Optimization Geometry of QAOA and Variational Quantum Algorithms

2026-10-04

AI Summary

This study addresses the lack of theoretical justification for classical optimizer selection in variational quantum algorithms. Based on VQE and QAOA models combined with parameterized circuit techniques, it systematically compares the optimization performance of BFGS and adaptive differential evolution (jSO) through visual analysis of objective function geometric properties. The work proposes a robust baseline comparison method employing multi-start matching budgets, revealing that the distribution of local minima and variations in solution quality decisively influence global search effectiveness. Ultimately, this research establishes a practical workflow connecting model construction, geometric diagnostics, and optimizer selection, clarifying the essential value of global search strategies in specific scenarios.

Abstract

Variational quantum algorithms turn choices of Hamiltonian, ansatz, and parameterization into a classical nonconvex optimization problem. We study how this objective function can be visualized and characterized in ways that help explain optimizer behavior. We distinguish two properties of the objective: the number of local minima encountered along sampled directions and the differences in quality among local-search endpoints. We then ask how a local optimizer, represented by BFGS, compares with adaptive differential evolution, represented by jSO. Rather than comparing jSO with a single local run, we allow BFGS multiple starts within the same function-evaluation budget. This gives local search repeated opportunities to explore different basins and provides a stronger baseline for asking when a global evolutionary solver is useful. We study these questions using VQE and QAOA, focusing on how frustra- tion, circuit depth, parameter tying, mixed locality, and nonlinear repa- rameterization change the Hamiltonian expectation-value objective seen by the classical optimizer. Increasing independent circuit depth raises the sampled local-minimum count without making global search more effective. Parameter tying, by contrast, produces both more repeated local structure and much larger differences in quality among local-search endpoints, and in this regime adaptive differential evolution outperforms function-evaluation-matched multistart BFGS. The comparison shows that the number of local minima alone does not determine whether global search is advantageous: the important distinction is whether different basins lead to similarly good solutions or to substantially different objective values. These results provide a practical workflow for connecting model construction, objective- function geometry, empirical diagnostics, and optimizer choice.

Problem

• Variational Quantum Algorithms
• QAOA
• Optimization Geometry
• Nonconvex Optimization
• Local Minima

Innovation

• Variational Quantum Algorithms
• Optimization Landscape
• QAOA
• Differential Evolution
• Parameter Tying