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arXiv:2609.23180

Adaptive Differential Evolution and Multistart Search for Noisy QAOA Optimization

Vojtěch Novák, Ivan Zelinka, Swagatam Das, Martin Beseda

quant-phcs.NESubmitted Sep 19, 2026• Announced Sep 22, 2026• 0 citations
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Abstract

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.

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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 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.
Efficient Job-Shop Scheduling (JSS) is still a challenging combinatorial optimisation problem in engineering systems because of machine-capacity constraints, operation precedence requirements and the exponential growth of the search space for the scheduling problem. The Quantum Approximate Optimization Algorithm (QAOA) has been shown to be a promising approach to solving these optimisation problems, however, when using realistic quantum noise models, the scheduling quality and reliability of the solution can be compromised. This work presents a noise-adaptive QAOA framework that automatically scales the number of shots measured, the classical optimiser, and the circuit depth, informed by an estimated quantum-noise level, and adaptively uses different error mitigation strategies. The scheduling problem is then formulated as a Quadratic Unconstrained Binary Optimisation (QUBO) problem and tested with small-scale engineering benchmark instances, under various simulated noise circumstances. Comparative simulation results show that the proposed adaptive strategy is able to keep the optimality gap low and feasible-solution rate high in every instance, and achieve competitive scheduling performance as compared to a fixed-depth noisy QAOA. The study shows how the adaptive selection of quantum parameters can enhance the robustness of QAOA for small systems of engineering scheduling problems in the presence of noisy quantum devices.
The optimization of Variational Quantum Eigensolver is severely challenged by finite-shot sampling noise, which distorts the cost landscape, creates false variational minima, and induces statistical bias called winner's curse. We investigate this phenomenon by benchmarking eight classical optimizers spanning gradient-based, gradient-free, and metaheuristic methods on quantum chemistry Hamiltonians H2_22​, H4_44​ chain, LiH (in both full and active spaces) using the truncated Variational Hamiltonian Ansatz. We analyze difficulties of gradient-based methods (e.g., SLSQP, BFGS) in noisy regimes, where they diverge or stagnate. We show that the bias of estimator can be corrected by tracking the \textit{population mean}, rather than the biased best individual when using population based optimizer. Our findings, which are shown to generalize to hardware-efficient circuits and condensed matter models, identify adaptive metaheuristics (specifically CMA-ES and iL-SHADE) as the most effective and resilient strategies. We conclude by presenting a set of practical guidelines for reliable VQE optimization under noise, centering on the co-design of physically motivated ansatz and the use of adaptive optimizers.
The Quantum Approximate Optimisation Algorithm (QAOA) has emerged as a leading candidate for near-term quantum advantage in combinatorial optimisation, but most evaluations remain confined to synthetic benchmarks such as random MaxCut instances. Translating QAOA to real-world problem instances -- with irregular graph structures, heterogeneous constraint types, and practical scale requirements -- introduces additional challenges including QUBO formulation overhead, hardware embedding costs, and the performance gap between QAOA on structured real instances vs. random benchmarks. This paper presents the Real-World QAOA (RW-QAOA) framework, a systematic methodology for applying QAOA to four real-world application domains: supply chain network optimisation, financial portfolio rebalancing, telecommunications frequency assignment, and traffic signal timing optimisation. RW-QAOA introduces three methodological contributions: a domain-adaptive QUBO formulator (DAQF) encoding real-world constraints as penalised QUBO terms with penalty coefficient auto-tuning; a warm-start initialisation strategy (WSIS) using classical heuristic solutions to seed QAOA parameter search; and a layered noise adaptation protocol (LNAP) adjusting circuit depth based on real-time device calibration data. RW-QAOA is evaluated on real problem instances from industry partners across all four domains on IBM Quantum Eagle (127-qubit). RW-QAOA achieves solution quality within 4.2% of classical exact solvers across all four domains, outperforming classical greedy heuristics by 12.4% (SD = 3.8%) on average. WSIS reduces QAOA parameter convergence iterations by 48.4%. LNAP improves approximation ratio by 8.4pp vs. fixed-depth QAOA under device noise variation.
Abstract Optimization of precision agriculture, a combinatorial optimization problem that has both dependent and independent binary variables associated with selecting the one-to-one mapping of crop planting, irrigation, fertilization, and land management practices will scale poorly as farm size increases. The recently proposed standard Quantum Approximate Optimization Algorithm (QAOA) falls into the category of hybrid quantum classically based optimization but has slow convergence rates and has been shown to become trapped in local minima through two p independent, static optimization variables with no run-time guidance. To address this, we propose an Adaptive Quantum Approximate Optimization Algorithm (A-QAOA) that implements a feedback-driven, layer-wise parameter scheduling approach based upon two adaptive runtime metrics of cost function change (C) and solution variance (Var(C)). A-QAOA improves convergence rates without compromising the optimum solution by reducing the classical optimizer's search space from O(p) — representing 2p independent variables — to just 2 adaptive base variables γ₀ and β₀. A-QAOA will provide a faster convergence rate (72.5%) and shorter runtime (72.1%) compared to the standard QAOA when tested within Qiskit Aer using the COBYLA optimization algorithm across five independent graph instances (10 trials/graph for 50 total trials per algorithm). Specifically, the A-QAOA converged 72.5% faster compared with the standard QAOA (95.4→26.2 iterations; t = 78.53, p < 0.0001, Cohen's d = 15.7) and exhibited a run-time reduction of 72.1% (30.0s→8.4s; t = 28.05, p < 0.0001, d = 5.6).To our knowledge, this is the first work utilizing runtime feedback based adaptive QAOA parameter scheduling methods for resource allocation in the field of precision agriculture and for demonstrating benchmark quantum-enhanced precision agriculture site optimization using NISQ (Noisy Intermediate-Scale Quantum) technology.