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arXiv:2610.05524

Optimization Geometry of QAOA and Variational Quantum Algorithms

Vojtěch Novák, Ivan Zelinka, Silvie Illésová, Swagatam Das, Martin Beseda

quant-phcs.NESubmitted Oct 4, 2026• Announced Oct 6, 2026• 0 citations
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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.

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Vojtěch Novák, Ivan Zelinka

Variational quantum algorithms recast state preparation as classical nonconvex optimization, but it is often unclear when multistart local search suffices and when population-based global search justifies its evaluation cost. Using a landscape-first design, controlled QAOA experiments identify two mechanisms making local search unreliable: parameter reuse across circuit layers and competition between 2-local and 3-local cost terms. Increasing QAOA depth alone does not replicate this effect. We test these mechanisms across eight unseen spin glasses, fresh $N=10$ and $N=12$ instances, and standard MaxCut, transverse-field Ising, and Heisenberg models. On all eight unseen spin glasses, adaptive differential evolution (DE) achieves lower median error than both multistart BFGS and multistart Powell on the two difficult constructions, whereas the independent-depth control does not. The parameter-reuse effect transfers to MaxCut at both sizes, while standard VQE models remain favorable to local search. A pre-benchmark landscape score combining random-start local outcomes and 1D parameter slices is fixed from confirmation data before evaluating new optimizers. It predicts whether adaptive DE outperforms multistart BFGS by over one spectral-range percentage point on 50 new quantum objectives with 80--86% condition-level accuracy. The clearest indicator of global-search utility is that local search frequently traps in inferior basins, rather than circuit depth or curvature anisotropy. The findings motivate whether quantum circuit complexity can be traded for harder classical optimization when robust global search is available.

quant-pharXiv:2609.14594•Sep 15, 2026•0 citations

Energy landscapes for the quantum approximate optimization algorithm

Choy Boy, David J. Wales

Variational quantum algorithms (VQAs) have demonstrated considerable potential in solving NP-hard combinatorial problems in the contemporary noisy intermediate-scale quantum (NISQ) era. The quantum approximate optimization algorithm (QAOA) is one such algorithm, used in solving the maximum cut (Max-Cut) problem for a given graph by successive implementation of L quantum circuit layers within a corresponding Trotterized ansatz. The challenge of exploring the cost function of VQAs arising from an exponential proliferation of local minima with increasing circuit depth has been well documented. However, fewer studies have investigated the impact of circuit depth on QAOA performance in finding the correct Max-Cut solution. Here we employ basin-hopping global optimization methods to navigate the energy landscapes for QAOA ansätze for various graphs, and analyze QAOA performance in finding the correct Max-Cut solution. The structure of the solution space is also investigated using discrete path sampling to build databases of local minima and the transition states that connect them, providing insightful visualizations using disconnectivity graphs. We find that the corresponding landscapes generally have a single funnel organization, which makes it relatively straightforward to locate low-lying minima with good Max-Cut solution probabilities. In some cases below the adiabatic limit the second lowest local minimum may even yield a higher solution probability than the global minimum. This important observation has motivated us to develop broader metrics in evaluating QAOA performance, based on collections of minima obtained from basin-hopping global optimization. Hence we establish expectation thresholds in elucidating useful solution probabilities from local minima, an approach that may provide significant gains in elucidating reasonable solution probabilities from local minima. Published by the American Physical Society 2024

Physical Review A•Jun 3, 2024•7 citations

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

Leonardo Lavagna, Massimo Panella

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.

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Adaptive Differential Evolution and Multistart Search for Noisy QAOA Optimization

Vojtěch Novák, Ivan Zelinka, Swagatam Das +1 more

We benchmark classical optimization of a fixed low-depth Quantum Approximate Optimization Algorithm (QAOA) ansatz across four cost-Hamiltonian families at $N=12$, $p=3$, and $D=6$. Ten optimizers are compared over 25 independent runs under common ceilings of 10\,000 and 30\,000 function evaluations (FEs), first with exact statevector objectives and then with two additive observation-noise levels. Exact objectives favor multistart BFGS and multistart CMA-ES. Under noisy feedback, adaptive population methods become more competitive, but the ranking depends on whether performance is measured by the best exact point visited or by the point selected from noisy observations. A targeted extension over all three pre-screened instances per family confirms this regime change while showing that named adaptive-DE winners are instance dependent: jSO-lite leads low-noise oracle search, iL-SHADE high-noise oracle search, and L-SRTDE high-noise selected solutions in the equal-instance summaries. Bootstrap analysis quantifies a non-negligible high-noise search--selection gap, and a retrospective fixed-budget verification proxy shows that reserving a small measurement budget for final re-evaluation improves selected quality across all ten methods at 30\,000 FEs. A supplementary structure-aware study further shows that QAOA cross-depth restriction and continuous basin refinement are more useful in these conditions than standalone Monte Carlo tree-search selection. Overall, optimizer choice depends jointly on landscape structure, observation noise, and final-point identification.

