of the problem of factorization of numbers in comparison with classical computers and pose a potential threat to information security systems. However, real QCs have a limited number of connections between them and problems with preserving a stable low temperature, which reduces the probability of detecting a global minimum. The joint use of QCs with classical computers based on hybrid cloud services is advisable when the search for the optimal solution by direct methods is a complex problem both in the theoretical sense and in the sense of the required amount of calculations for tasks with specific data. The article proposes a method for improving the accuracy of solving the factorization problem based on multiple minimum search by the method of hardware reverse quantum annealing with a variation of its parameters. The results of numerical experiments for two different QC processors and a hybrid quantum-classical computer by D-Wave are presented, it is shown that the maximum number that can be factorized exclusively by direct annealing is 143, and with a combination of direct and reverse annealing 255. The purpose. Examination of the influence of the parameters of quantum annealing and the corresponding solutions for the adiabatic CC, developed by D-Wave, on the quality of the solution of the factorization problem. To give recommendations for improving the accuracy of solving the factorization problem and increasing the statistical frequency of the appearance of correct pairs of multipliers. Results. Numerical experiments have shown that for the problem of factorization of numbers, the successive application of direct and reverse annealing makes it possible to improve the probability of obtaining the correct pair of multipliers and to more than double the statistical frequency of its occurrence. Quantum annealing modes: pause and quenching reduce the probability of obtaining the correct solution and worsen the statistical frequency of the appearance of correct pairs of multipliers. Conclusions. The use of direct and reverse annealing makes it possible to increase the probability of obtaining the correct solution of the factorization problem for the adiabatic QC of D-Wave. Increasing the calculation time of the problem is justified, since it allows increasing the probability of a correct solution. The use of hybrid quantum-classical computing and cloud services allows factorization for numbers with a bit depth of up to twenty-two bits. Keywords: quantum annealing, factorization of natural numbers, asymmetric shifts, hardening, reverse annealing, combinatorial optimization.
Quantum computing has emerged as a promising computational paradigm with the potential to solve problems that are intractable for classical computers. This paper provides an overview of the current state of quantum computing, including its historical development, hardware advancements, applications, and challenges. Special attention is given to combinatorial optimization, which is considered one of the most promising application domains. The limitations of current quantum systems and future research directions are also discussed. This paper presents a comparative study between a conventional D-Wave-based simulated annealing approach and a proposed hybrid optimization model for solving largescale combinatorial optimization problems. The study focuses on benchmark problems, including AES, TSP gr24, and a real-world load balancing problem, all formulated as QUBO models. The D-Wave approach employs the dwave-neal simulated annealing engine with various parameter settings, while the proposed model introduces a decomposition-based framework combined with adaptive parameter control and iterative refinement. The experimental results demonstrate that the proposed model consistently outperforms the conventional approach in terms of solution quality, achieving significantly lower energy values across all problem instances. In particular, for the TSP gr24 problem, the proposed model achieves near-optimal solutions with substantially reduced computation time. For large-scale problems such as AES and load balancing, the proposed model maintains scalability and robustness, whereas the conventional approach suffers from increased computational cost and performance degradation. Although the proposed method introduces additional computational steps, its hybrid structure effectively mitigates limitations related to local minima, parameter sensitivity, and scalability. These findings indicate that integrating classical optimization techniques with quantum-inspired methods is essential for practical applications. The proposed model provides a promising framework for solving complex real-world optimization problems and highlights the importance of hybrid approaches in advancing quantum annealing technologies.
Quantum annealing is a promising heuristic for combinatorial optimization, but on current hardware its performance degrades for larger and more complex problems due to noise and small energy gaps. Reverse annealing has been proposed as a refinement strategy, yet it remains unclear when it provides systematic advantages over standard forward annealing or simply increasing annealing time. We find that combining forward and reverse annealing consistently improves solution quality and efficiency across multiple problem classes. The benefits of reverse annealing increase with problem complexity and are strongest in regimes where forward annealing is increasingly limited. Moreover, reverse annealing yields larger efficiency gains than simply extending forward annealing times. We establish these results through a systematic experimental study on a D-Wave Advantage system, benchmarking reverse annealing across Max-Cut, Number Partitioning, and sparse clustering problems while varying reverse distance, pause duration, and annealing time. We identify a narrow optimal regime for reverse annealing parameters linked to the location of freeze-out points and energy-level crossings in the annealing schedule. These findings demonstrate that reverse annealing is most valuable for large, high-complexity optimization problems and is likely to gain importance as quantum annealing hardware scales toward more realistic applications.
