From Problem-Structured QAOA to Problem-Agnostic Variational MaxCut Solvers: Representation, Redundancy, and Optimization Landscapes

Variational quantum optimization provides a useful setting for examining how circuit structure, parameterization, and trainability interact in practice. Objectives: Among the most promising variational quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA) depends critically on circuit architecture, but greater circuit complexity does not necessarily lead to easier optimization or better solutions. Using the Maximum Cut (MaxCut) problem on graphs as a controlled setting, we investigate how variational behavior changes as problem-specific QAOA structure is progressively removed, with particular attention to representational capacity, optimization geometry, and finite-budget trainability. Methods: We analytically characterize a graph-agnostic product-state formulation of MaxCut and derive exact equivalences, redundancies, and local-landscape properties. These results motivate a controlled comparison among separable product ansätze, a representative graphagnostic entangling circuit, and graph-structured QAOA at several depths, including a parameter-count-matched control. The numerical benchmark comprises 95 small unweighted graph instances optimized under a common objective-evaluation budget. Results: The product-state formulation preserves the exact discrete MaxCut optimum, while its strict local maxima correspond to classical cuts that are stable under singlevertex flips. Redundant phase parameters increase optimization effort without enlarging the objective-relevant family. Under the prescribed optimization budget, the productstate ansatz attains higher average solution quality than the tested QAOA depths while requiring substantially fewer objective evaluations. The graph-agnostic entangling comparator provides no consistent advantage over the separable baseline, and a simple classical local-search method remains highly competitive. Conclusion: In the regime considered here, objective-relevant parameterization and optimization geometry can be as important as circuit richness.