Superpositional Gradient Descent: Harnessing Quantum Principles for Model Training

TL;DR

Problem: classical optimizers struggle to explore high-dimensional, non-convex loss landscapes in large language model training. Approach: Superpositional Gradient Descent fuses momentum-based updates with quantum-inspired perturbations and a quantum attention mechanism within a hybrid PyTorch–Qiskit workflow, formalized by $\theta_{t+1} = \theta_t - \alpha\left( \frac{m_t}{\sqrt{v_t} + \epsilon} + \lambda \cdot \mathcal{Q}(\theta_t, \nabla_{\theta_t} L) \right)$ and an oscillatory perturbation $\mathcal{Q}(\theta, \nabla L)$. Key contributions: a sine-based perturbation design, a quantum transformer attention substitute, and empirical results on synthetic sequence classification and GSM8K fine-tuning showing faster convergence and lower final losses. Significance: demonstrates a viable pathway for leveraging quantum principles to enhance deep learning optimization, with practical insights into hardware and scalability limitations for future real-quantum implementations.

Abstract

Large language models (LLMs) are increasingly trained with classical optimization techniques like AdamW to improve convergence and generalization. However, the mechanisms by which quantum-inspired methods enhance classical training remain underexplored. We introduce Superpositional Gradient Descent (SGD), a novel optimizer linking gradient updates with quantum superposition by injecting quantum circuit perturbations. We present a mathematical framework and implement hybrid quantum-classical circuits in PyTorch and Qiskit. On synthetic sequence classification and large-scale LLM fine-tuning, SGD converges faster and yields lower final loss than AdamW. Despite promising results, scalability and hardware constraints limit adoption. Overall, this work provides new insights into the intersection of quantum computing and deep learning, suggesting practical pathways for leveraging quantum principles to control and enhance model behavior.

Figures (2)

• Figure 1: Learning curves for text classification task. Superpositional Gradient Descent achieves faster convergence and higher final accuracy.
• Figure 2: LLM fine-tuning loss curves on GSM8K. Superpositional Gradient Descent achieves lower loss and more stable convergence.