qubo formulation

This paper addresses the challenge of efficiently solving the Quadratic Knapsack Problem (QKP) on conventional Ising machines. To overcome the limitations of the standard Quadratic Unconstrained Binary Optimization (QUBO) paradigm—which restricts variables to binary values and lacks native support for constraints—we propose the Extended Ising Machine (EIM) framework. EIM introduces real-valued dependent variables and explicit constraint modeling, significantly enhancing expressive power and modeling flexibility for constrained combinatorial optimization. By designing a QKP-specific constraint encoding scheme, EIM embeds the original problem directly into a continuous-discrete hybrid energy function, eliminating the need for large numbers of auxiliary variables. Experimental results on multiple benchmark instances demonstrate that EIM achieves superior solution quality and runtime efficiency compared to both standard Ising models and state-of-the-art baselines, including commercial exact solvers (e.g., Gurobi) and mainstream heuristic algorithms.

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 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.

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 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 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 addresses the challenge of training binary neural networks (BNNs), whose discrete parameters hinder optimization and limit deployment on resource-constrained devices. The authors unify BNN training as a Quadratic Unconstrained Binary Optimization (QUBO) problem, enabling support for arbitrary network topologies for the first time. They introduce two novel regularization strategies: one that maximizes neuronal margins to enhance discriminative capacity, and another that employs a Dropout-like iterative subnetwork parameter penalization mechanism to improve generalization. Implemented on a GPU-accelerated Ising solver, the proposed approach significantly improves training dynamics and achieves substantial gains in classification accuracy across multiple benchmark datasets.

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.