quant-pharXiv:2609.23180•Sep 22, 2026•0 citations

Energy Landscapes for the Quantum Approximate Optimisation Algorithm

Choy Boy, David J. Wales

Variational quantum algorithms (VQAs) have demonstrated considerable potential in solving NP-hard combinatorial problems in the contemporary near intermediate-scale quantum (NISQ) era. The quantum approximate optimisation algorithm (QAOA) is one such algorithm, used in solving the maximum cut (Max-Cut) problem for a given graph by successive implementation of $L$ quantum circuit layers within a corresponding Trotterised ansatz. The challenge of exploring the cost function of VQAs arising from an exponential proliferation of local minima with increasing circuit depth has been well-documented. However, fewer studies have investigated the impact of circuit depth on QAOA performance in finding the correct Max-Cut solution. Here, we employ basin-hopping global optimisation methods to navigate the energy landscapes for QAOA ansätze for various graphs, and analyse QAOA performance in finding the correct Max-Cut solution. The structure of the solution space is also investigated using discrete path sampling to build databases of local minima and the transition states that connect them, providing insightful visualisations using disconnectivity graphs. We find that the corresponding landscapes generally have a single funnel organisation, which makes it relatively straightforward to locate low-lying minima with good Max-Cut solution probabilities. In some cases below the adiabatic limit the second lowest local minimum may even yield a higher solution probability than the global minimum. This important observation has motivated us to develop broader metrics in evaluating QAOA performance, based on collections of minima obtained from basin-hopping global optimisation. Hence we establish expectation thresholds in elucidating useful solution probabilities from local minima, an approach that may provide significant gains in elucidating reasonable solution probabilities from local minima.

arXiv:2401.04784•Jan 9, 2024•1 citation

Optimization Landscape Geometry in VQE for Frustrated Quantum Spin Models

Vojtěch Novák, Ivan Zelinka, Swagatam Das +1 more

We benchmark eight classical optimizers for exact-statevector VQE calculations on a controlled hierarchy of frustrated spin models, ranging from a diagonal Ising glass to transverse-field Ising and anisotropic Heisenberg models. The benchmark includes local, stochastic-gradient, evolutionary, covariance-adaptation, and swarm-based optimization methods under matched function-evaluation budgets. To understand their performance beyond final energies, we characterize the underlying Hamiltonian--ansatz landscapes in terms of local minima, gradients, curvature, and ground-state reachability. We use simple variational circuits, from an $R_y$ product-state ansatz for the diagonal model to shallow $R_y$--CNOT hardware-efficient circuits for the noncommuting models, and study how increasing circuit depth changes their expressivity, reachability, and optimization geometry. We find that optimizer performance changes substantially across the model hierarchy and is closely connected to landscape structure, while the variational gap represents a separate source of error. These results show how classical optimization, variational expressivity, and landscape geometry jointly determine VQE performance for frustrated spin models.

