This document details the methodology and steps taken to convert Higher Order Unconstrained Binary Optimization (HUBO) models into Quadratic Unconstrained Binary Optimization (QUBO) models. The focus is primarily on prime factorization problems; a critical and computationally intensive task relevant in various domains including cryptography, optimization, and number theory. The conversion from Higher-Order Binary Optimization (HUBO) to Quadratic Unconstrained Binary Optimization (QUBO) models is crucial for harnessing the capabilities of advanced computing methodologies, particularly quantum computing and DYNEX neuromorphic computing. Quantum computing offers potential exponential speedups for specific problems through its intrinsic parallelism capabilities. Conversely, DYNEX neuromorphic computing enhances efficiency and accelerates the resolution of intricate, pattern-oriented tasks by simulating memristors in GPUs, employing a highly decentralized approach, via Blockchain technology. This transformation enables the exploitation of these cutting-edge computing paradigms to address complex optimization challenges effectively. Through detailed explanations, mathematical formulations, and algorithmic strategies, this document aims to provide a comprehensive guide to understanding and implementing the conversion process from HUBO to QUBO. It underscores the importance of such transformations in making prime factorization computationally feasible on both existing classical computers and emerging computing technologies.
This document details the methodology and steps taken to convert Higher Order Unconstrained Binary Optimization (HUBO) models into Quadratic Unconstrained Binary Optimization (QUBO) models. The focus is primarily on prime factorization problems; a critical and computationally intensive task relevant in various domains including cryptography, optimization, and number theory. The conversion from Higher-Order Binary Optimization (HUBO) to Quadratic Unconstrained Binary Optimization (QUBO) models is crucial for harnessing the capabilities of advanced computing methodologies, particularly quantum computing and DYNEX neuromorphic computing. Quantum computing offers potential exponential speedups for specific problems through its intrinsic parallelism capabilities. Conversely, DYNEX neuromorphic computing enhances efficiency and accelerates the resolution of intricate, pattern-oriented tasks by simulating memristors in GPUs, employing a highly decentralized approach, via Blockchain technology. This transformation enables the exploitation of these cutting-edge computing paradigms to address complex optimization challenges effectively. Through detailed explanations, mathematical formulations, and algorithmic strategies, this document aims to provide a comprehensive guide to understanding and implementing the conversion process from HUBO to QUBO. It underscores the importance of such transformations in making prime factorization computationally feasible on both existing classical computers and emerging computing technologies.
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.
This article explores the comparative strengths of Higher-Order Unconstrained Binary Optimization (HUBO) and Quadratic Unconstrained Binary Optimization (QUBO) models in the context of probability optimization using tensor sampling techniques. HUBO can represent interactions beyond pairwise terms without introducing auxiliary variables, while QUBO benefits from a simpler algebraic structure and a wide range of established solvers. We combine a theoretical analysis of formulation expressiveness and reduction overhead with empirical experiments on synthetic polynomial objectives, RSA factorization, and Max-Cut instances. The results show that native HUBO formulations are often preferable when the original problem contains high-order interactions, because reducing such problems to QUBO increases the effective dimension seen by tensor samplers. At the same time, QUBO remains competitive for naturally quadratic problems. These findings provide practical guidance for selecting an optimization framework that balances model fidelity, dimension growth, and solver performance.
Solving the eigenvalue problem is particularly important in almost all fields of science and engineering. With the development of quantum computers, multiple algorithms have been proposed for this purpose. However, such methods are usually only applicable to matrices of specific types, such as unitary or Hermitian matrices. The quantum annealer of the D-Wave, a quantum computer, returns the minimum value of the quadratic unconstrained binary optimization (QUBO) model. Thus, quantum annealers can be leveraged to solve arbitrary eigenvalue problems by formulating corresponding QUBO models. In this paper, we propose two higher-order unconstrained optimization (HUBO) formulations to solve eigenvalue problems involving n×n general matrices. In addition, we use a formula to reduce the order and convert the HUBO model into a QUBO model. Further, by using a quantum approximate optimization algorithm, this method can be extended to a gate-model quantum computer.
Optimization is a crucial challenge across various domains, including finance, resource allocation, and mobility. Quantum computing has the potential to redefine the way we handle complex problems by reducing computational complexity and enhancing solution quality. Optimization, particularly of objective functions, stands to benefit significantly from quantum solvers, which leverage principles of quantum mechanics like superposition, entanglement, and tunneling. The Ising and Quadratic Unconstrained Binary Optimization (QUBO) models are the most suitable formulations for these solvers, involving binary variables and constraints treated as penalties within the overall objective function. To harness quantum approaches for optimization, two primary strategies are employed: exploiting quantum annealers—special-purpose optimization devices—and designing algorithms based on quantum circuits. This review provides a comprehensive overview of quantum optimization methods, examining their advantages, challenges, and limitations. It demonstrates their application to real-world scenarios and outlines the steps to convert generic optimization problems into quantum-compliant models. Lastly, it discusses available tools and frameworks that facilitate the exploration of quantum solutions for optimization tasks.
The increasing complexity of industrial scheduling and transport routing problems motivates the study of alternative optimization formulations and computational paradigms. In this work, we study how higher-order unconstrained binary optimization (HUBO) formulations of such problems map onto quantum optimization workflows in both noisy and fault-tolerant regimes. We consider three representative logistics and manufacturing use cases and formulate each as a HUBO problem. This captures process intricacies, such as highly correlated assembly-line scheduling rules, which are difficult to express faithfully with the standard quadratic (QUBO) form, while at the same time reducing the number of binary variables required in the quantum mapping, thus lowering qubit demand. We compare the HUBO formulations with corresponding QUBO encodings, highlighting a key trade-off: while HUBO reduces qubit requirements through compact binary encoding, it introduces higher-order interaction terms that increase circuit depth, limiting feasibility on current quantum hardware. The proposed formulations are validated using classical solvers across several problem instances and benchmark small routing problem instances using bias-field digitized counterdiabatic quantum optimization in classical simulation. We complement these results with a resource and scalability analysis, focusing on the capacitated vehicle routing problem as a representative large-scale industrial use case. Our analysis indicates that while HUBO formulations offer advantages in qubit scaling compared to QUBO encodings, their practical implementation is constrained by gate fidelity, coherence, and circuit depth, making hybrid quantum-classical workflows and early fault-tolerant quantum hardware the most plausible settings for their practical use.
Quadratic Unconstrained Binary Optimization (QUBO) models are defined by quadratic expressions composed of binary variables, each taking a value of either 0 or 1. The objective of the QUBO problem is to find an optimal assignment of these binary variables that minimizes the value of the QUBO model. The QUBO problem has attracted significant research interest due to its versatility in reformulating various NP-complete problems and its potential for efficiently finding heuristic solutions through quantum annealing. This paper focuses on converting Cubic Unconstrained Binary Optimization (CUBO) problems, defined by cubic expressions, into equivalent QUBO models. Through this conversion, CUBO problems can be solved using quantum annealing. Since the important NP-hard problem 3-SAT can be straightforwardly represented as a CUBO problem, investigating the conversion from CUBO to QUBO models is a meaningful endeavor.