Classical \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SU}(2)$$\end{document} models match or exceed shallo

Abstract

Quaternion-valued neural networks and variational quantum circuits (VQCs) both derive local transformations from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\textrm{SU}(2)}$$\end{document} geometry, yet their relative performance on classical supervised learning tasks remains poorly understood. We present a controlled comparison in which real-valued, quaternion-valued, and quantum classification heads operate on identical frozen feature representations across MNIST, FashionMNIST, and CIFAR-10. CIFAR-10 experiments include both a learned 16-dimensional bottleneck and frozen ImageNet-pretrained ResNet18 features to separate architectural effects from representation quality. Quaternion classifiers consistently match or closely approach real-valued baselines while substantially outperforming shallow VQCs. On MNIST and FashionMNIST, quaternion networks achieve near-equivalence with real-valued multilayer perceptrons, whereas product-state VQCs exhibit lower accuracy and substantially higher computational cost. On CIFAR-10, quaternion networks retain 94–97% of real-valued performance across both feature regimes and remain stable under a 32-fold increase in feature dimensionality. Product-state quantum circuits underperform quaternion classifiers across all benchmarks, while entanglement provides only modest gains on grayscale datasets and reverses under pretrained CNN features, corresponding to a 9.25 percentage-point degradation relative to the product-state circuit. Optimization diagnostics show that Fubini–Study / quantum Fisher information natural-gradient methods improve geometric alignment but do not materially improve short-horizon loss reduction relative to Adam. A Friedman test on the five-seed MNIST evaluation provides evidence of a non-random model ordering (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\chi }^{\varvec{2}} \varvec{=} \varvec{12.796}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{p = 0.0051}$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{n = 5}$$\end{document}), with post-hoc Wilcoxon signed-rank tests yielding large effect sizes (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{d > 5}$$\end{document}) for all QuatNet vs. quantum comparisons on MNIST. For FashionMNIST and CIFAR-10, large effect sizes (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{d > 2.0}$$\end{document}) serve as the primary inferential statistic given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{n = 3}$$\end{document}. These results indicate that, on classical vision benchmarks lacking intrinsic quantum structure, quaternion networks provide efficient and stable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\textrm{SU}(2)}$$\end{document} alternatives to shallow variational quantum circuits. The findings suggest that shared local \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\textrm{SU}(2)}$$\end{document} geometry and shallow entanglement are insufficient, within the shallow circuit regime studied here, to confer practical quantum advantage on classical image-classification tasks. The conclusions are bounded to shallow, measurement-limited variational circuits operating on classical image-classification tasks without intrinsic quantum structure.