quant-pharXiv:2609.00235•Sep 2, 2026•0 citations
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.
Variational quantum algorithms recast state preparation as classical nonconvex optimization, but it is often unclear when multistart local search suffices and when population-based global search justifies its evaluation cost. Using a landscape-first design, controlled QAOA experiments identify two mechanisms making local search unreliable: parameter reuse across circuit layers and competition between 2-local and 3-local cost terms. Increasing QAOA depth alone does not replicate this effect. We test these mechanisms across eight unseen spin glasses, fresh N=10N=10N=10 and N=12N=12N=12 instances, and standard MaxCut, transverse-field Ising, and Heisenberg models. On all eight unseen spin glasses, adaptive differential evolution (DE) achieves lower median error than both multistart BFGS and multistart Powell on the two difficult constructions, whereas the independent-depth control does not. The parameter-reuse effect transfers to MaxCut at both sizes, while standard VQE models remain favorable to local search. A pre-benchmark landscape score combining random-start local outcomes and 1D parameter slices is fixed from confirmation data before evaluating new optimizers. It predicts whether adaptive DE outperforms multistart BFGS by over one spectral-range percentage point on 50 new quantum objectives with 80--86% condition-level accuracy. The clearest indicator of global-search utility is that local search frequently traps in inferior basins, rather than circuit depth or curvature anisotropy. The findings motivate whether quantum circuit complexity can be traded for harder classical optimization when robust global search is available.
Variational quantum algorithms (VQAs) have demonstrated considerable potential in solving NP-hard combinatorial problems in the contemporary noisy intermediate-scale quantum (NISQ) era. The quantum approximate optimization algorithm (QAOA) is one such algorithm, used in solving the maximum cut (Max-Cut) problem for a given graph by successive implementation of L quantum circuit layers within a corresponding Trotterized ansatz. The challenge of exploring the cost function of VQAs arising from an exponential proliferation of local minima with increasing circuit depth has been well documented. However, fewer studies have investigated the impact of circuit depth on QAOA performance in finding the correct Max-Cut solution. Here we employ basin-hopping global optimization methods to navigate the energy landscapes for QAOA ansätze for various graphs, and analyze QAOA performance in finding the correct Max-Cut solution. The structure of the solution space is also investigated using discrete path sampling to build databases of local minima and the transition states that connect them, providing insightful visualizations using disconnectivity graphs. We find that the corresponding landscapes generally have a single funnel organization, which makes it relatively straightforward to locate low-lying minima with good Max-Cut solution probabilities. In some cases below the adiabatic limit the second lowest local minimum may even yield a higher solution probability than the global minimum. This important observation has motivated us to develop broader metrics in evaluating QAOA performance, based on collections of minima obtained from basin-hopping global optimization. Hence we establish expectation thresholds in elucidating useful solution probabilities from local minima, an approach that may provide significant gains in elucidating reasonable solution probabilities from local minima. Published by the American Physical Society 2024
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.
We benchmark classical optimization of a fixed low-depth Quantum Approximate Optimization Algorithm (QAOA) ansatz across four cost-Hamiltonian families at N=12N=12N=12, p=3p=3p=3, and D=6D=6D=6. Ten optimizers are compared over 25 independent runs under common ceilings of 10\,000 and 30\,000 function evaluations (FEs), first with exact statevector objectives and then with two additive observation-noise levels. Exact objectives favor multistart BFGS and multistart CMA-ES. Under noisy feedback, adaptive population methods become more competitive, but the ranking depends on whether performance is measured by the best exact point visited or by the point selected from noisy observations. A targeted extension over all three pre-screened instances per family confirms this regime change while showing that named adaptive-DE winners are instance dependent: jSO-lite leads low-noise oracle search, iL-SHADE high-noise oracle search, and L-SRTDE high-noise selected solutions in the equal-instance summaries. Bootstrap analysis quantifies a non-negligible high-noise search--selection gap, and a retrospective fixed-budget verification proxy shows that reserving a small measurement budget for final re-evaluation improves selected quality across all ten methods at 30\,000 FEs. A supplementary structure-aware study further shows that QAOA cross-depth restriction and continuous basin refinement are more useful in these conditions than standalone Monte Carlo tree-search selection. Overall, optimizer choice depends jointly on landscape structure, observation noise, and final-point identification.
Variational quantum algorithms (VQAs) have demonstrated considerable potential in solving NP-hard combinatorial problems in the contemporary near intermediate-scale quantum (NISQ) era. The quantum approximate optimisation algorithm (QAOA) is one such algorithm, used in solving the maximum cut (Max-Cut) problem for a given graph by successive implementation of LLL quantum circuit layers within a corresponding Trotterised ansatz. The challenge of exploring the cost function of VQAs arising from an exponential proliferation of local minima with increasing circuit depth has been well-documented. However, fewer studies have investigated the impact of circuit depth on QAOA performance in finding the correct Max-Cut solution. Here, we employ basin-hopping global optimisation methods to navigate the energy landscapes for QAOA ansätze for various graphs, and analyse QAOA performance in finding the correct Max-Cut solution. The structure of the solution space is also investigated using discrete path sampling to build databases of local minima and the transition states that connect them, providing insightful visualisations using disconnectivity graphs. We find that the corresponding landscapes generally have a single funnel organisation, which makes it relatively straightforward to locate low-lying minima with good Max-Cut solution probabilities. In some cases below the adiabatic limit the second lowest local minimum may even yield a higher solution probability than the global minimum. This important observation has motivated us to develop broader metrics in evaluating QAOA performance, based on collections of minima obtained from basin-hopping global optimisation. Hence we establish expectation thresholds in elucidating useful solution probabilities from local minima, an approach that may provide significant gains in elucidating reasonable solution probabilities from local minima.
We benchmark eight classical optimizers for exact-statevector VQE calculations on a controlled hierarchy of frustrated spin models, ranging from a diagonal Ising glass to transverse-field Ising and anisotropic Heisenberg models. The benchmark includes local, stochastic-gradient, evolutionary, covariance-adaptation, and swarm-based optimization methods under matched function-evaluation budgets. To understand their performance beyond final energies, we characterize the underlying Hamiltonian--ansatz landscapes in terms of local minima, gradients, curvature, and ground-state reachability. We use simple variational circuits, from an RyR_yRy​ product-state ansatz for the diagonal model to shallow RyR_yRy​--CNOT hardware-efficient circuits for the noncommuting models, and study how increasing circuit depth changes their expressivity, reachability, and optimization geometry. We find that optimizer performance changes substantially across the model hierarchy and is closely connected to landscape structure, while the variational gap represents a separate source of error. These results show how classical optimization, variational expressivity, and landscape geometry jointly determine VQE performance for frustrated spin models.