qubo formulation

To address the quantum-bit (qubit) resource constraints of Noisy Intermediate-Scale Quantum (NISQ) devices, this paper proposes a Quadratic Unconstrained Binary Optimization (QUBO) encoding method based on a generalized exponential penalty function for combinatorial optimization problems with inequality constraints. Unlike conventional linear or quadratic penalties, our framework systematically analyzes the theoretical properties and embedding efficiency of multiple exponential penalty functions, directly incorporating constraints into the objective function and thereby drastically reducing the number of auxiliary binary variables. Experimental evaluation on bin packing and traveling salesman problems achieves 57% and 83% qubit reduction, respectively; on problem instances requiring 8–12 qubits, solution quality matches that of classical solvers. The method balances modeling simplicity with hardware compatibility, offering a scalable, low-overhead paradigm for constrained optimization on NISQ-era quantum hardware.

This work addresses the issue that quadratic penalty relaxations of binary linear programs often yield spurious or infeasible local minima. To overcome this, we propose a class of QUBO relaxation models satisfying specific structural conditions that guarantee all local minima are feasible and strictly binary. Leveraging these conditions, we derive novel differentiable relaxations for classical combinatorial optimization problems—including open-pit mining, the 0–1 knapsack problem, and the traveling salesman problem—and solve them using gradient-based optimizers such as projected gradient descent and Adam. Experimental results demonstrate that the proposed approach reliably converges to valid binary solutions, thereby establishing clear theoretical guarantees and delineating the applicability boundaries of differentiable optimization as a local solver for combinatorial problems.

This work addresses the Dominating Set (DS) problem in graph theory and its key variants—Independent Dominating Set, Total Dominating Set, and k-Dominating Set—by introducing the first systematic Quadratic Unconstrained Binary Optimization (QUBO) modeling framework. The method reformulates problem constraints and the objective function to drastically reduce the number of binary variables, cutting required qubits by 40–70% compared to conventional QUBO encodings. Designed for both modeling fidelity and hardware compatibility, it enables efficient execution on current noisy intermediate-scale quantum (NISQ) devices. Key contributions include: (1) the first unified, compact QUBO formulation covering multiple DS variants; (2) elimination of structural redundancy inherent in prior approaches, thereby lowering quantum resource requirements; and (3) advancing DS problems from theoretical QUBO mapping toward practical quantum solvers. Empirical evaluation confirms the model’s correctness and efficacy on both real quantum annealers and classical QUBO simulators.

Quadratic Unconstrained Binary Optimization (QUBO) formulations for Bayesian network structure learning suffer from exponential blowup in binary variables, severely limiting scalability. Method: This paper introduces a decomposition-based quadraticization paradigm that combines constraint decomposition, divide-and-conquer transformation of higher-order terms, and sparse structural modeling—while preserving exact equivalence to the original problem. Contribution/Results: Evaluated on 16 benchmark instances with 37–223 variables, the approach reduces the number of binary variables by up to an order of magnitude, effectively overcoming the scalability bottleneck of conventional quadraticization methods. The resulting lightweight QUBO formulation exhibits enhanced hardware compatibility and solver robustness, leading to significant improvements in both performance and efficiency of quantum annealers and hybrid optimizers for score-maximization tasks.

This paper addresses tridiagonal-structured discrete optimization problems—including QUBO, QUDO, and generalized tensorial T-QUDO—where the objective function involves only quadratic couplings between adjacent variables. Method: We propose the first rigorous polynomial-time quantum-inspired algorithm, based on tensor network modeling: (i) constructing a quantum state encoding the objective function via imaginary-time evolution; (ii) iteratively extracting the configuration with maximal amplitude through partial trace contraction and matrix product state (MPS) optimization. Contribution/Results: The algorithm achieves time complexity O(nχ³), where χ is the bond dimension (tensor rank at boundaries), and provably identifies degenerate global optima. It is the first exact polynomial-time solver for tridiagonal discrete optimization, unifying treatment of both binary and multi-level discrete variables. Numerical experiments confirm correctness, efficiency, and scalability across problem sizes.

This study addresses the challenge of constructing credit scoring scales that satisfy regulatory constraints—a complex constrained combinatorial optimization problem in financial credit rating. For the first time, this work formulates the problem as a Quadratic Unconstrained Binary Optimization (QUBO) model, making it compatible with quantum computing frameworks while enabling efficient solution via classical heuristic algorithms. The proposed approach achieves solution quality equivalent to exhaustive search but with significantly improved scalability, thereby supporting applications involving more intricate regulatory constraints. By bridging regulatory compliance and computational efficiency, this method establishes a novel paradigm for designing credit scoring systems that are both compliant and scalable.

This work proposes a novel QUBO encoding method for permutation problems based on comparison-swap networks. Requiring only $O(n \log^2 n)$ binary variables, the approach substantially reduces both the number of variables and the density of the interaction graph while ensuring a bijective mapping to the space of permutations and enabling unbiased sampling. It is the first to integrate oblivious comparison-swap networks with QUBO modeling, thereby supporting constraints such as fixed points and parity, as well as algebraic operations including permutation multiplication, inversion, and order detection. Compared to conventional permutation matrix formulations, the resulting model is significantly more compact and sparse, facilitating efficient generation of solutions with prescribed properties—such as a given order or commutativity with a target permutation—thus offering promising applications in cryptography and combinatorial design.

This work proposes a quantum-inspired tensor network algorithm based on the MeLoCoToN framework for combinatorial optimization problems involving k-local interactions, including QUBO, QUDO, and their higher-order tensor formulations. The approach integrates superposition-state initialization, imaginary-time evolution, and projective measurement, and introduces two efficient implementation strategies: one leveraging fourth-order tensor contraction and the other combining sparse matrix-vector multiplication with a novel “Waterfall” computational technique. As the first study to apply tensor networks to k-local QUBO/QUDO-type problems, the method demonstrates significant performance advantages over conventional quadratic optimization solvers across multiple benchmark instances, thereby validating its efficacy and computational superiority.

Gradient-based methods for large-scale combinatorial optimization often become trapped in local optima, particularly under non-massively-parallel settings where their performance is constrained. This work identifies the core bottleneck as optimization stagnation rather than limitations in model capacity or computational resources, and introduces a differentiable global reset algorithm incorporating a mutation mechanism. By synergistically combining local search strategies with a newly designed quadratic objective function for MaxCut within a relaxed QUBO framework, the proposed approach effectively escapes local optima. Experimental results demonstrate that the method significantly outperforms state-of-the-art heuristics, commercial integer programming solvers, and recent GPU-accelerated techniques on large-scale graphs, all without relying on extensive parallel initialization.

This study addresses the suboptimal splitting problem in Classification and Regression Trees (CART) when handling categorical predictors, which arises from inadequate optimization over combinatorial split scores. The authors propose, for the first time, formulating the split selection task as a Quadratic Unconstrained Binary Optimization (QUBO) problem and solving it exactly via the Dinkelbach algorithm. This approach transcends the limitations of conventional greedy heuristics, yielding significantly higher-quality splits while opening new avenues for hybrid classical-quantum computation in tree-based models. Experimental results demonstrate that the proposed QUBO regression tree achieves superior split solutions compared to standard CART, without compromising predictive performance.