Comparative Analysis of Classical and Quantum Runge-Kutta Methods in Classification Tasks

Deep learning models have driven remarkable progress across image classification, natural language processing, and generative modelling. Yet their reliance on millions of trainable parameters brings substantial computational and environmental costs. This paper investigates whether Ordinary Differential Equation inspired Transformer architectures, built on first through fourth order Runge-Kutta numerical solvers, can reduce parameter counts while preserving classification quality. I implement both classical and quantum variants of these ODE-Transformer blocks inside Vision Transformer encoders and evaluate them on MNIST digit classification, CIFAR-10 object classification, and IMDb sentiment analysis. Classical fourth order Runge-Kutta blocks achieve 95 percent accuracy on MNIST and 42 percent on CIFAR-10, outperforming lower order methods by a small margin while training time grows with method complexity. An optimized RK4 variant that removes intermediate stages converges faster and uses fewer parameters without sacrificing performance. Quantum implementations match 91 percent accuracy on MNIST with roughly 40 percent fewer trainable parameters, though they require significantly more wall-clock training time due to circuit simulation overhead. On IMDb, the quantum enhanced model matches the classical 85 percent accuracy while halving the parameter count. Finally, a Quantum Runge-Kutta Transformer GAN called QRKT-GAN achieves an Inception Score of 74.89 and a Frechet Inception Distance of 6.78 on CIFAR-10, substantially outperforming the classical TransGAN baseline. These results suggest that quantum ODE-Transformer hybrids offer a viable path toward more parameter efficient deep learning, with practical advantages likely to grow as fault tolerant quantum hardware matures.