Introduction. Quantum computers provide several times faster solutions to several NP-hard combinatorial optimization problems in comparison with computing clusters. The trend of doubling the number of qubits of quantum computers every year suggests the existence of an analog of Moore's law for quantum computers, which means that soon they will also be able to get a significant acceleration of solving many applied large-scale problems. The purpose of the article is to review methods for creating algorithms of quantum computer mathematics for combinatorial optimization problems and to analyze the influence of the qubit-to-qubit coupling and connections strength on the performance of quantum data processing. Results. The article offers approaches to the classification of algorithms for solving these problems from the perspective of quantum computer mathematics. It is shown that the number and strength of connections between qubits affect the dimensionality of problems solved by algorithms of quantum computer mathematics. It is proposed to consider two approaches to calculating combinatorial optimization problems on quantum computers: universal, using quantum gates, and specialized, based on a parameterization of physical processes. Examples of constructing a half-adder for two qubits of an IBM quantum processor and an example of solving the problem of finding the maximum independent set for the IBM and D-wave quantum computers are given. Conclusions. Today, quantum computers are available online through cloud services for research and commercial use. At present, quantum processors do not have enough qubits to replace semiconductor computers in universal computing. The search for a solution to a combinatorial optimization problem is performed by achieving the minimum energy of the system of coupled qubits, on which the task is mapped, and the data are the initial conditions. Approaches to solving combinatorial optimization problems on quantum computers are considered and the results of solving the problem of finding the maximum independent set on the IBM and D-wave quantum computers are given. Keywords: quantum computer, quantum computer mathematics, qubit, maximal independent set for a graph.
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.
Introduction. The hope of solving the problem of the avalanche-like growth of requirements for computing power, essential for solving complex routing problems and other problems of combinatorial optimization, relies on the latest quantum computers, in the development of which governments and corporations invest multi-billion investments. The article examines modern routing algorithms and performs their analysis and verification, if the authors of the algorithm provided appropriate test programs. The purpose of the article is to review the current state of development in the field of development of routing algorithms for hybrid quantum-classical clouds, analyze them and propose a classification of algorithms. Results. Modern quantum computers (QCs) make it possible to find approximate solutions to some of mathematical problems faster than classical computers. The inaccuracy of the solutions obtained by the QC is a consequence of physical and technological limitations: calculation errors are caused by thermal noise, a small number of computational elements - qubits and connections between them, which requires the decomposition of the problem and the use of heuristic algorithms. The analysis of approaches to the solution of optimization problems on QC allows us to single out: quantum response and variational search of eigenvalues based on quantum logic gates as the general directions of development of the vast majority of algorithms for solving routing problems. The considered algorithms reduce the vehicle routing problem to a quadratic unconstrained binary optimization problem, which is isomorphic to the Hamilton-Ising model. In this form, the problem is suitable for embedding in QC, which finds an approximate solution that has the best statistical reliability or corresponds to the quantum state with the lowest energy. As a separate class, vehicle routing algorithms for classical computers that use quantum computing to accelerate problem solving can be distinguished. For example, neural networks that calculate weighting factors using QC or an ant algorithm that calculates a pheromone trail in a hybrid cloud. It should be mentioned the quantum-inspired algorithms, which are based on software tools for the simulation of QC and the corresponding libraries and allow creating an effective class of algorithms for solving problems of vehicle routing. Conclusions. Combining hardware quantum annealing with a number of software tools for calculating optimization problems for classical computers in a hybrid quantum-classical cloud service allows to obtain advantages in speed and accuracy of some types of complex optimization problems of a commercial scale, in particular, routing vehicles, which is already bringing substantial profits to a number of corporations. Keywords: vehicle routing problem, quantum computer, annealing, combinatorial optimization, traveling salesman problem, clustering, qubit.
Quantum computers use quantum resources to carry out computational tasks and may outperform classical computers in solving certain computational problems. Special-purpose quantum computers such as quantum annealers employ quantum adiabatic theorem to solve combinatorial optimization problems. In this paper, we compare classical annealings such as simulated annealing and quantum annealings that are done by the D-Wave machines both theoretically and numerically. We show that if the classical and quantum annealing are characterized by equivalent Ising models, then solving an optimization problem, i.e., finding the minimal energy of each Ising model, by the two annealing procedures, are mathematically identical. For quantum annealing, we also derive the probability lower-bound on successfully solving an optimization problem by measuring the system at the end of the annealing procedure. Moreover, we present the Markov chain Monte Carlo (MCMC) method to realize quantum annealing by classical computers and investigate its statistical properties. In the numerical section, we discuss the discrepancies between the MCMC based annealing approaches and the quantum annealing approach in solving optimization problems.