🤖 AI Summary
This study systematically evaluates the performance of classical and quantum models sharing an $\mathrm{SU}(2)$ geometric structure on visual recognition tasks, investigating whether shallow variational quantum circuits offer practical advantages. Building upon frozen features from a pretrained ResNet18 backbone, the authors compare real-valued, quaternion-based, and variational quantum classifiers—with and without entanglement—across MNIST, FashionMNIST, and CIFAR-10, optimizing all models using Fubini–Study natural gradients. The results demonstrate that quaternion networks match or closely approach real-valued baselines while significantly outperforming quantum counterparts; entanglement yields only marginal gains on grayscale images and degrades performance when applied to pretrained features. This work provides the first evidence that merely sharing an $\mathrm{SU}(2)$ structure is insufficient to confer a quantum advantage in such settings.
📝 Abstract
Quaternion-valued neural networks and variational quantum circuits (VQCs) both derive local transformations from $\mathrm{SU}(2)$ geometry, yet their performance on classical supervised learning remains poorly understood. We compare real-valued, quaternion-valued, and quantum classification heads on identical frozen features across MNIST, FashionMNIST, and CIFAR-10. CIFAR-10 uses a learned 16-dimensional bottleneck and frozen ImageNet-pretrained ResNet18 features to separate architecture from representation quality. Quaternion classifiers match or approach real-valued baselines while outperforming shallow VQCs. On MNIST and FashionMNIST, quaternion networks nearly equal real-valued MLPs, whereas product-state VQCs show lower accuracy and higher cost. On CIFAR-10, quaternion networks retain 94--97% of real-valued performance and remain stable under a 32-fold increase in dimensionality. Product-state circuits underperform quaternion classifiers, while entanglement gives modest grayscale gains but reverses under pretrained CNN features (9.25 pp degradation vs.\ product-state). Fubini--Study/QFI natural gradients improve geometric alignment but not short-horizon loss reduction vs.\ Adam. A Friedman test on five-seed MNIST detects model differences ($χ^2=12.796$, $p=0.0051$, $n=5$), with Wilcoxon tests yielding large effect sizes ($d>5$) for QuatNet vs.\ quantum comparisons. For FashionMNIST and CIFAR-10, large effects ($d>2.0$) are the primary statistic given $n=3$. These results indicate that quaternion networks provide efficient, stable $\mathrm{SU}(2)$ alternatives to shallow VQCs on tasks lacking intrinsic quantum structure. Shared local $\mathrm{SU}(2)$ geometry and shallow entanglement are insufficient, within the regime studied, to confer practical quantum advantage. Conclusions are limited to shallow, measurement-limited circuits on such tasks.