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Quantum Machine Intelligence

A cross-domain empirical benchmark of quantum-inspired and classical optimization algorithms for machine learning

Görkem Yılmaz

Published Sep 11, 2026• 0 citations
Publisher

Abstract

Abstract Quantum-inspired (QI) optimizers are increasingly proposed as alternatives to classical gradient-based training, yet are frequently benchmarked through classical stand-ins rather than themselves. Across tabular, vision, and language tasks we evaluate three classes: direct implementations of published update rules (Spall’s SPSA; quantum-behaved particle swarm optimization, QPSO, adapted to stochastic minibatches), a documented classical analog of quantum natural gradient (an NGD-style diagonal second-moment preconditioner), and the hyperparameter-control configurations under which such methods are often reported. The gap between these classes is our central result: on the tabular benchmark the hyperparameter control trains with the classical baselines while the direct algorithms fall $$\approx $$ 10–24 points below it, and on CIFAR-10/ResNet18, where v1’s “SPSA” arm was a learning-rate-scaled Adam, Adam itself stands $$\approx $$ 59 points above genuine SPSA. The QPSO adaptation stays at chance on LSTM tasks a legacy “QPSO” proxy solves at 79–81%, at 18–38 $$\times $$ Adam’s cost there and up to 74 $$\times $$ on the small CV model. A seed-paired $$2\times 2$$ factorial on RoBERTa identifies the smaller learning rate as the dominant observed component of the earlier “QNG advantage” ( $$+0.50$$ points at the dataset level); with only three datasets, modified Hartung–Knapp uncertainty spans zero. The configured NGD arm shows no detectable advantage over a learning-rate-matched AdamW (descriptive pooled $$p{=}0.23$$ ) at 6– $$7\%$$ extra cost, and on ResNet18 its trained endpoints have larger dominant-curvature estimates than the Adam family’s. Headline comparisons use five seeds, paired tests, Holm correction, and stated detection limits. We conclude with an auditing taxonomy (direct algorithm / analog / hyperparameter control): within the arms, protocols, and scales evaluated, no advantage attributable to a QI mechanism is detected after these controls.

