The Quadratic Unconstrained Binary Optimization (QUBO) model is widely used to solve combinatorial optimization problems by determining the optimal assignment of binary variables in a quadratic formulation. Given its versatility, extensive research has been conducted on developing QUBO solvers and converting combinatorial optimization problems into equivalent QUBO formulations. This paper introduces QUBO++, a novel C++ symbolic mathematical computation library specifically designed for QUBO modeling. It provides a comprehensive set of operators and functions for symbolic computation, enabling seamless transformation of optimization problems into QUBO models. While several Python-based libraries exist for similar purposes, QUBO++ is the only dedicated C++ library that supports multithreaded acceleration via Intel Threading Building Blocks (oneTBB). In addition to its modeling capabilities, QUBO++ includes two fundamental QUBO solvers: the Exhaustive Solver, which is useful for verifying small-scale problems, and the Easy Solver, designed for efficient testing and debugging. Compared to existing Python-based QUBO modeling tools such as PyQUBO, Fixstars Amplify, and Tytan, QUBO++ demonstrates significantly higher performance, achieving speedups of over 1,700 times in certain scenarios. By leveraging QUBO-based problem-solving techniques, QUBO++ serves as a powerful and versatile tool for addressing complex optimization challenges in both industrial and societal applications.
The Quadratic Unconstrained Binary Optimization (QUBO) model is widely used to solve combinatorial optimization problems by determining the optimal assignment of binary variables in a quadratic formulation. Given its versatility, extensive research has been conducted on developing QUBO solvers and converting combinatorial optimization problems into equivalent QUBO formulations. This paper introduces QUBO++, a novel C++ symbolic mathematical computation library specifically designed for QUBO modeling. It provides a comprehensive set of operators and functions for symbolic computation, enabling seamless transformation of optimization problems into QUBO models. While several Python-based libraries exist for similar purposes, QUBO++ is the only dedicated C++ library that supports multithreaded acceleration via Intel Threading Building Blocks (oneTBB). In addition to its modeling capabilities, QUBO++ includes two fundamental QUBO solvers: the Exhaustive Solver, which is useful for verifying small-scale problems, and the Easy Solver, designed for efficient testing and debugging. Compared to existing Python-based QUBO modeling tools such as PyQUBO, Fixstars Amplify, and Tytan, QUBO++ demonstrates significantly higher performance, achieving speedups of over 1,700 times in certain scenarios. By leveraging QUBO-based problem-solving techniques, QUBO++ serves as a powerful and versatile tool for addressing complex optimization challenges in both industrial and societal applications.
We present PyQUBO, an open-source Python library for constructing quadratic unconstrained binary optimizations (QUBOs) from the objective functions and the constraints of optimization problems. PyQUBO enables users to prepare QUBOs or Ising models for various combinatorial optimization problems with ease thanks to the abstraction of expressions and the extensibility of the program. QUBOs and Ising models formulated using PyQUBO are solvable by Ising machines, including quantum annealing machines. We introduce the features of PyQUBO with applications in the number partitioning problem, knapsack problem, graph coloring problem, and integer factorization using a binary multiplier. Moreover, we demonstrate how PyQUBO can be applied to production-scale problems through integration with quantum annealing machines. Through its flexibility and ease of use, PyQUBO has the potential to make quantum annealing a more practical tool among researchers.
