QUBO-to-MaxCut Transformation

Updated 20 September 2025

QUBO-to-MaxCut transformation is a method that converts quadratic binary optimization problems into equivalent graph partitioning forms for solving NP-hard challenges.
Preprocessing and variable fixing rules simplify the mapping process by reducing redundant variables and compressing the problem size for efficient optimization.
Advanced techniques like community decomposition, eigenvalue-guided transformation, and penalty quadratization enable effective embedding on hybrid quantum-classical systems.

The QUBO-to-MaxCut transformation refers to the suite of mathematical, algorithmic, and computational methods developed to map quadratic unconstrained binary optimization (QUBO) problems onto equivalent instances of the MaxCut problem. This transformation is central to combinatorial optimization, quantum computing, and associated fields because of the prevalence of both QUBO and MaxCut as canonical NP-hard archetypes, and the frequent need to interface between algorithmic methods or hardware platforms tailored for one or the other. Theoretical mappings, practical reduction rules, embedding strategies, and recent advances in quantum-inspired and hybrid classical/quantum algorithms all play crucial roles in this process.

1. Formal QUBO-to-MaxCut Mapping and Variable Encoding

The QUBO-to-MaxCut transformation typically starts from the standard QUBO representation: $\max_{x \in \{0,1\}^n} x^\top Q x,$ where $Q$ is a symmetric real matrix. The mapping capitalizes on the equivalence between QUBO and Ising models, and subsequently their connection to MaxCut. By substituting binary variables $x_i \in \{0,1\}$ with spin variables $z_i = 1 - 2x_i \in \{-1, +1\}$ , the quadratic form is recast as

$H(z) = h_0 - \sum_{i<j} J_{ij} z_i z_j - \sum_i h_i z_i,$

where

$h_0 = \frac{1}{2}\sum_{i \leq j} Q_{ij}, \quad h_i = \frac{1}{2}\sum_j Q_{ij}, \quad J_{ij} = -\frac{1}{2} Q_{ij}.$

In MaxCut, the objective is

$\max_{z \in \{-1,1\}^n} \sum_{(i,j) \in E} w_{ij} (1 - z_i z_j)/2,$

and so, up to constant shifts and scaling, any QUBO may be rewritten (possibly after embedding with an auxiliary variable to absorb local fields) as a MaxCut cost function over an appropriate graph (Wu et al., 13 Aug 2024).

2. Preprocessing, Reduction, and Logical Implications

Significant improvements in practical solution quality and computational efficiency are obtained by preprocessing QUBOs before or during the QUBO-to-MaxCut transformation. Rule-based preprocessing analyzes the QUBO matrix to identify variables that must be fixed (eliminated) or substituted prior to full optimization. For instance, rules using coefficient sums and lower/upper bounds on variable contributions (such as $c_i + D_i^-$ and $c_i + D_i^+$ ) allow immediate assignment of variables (Glover et al., 2017, Lewis et al., 2017). More advanced implications involve logical inequalities and substitution, such as deducing $x_i \geq x_j$ or $x_i + x_j \leq 1$ when pairwise conditions are satisfied. These deductions are vital for reducing the effective problem dimension and for simplifying the structure of the MaxCut instance post-transform.

Preprocessing benefits are particularly acute for quantum and quantum-inspired hardware, where node and edge count strongly affect embedding cost and resource requirements, and for metaheuristic algorithms, where reduced instance size and implicit constraints accelerate convergence and enhance solution quality (Glover et al., 2017, Lewis et al., 2017).

3. Algorithmic and Structural Pathways

Multiple algorithmic techniques underpin the QUBO-to-MaxCut mapping, including:

Explicit Construction: Forming the MaxCut instance graph by setting edge weights according to nonzero quadratic terms in QUBO, interpreting local fields as connections to an auxiliary node so as to capture linear terms.
Community Detection and Decomposition: Partitioning the QUBO’s interaction graph into communities, eliminating internal degrees of freedom (core spins), and expressing the reduced PUBO (polynomial unconstrained binary optimization) on the boundary spins. This leads to a smaller MaxCut or QUBO instance, often with higher-degree terms which are then handled by further quadraticization (Guerreschi, 2021). Such divide-and-conquer approaches result in substantial resource reductions and improved approximation ratios for quantum optimization algorithms.
Eigenvalue-Guided Transformation: Augmenting the QUBO matrix $Q$ with a low-rank matrix constructed from dominant eigenvectors to guide local search or heuristic optimization toward high-quality MaxCut solutions. Caution is advised when using this method as it increases density and can degrade performance if dominant modes are overemphasized (Verma et al., 2021).
Penalty-Based Quadratization: For pseudo-Boolean functions of degree higher than two, systematic reduction strategies based on frequency counts and integer programming minimize the number of auxiliary variables and penalty coefficients needed in transformation to QUBO, yielding more tractable downstream MaxCut instances (Verma et al., 2021).