DOI
10.1007/s42484-026-00444-y

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Abstract Quantum-inspired (QI) optimizers are increasingly proposed as alternatives to classical gradient-based training, yet are frequently benchmarked through classical stand-ins rather than themselves. Across tabular, vision, and language tasks we evaluate three classes: direct implementations of published update rules (Spall’s SPSA; quantum-behaved particle swarm optimization, QPSO, adapted to stochastic minibatches), a documented classical analog of quantum natural gradient (an NGD-style diagonal second-moment preconditioner), and the hyperparameter-control configurations under which such methods are often reported. The gap between these classes is our central result: on the tabular benchmark the hyperparameter control trains with the classical baselines while the direct algorithms fall ≈\approx ≈ 10–24 points below it, and on CIFAR-10/ResNet18, where v1’s “SPSA” arm was a learning-rate-scaled Adam, Adam itself stands ≈\approx ≈ 59 points above genuine SPSA. The QPSO adaptation stays at chance on LSTM tasks a legacy “QPSO” proxy solves at 79–81%, at 18–38 ×\times × Adam’s cost there and up to 74 ×\times × on the small CV model. A seed-paired 2×22\times 22×2 factorial on RoBERTa identifies the smaller learning rate as the dominant observed component of the earlier “QNG advantage” ( +0.50+0.50+0.50 points at the dataset level); with only three datasets, modified Hartung–Knapp uncertainty spans zero. The configured NGD arm shows no detectable advantage over a learning-rate-matched AdamW (descriptive pooled p=0.23p{=}0.23p=0.23 ) at 6– 7%7\%7% extra cost, and on ResNet18 its trained endpoints have larger dominant-curvature estimates than the Adam family’s. Headline comparisons use five seeds, paired tests, Holm correction, and stated detection limits. We conclude with an auditing taxonomy (direct algorithm / analog / hyperparameter control): within the arms, protocols, and scales evaluated, no advantage attributable to a QI mechanism is detected after these controls.
Quantum Machine Learning is gaining traction by leveraging quantum advantage to outperform classical Machine Learning. Many classical and quantum optimizers have been proposed to train Parameterized Quantum Circuits in the simulation environment, achieving high accuracy and fast convergence speed. However, to the best of our knowledge, currently there is no related work investigating these optimizers on multiple algorithms, which may lead to the selection of suboptimal optimizers. In this paper, we first benchmark the most popular classical and quantum optimizers, such as Gradient Descent (GD), Adaptive Moment Estimation (Adam), and Quantum Natural Gradient Descent (QNG), through the Quantum Compilation algorithm. Evaluated metrics include the lowest cost value and the wall time. The results indicate that Adam outperforms other optimizers in terms of convergence speed, cost value, and stability. Furthermore, we conduct additional experiments on multiple algorithms with Adam variants, demonstrating that the choice of hyperparameters significantly impacts the optimizer's performance.
Training variational quantum models requires choosing between parameter-shift gradients, which are exact but cost O(P)O(P)O(P) forward evaluations, and simultaneous perturbation stochastic approximation (SPSA), which uses only two samples but produces high-variance estimates that can degrade optimisation on small supervised tasks. Whether the cheap gradient is usable depends on the variance that results from different choices of the SPSA perturbation scale, learning rate, and gain-decay schedule. We varied those quantities across a broad grid on a 6-qubit, 60-parameter QNLI classifier and compared the best configurations to parameter-shift AdamW and BuresQNG. AdamW-style SPSA with c0=0.01c_0=0.01c0​=0.01, η=0.10η=0.10η=0.10, γ=0.10γ=0.10γ=0.10 reached 55%±11%55\% \pm 11\%55%±11% test accuracy, improving over the default configuration (49%±6%49\% \pm 6\%49%±6%) but remaining 16-19 percentage points below the parameter-shift baselines because the two-sample SPSA gradient estimate has too much variance for reliable optimisation of 60 parameters in 40 epochs. Classical-gain SPSA and Bures-preconditioned SPSA performed worse, at 51%51\%51% and 46%46\%46% respectively. Bures-preconditioning a noisy two-sample SPSA gradient amplifies perturbation noise.
Abstract Hybrid quantum machine learning (QML) relies on quantum variational circuits (QVCs) trained together with classical components. While gradient-based training is standard, it is costly on near-term hardware and vulnerable to barren plateaus, motivating gradient-free alternatives. However, there is limited systematic evidence comparing gradient-free methods to strong gradient baselines under matched quantum-evaluation budgets. A benchmark of 15 gradient-based, gradient-free, and hybrid optimizers is conducted on four binary classification datasets and two three-qubit QVC architectures using a unified experimental protocol under a fixed per-step quantum-evaluation budget. Across tasks, Adam is the strongest gradient-based method. Among the purely gradient-free schemes, Asexual Reproduction Optimization is the strongest optimizer; tournament-selection and elite-pool genetic algorithms (GA-Tourn, GA-Elite) also frequently match the best gradient-based configurations. Among the hybrid schemes, Gradient Lexicase Selection with Adam (GLS- Adam ) performs comparably. In contrast, the two gradient-free methods big bang–big crunch and a cross-entropy method variant often fail to improve beyond chance. Additional sensitivity analyses show that circuit depth can change the preferred optimizer, whereas finite-shot sampling budget and post-training backend-specific noise do not materially change the main conclusions. Overall, gradient-based methods remain robust defaults for hybrid QML, while a small subset of gradient-free and hybrid approaches provides viable alternatives under realistic quantum-evaluation constraints.
Adaptive quantum machine learning (QML) represents an exciting frontier at the intersection of quantum computing and classical bio-inspired optimization. The constraints of noisy intermediate-scale quantum (NISQ) devices—limited qubit counts, significant decoherence, and measurement uncertainties—pose formidable challenges to training parameterized quantum circuits (PQCs) using traditional gradient-based methods. To overcome these hurdles, we propose a novel Adaptive PSO–QML framework that integrates a particle swarm optimization (PSO)–inspired algorithm with dynamic hyperparameter adjustment, enabling real-time tuning of PQC parameters based on performance feedback. Unlike static optimizers, our approach continuously adapts its inertia weight and cognitive/social coefficients to balance exploration and exploitation, responding to stagnation or rapid improvement in measured fitness. We evaluate this hybrid algorithm on two benchmark classification tasks: the classical Iris dataset encoded into a 2-qubit circuit and a quantum-encoded subset of the MNIST dataset on 4 qubits. Our experiments, conducted on the Qiskit Aer simulator under realistic noise models (depolarizing, amplitude damping, and mixed), demonstrate that Adaptive PSO–QML achieves an average accuracy improvement of 12% over fixed-parameter PSO and 15% over Adam-based gradient descent, with p-values below 0.01 across paired t-tests. Statistical analysis summarized in Table 1 confirms the significance of these gains. Detailed simulation research further reveals that the adaptive mechanism not only accelerates convergence by approximately 30%—reducing total circuit evaluations—but also exhibits robust performance across varying circuit depths (1–3 entangling layers) and noise intensities. Our findings suggest that bio-inspired adaptation is a powerful tool for mitigating barren plateaus and noise effects in NISQ-era QML, paving the way for scalable quantum advantage in near-term hardware.
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This study explores quantum-inspired optimization algorithms (QIOAs) as practical tools for solving constrained combinatorial problems on classical machines. We analyze tasks that can be formulated in Ising or QUBO forms and benchmark three representative approaches: Simulated Bifurcation (SB), Quantum-Inspired Particle Swarm Optimization (QPSO), and a tensor-network (TN) method. A unified evaluation protocol is introduced that captures solution quality, feasibility, runtime, energy consumption, parameter sensitivity, and scalability. To ensure reliable performance, we develop automated penalty-adjustment and schedule-calibration strategies. Experimental results across diverse benchmarks show that SB and TN together yield the best trade-offs between accuracy and runtime: SB is efficient on dense instances with moderate penalties and low energy use, while TN is advantageous for sparse or modular networks, especially on GPUs. QPSO is competitive in smoother landscapes but more sensitive to hyperparameters. By linking performance to structural properties such as spectral gaps and constraint activity, we provide guidance for selecting suitable algorithms. Overall, QIOAs emerge as effective intermediaries between existing classical heuristics and emerging quantum devices, supported by a reproducible, energy-aware evaluation framework.