The field of Combinatorial Optimization (CO) is one of the most important
areas in the general field of optimization, with important applications found
in every industry, including both the private and public sectors. It is also
one of the most active research areas pursued by the research communities of
Operations Research, Computer Science, and Analytics as they work to design and
test new methods for solving real world CO problems. In recent years, we have
discovered that a mathematical formulation known as QUBO, an acronym for a
Quadratic Unconstrained Binary Optimization problem, can embrace a large
variety of important CO problems found in industry and government. Through
special reformulation techniques that are easy to apply, the power of QUBO
solvers can be used to efficiently solve many important problems once they are
put into the QUBO framework. This two-step process of first re-casting an
original model into the form of a QUBO model and then solving it with
appropriate software enables the QUBO model to become a unifying framework for
combinatorial optimization. The alternative path that results for effectively
modeling and solving many important problems is a new development in the field
of combinatorial optimization. The materials provided in subsequent sections
illustrate the process of reformulating important optimization problems as QUBO
models through a series of explicit examples. Collectively these examples
highlight the application breadth of the QUBO model. We do not focus here on
modeling a given problem in the form classically adopted in the optimization
community. Rather, our examples disclose the methods that can be used to recast
a given problem, perhaps existing in a classical mathematical form, into an
equivalent QUBO model.
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.
Quadratic Unconstrained Binary Optimization (QUBO) is a versatile framework for modeling combinatorial optimization problems. This study benchmarks five software-based QUBO solvers: Neal, PyTorch (CPU), PyTorch (GPU), JAX, and SciPy, on randomly generated QUBO matrices ranging from 1000x1000 to 45000x45000, under six convergence thresholds from 10^-1 to 10^-6. We evaluate their performance in terms of solution quality (energy) and computational time. Among the solvers tested, Neal achieved the lowest energy values but was limited to problems with up to 6000 variables due to high memory consumption. PyTorch produced slightly higher energy results than Neal but demonstrated superior scalability, solving instances with up to 45000 variables. Its support for GPU acceleration and CPU multi-threading also resulted in significantly shorter runtimes. JAX yielded energy values slightly above those of PyTorch and was limited to 25000 variables, with runtimes comparable to PyTorch on GPU. SciPy was the most constrained solver, handling only up to 6000 variables and consistently producing the highest energy values with the longest computation times. These findings highlight trade-offs between solution quality, scalability, and runtime efficiency, and suggest that PyTorch is the most balanced choice for large-scale QUBO problems when computational resources permit.
With the advances in customized hardware for quantum annealing and digital/CMOS Annealing, Quadratic Unconstrained Binary Optimization (QUBO) models have received growing attention in the optimization literature. Motivated by an existing general-purpose approach that derives QUBO models from binary linear programs (BLP), we propose a novel Multilevel Constraint Transformation Scheme (MLCTS) that derives QUBO models with fewer ancillary binary variables. We formulate sufficient conditions for the existence of a compact QUBO formulation (i.e., in the original BLP decision space) in terms of constraint levelness and demonstrate the flexibility and applicability of MLCTS on synthetic examples and several well-known combinatorial optimization problems, i.e., the Maximum 2-Satisfiability Problem, the Linear Ordering Problem, the Community Detection Problem, and the Maximum Independence Set Problem. For a proof-of-concept, we compare the performance of two QUBO models for the latter problem on both a general-purpose software-based solver and a hardware-based QUBO solver. The MLCTS-derived models demonstrate significantly better performance for both solvers, in particular, solving up to seven times more instances with the hardware-based approach.
Quadratic Unconstrained Binary Optimization (QUBO) is a central formulation for combinatorial optimization and has gained increasing attention due to its compatibility with quantum, hybrid quantum-classical, and quantum-inspired solvers. However, translating natural-language problem descriptions into correct QUBO formulations remains difficult, requiring the identification of binary variables, constraints, objective functions, penalty terms, and suitable penalty weights. This process is time-consuming and often demands substantial domain expertise. To address this challenge, we propose an end-to-end multi-agent framework that automatically generates QUBO formulations from natural-language problem descriptions, supported by structured or unstructured test cases. To evaluate its performance, We also introduce QUBOBench, a benchmark containing 100 combinatorial optimization problems across 12 application domains, curated from peer-reviewed literature, competitions, and canonical NP-hard problems. Experimental results show that our framework achieves 68% accuracy on QUBOBench, outperforming a direct single-call baseline by 22%. Further analysis identifies iterative self-repair as the most important component contributing to improved performance. The data and code are open-sourced at https://quitttcat.github.io/QuantumQUBOAgent.