4. Embedding and Solution Strategies in Quantum Computing

As QUBO and MaxCut underlie prominent quantum optimization regimes, particularly the Quantum Approximate Optimization Algorithm (QAOA) and quantum annealing, their inter-translation is vital. Key insights include:

Variable Reduction and Qubit Efficiency: Amplitude encoding and variational approaches (e.g., LogQ, QEMC) compress QUBO/MaxCut problems for quantum circuits, requiring only $\lceil \log_2 n \rceil$ qubits as opposed to the canonical $n$ qubits used in standard QAOA. These methods use continuous parameterizations (e.g., smooth sigmoid mappings) to facilitate gradient-based classical optimization, dramatically reducing both quantum device and classical optimizer resource requirements. The cost function for MaxCut is typically constructed as the expectation value of the graph Laplacian operator in the quantum state parameterized by the ansatz (Chatterjee et al., 11 Jul 2025, Tene-Cohen et al., 2023, Rančić, 2021).
Divide-and-Conquer plus Quantum Heuristics: By decoupling communities in QUBO/MaxCut transformations and then applying QAOA or variational quantum algorithms to the compressed, reduced instances, significant qubit/resource reduction (∼42% for random regular graphs) and enhanced approximation quality have been realized (Guerreschi, 2021).
Hybrid Quantum-Classical SCF in Quantum Chemistry: In Hartree-Fock optimization, each iteration’s discrete subproblem can be cast as a QUBO and mapped to a MaxCut instance, which is then attacked via semidefinite programming, quantum annealing, QAOA, or other specialized quantum subroutines, providing robustness and approximation guarantees at every step (Ralli et al., 4 Jun 2025).

5. Advanced Integer and SDP Programming Links

SDP relaxations, most notably the Goemans–Williamson algorithm, yield provable approximation ratios (approximately 0.878 for nonnegative-weight MaxCut) and naturally operate on the quadratic forms produced by QUBO-to-MaxCut reduction. In many pipeline architectures—e.g., for large industrial instances, real-time hybrid optimization, or combinatorial quantum chemistry—the QUBO-to-MaxCut transformation is not merely formal but enables practical deployment of mature SDP frameworks, branch-and-cut integer solvers augmented by odd-cycle inequalities, and hybrid algorithms exploiting the underlying graph structure (Rehfeldt et al., 2022, Ralli et al., 4 Jun 2025).

6. Scalability, Sparsification, and Preprocessing for Cloud and NISQ Systems

In practical settings (especially for cloud-based quantum(-inspired) solvers and NISQ hardware), preprocessing is essential for reducing communication overhead and computational cost. Graph sparsification, typically by effective resistance sampling, reduces the number of QUBO nonzeros from $O(n^2)$ to $O(n \log n)$ for dense graphs while preserving all cut values to within an $(1+\epsilon)$ factor. Empirical evidence demonstrates that sparsified QUBO-to-MaxCut instances achieve objective values within 90% of optimum while shrinking data transfer and solve times by over 80% (Suppakitpaisarn et al., 23 Jan 2024). Preprocessing methods, such as those in (Glover et al., 2017, Lewis et al., 2017), further complement sparsification by removing determinable or redundant variables, accelerating overall solution workflows.

7. Limitations, Extensions, and Multi-Problem Reductions

Not all QUBOs map directly onto “canonical” MaxCut (i.e., with only pairwise terms and certain weight signs), especially when higher-order constraints, dense auxiliary variable interconnections, or strict structure are present. Penalty-based quadraticization (as in higher-degree pseudo-Boolean to QUBO) can induce dense or fully connected MaxCut instances, leading to practical modeling and embedding challenges (Verma et al., 2021, Gabor et al., 2022). Additionally, QUBO-to-MaxCut techniques serve as stepping stones for further transformations—e.g., from MaxCut/Ising to Maximum Independent Set (MIS)—broadening the impact of these methodologies in combinatorial optimization and coding theory (Wu et al., 13 Aug 2024).

Summary Table: Canonical Steps in QUBO-to-MaxCut Transformation

Step	Key Formula/Principle	Reference
Binary-to-Spin Mapping	$z = 1-2x$	(Wu et al., 13 Aug 2024, Glover et al., 2017)
MaxCut Objective	$\sum_{(i,j)} w_{ij} (1-z_iz_j)/2$	(Ralli et al., 4 Jun 2025, Gabor et al., 2022)
Preprocessing/Fixing	Variable assignment rules (e.g., $c_i + D^-_i \geq 0$ )	(Glover et al., 2017, Lewis et al., 2017)
Community Decomposition	Core-boundary splitting	(Guerreschi, 2021)
Amplitude Encoding	Parameterization $R(\theta)$ , state compression	(Chatterjee et al., 11 Jul 2025, Tene-Cohen et al., 2023)
Penalty Quadratization	Auxiliary + Penalty ( $M$ )	(Verma et al., 2021)
Laplacian-based Quantum	$\|\Psi(\theta)\rangle = U(\theta) H^{\otimes N} \|0\rangle$	(Chatterjee et al., 11 Jul 2025)

In conclusion, the QUBO-to-MaxCut transformation encompasses a mathematically principled mapping from quadratic binary objectives to graph partition formalisms, operationalized by advanced preprocessing, logical deduction, decomposition, and embedding strategies. These allow practitioners to exploit a wide array of classical, quantum, and hybrid optimization architectures, each leveraging the fundamental structure preserved through this transformation. The approach continues to evolve, especially as resource-aware and quantum-enabled platforms propel the optimization of ever larger and more complex instances (Glover et al., 2017, Guerreschi, 2021, Chatterjee et al., 11 Jul 2025, Tene-Cohen et al., 2023, Rančić, 2021, Ralli et al., 4 Jun 2025, Suppakitpaisarn et al., 23 Jan 2024, Lewis et al., 2017, Rehfeldt et al., 2022, Gabor et al., 2022, Wu et al., 13 Aug 2024, Verma et al., 2021, Lai et al., 16 Oct 2024, Verma et al., 2021, Semenov et al., 28 Oct 2024).