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Superpositional Gradient Descent: Harnessing Quantum Principles for Model Training

Ahmet Erdem Pamuk, Emir Kaan Özdemir, Şuayp Talha Kocabay

arXiv:2511.01918v2cs.LGquant-ph

TL;DR

Classical gradient methods face challenges navigating the high-dimensional, non-convex landscapes of transformer training. The paper introduces Superpositional Gradient Descent, which adds quantum-inspired perturbations to gradient updates, and reports faster convergence and improved performance against classical baselines.

  • Problem

    Classical optimization can struggle with high-dimensional, non-convex transformer loss landscapes, including poor local minima and slow convergence [1],.

    Source: §I · p. 1

  • Method

    Superpositional Gradient Descent incorporates quantum-inspired perturbations into classical gradient-based optimization to explore parameter configurations.

    Source: §I · p. 1

  • Results

    Superpositional Gradient Descent variants outperform AdamW in convergence speed and final loss, while λ = 0.5 reaches 93.8% ± 0.7% final accuracy, 2.3 percentage points above standard Adam.

    Source: B. Results · p. 3

  • Takeaways & Limitations

    The findings suggest quantum-inspired perturbations may help escape poor local minima by exploring multiple parameter configurations.

    Source: §V · p. 4

Abstract

from arXiv · show

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.

I. INTRODUCTION

The introduction frames difficult optimization in high-dimensional neural-network landscapes as a training challenge and proposes quantum-inspired perturbations as a way to explore them. It presents Superpositional Gradient Descent as a hybrid approach intended to improve convergence and model performance.

I. INTRODUCTION

  • Classical gradient methods can require many iterations or become trapped in poor local minima in the high-dimensional, non-convex landscapes of transformer training [1],.The introduction identifies effective exploration of the parameter space as a central challenge.Source: §I · p. 1
  • Quantum superposition motivates exploring multiple parameter configurations to help gradient descent escape poor local minima and find better solutions.The paper presents this as a hypothesis for improving exploration, not as a demonstrated mechanism.Source: §I · p. 1
  • Superpositional Gradient Descent adds quantum-inspired perturbations to classical gradient updates and is reported to improve convergence speed and final performance across tasks.The paper also points to hybrid quantum-classical tools for implementation without requiring quantum hardware.Source: §I · p. 1

A. Gradient Descent in Large Language Models

The background reviews classical gradient methods for LLM training and quantum optimization concepts that inform the paper’s approach.

A. Gradient Descent in Large Language Models

  • Gradient descent iteratively updates parameters to minimize loss, while SGD, Adam, and AdamW are used for large-scale models with complex architectures [1],, [8], [9].Prior work also describes transformers as performing a form of metaoptimization during in-context learning [10].Source: A. Gradient Descent in Large Language… · pp. 1–2

B. Quantum Superposition in Optimization

  • Quantum superposition allows multiple states to coexist until measurement, a principle used by QAOA to explore solution spaces [4].QAOA initializes a superposition and applies parameterized gates to evolve the system toward an optimum.Source: B. Quantum Superposition in Optimization · p. 2
  • Quantum annealing uses quantum fluctuations to seek global minima in combinatorial optimization, and quantum techniques have inspired proposals for classical optimization –[14].Source: B. Quantum Superposition in Optimization · p. 2

III. SUPERPOSITIONAL GRADIENT DESCENT: A QUANTUM-INSPIRED OPTIMIZATION APPROACH

Superpositional Gradient Descent combines classical momentum-based optimization with quantum-inspired perturbations. Its sine-based perturbations vary with parameter values and are intended to alter gradient signals during exploration.

A. Mathematical Formulation

1) Rationale for Sine-Based Perturbations:

  • The perturbation function uses sinusoidal modulation, drawing on sine and cosine components associated with quantum wave functions.Source: 1) Rationale for Sine-Based… · p. 2
  • For i<nqubits, the perturbation is sin(π · θ_i) times the gradient; it is zero otherwise, so its sign can boost or dampen the gradient as θ_i changes.The paper says this wave-like modulation helps the optimizer escape shallow local minima.Source: 1) Rationale for Sine-Based… · p. 2

2) Hyperparameter Selection:

The selected hyperparameters balance convergence behavior, perturbation strength, computational cost, and simulation overhead.

2) Hyperparameter Selection:

  • The learning rate is set to 1×10−3 for text tasks and 2×10−5 for large-scale fine-tuning; λ controls perturbation strength, with 0.5 linked to faster convergence but slightly higher per-iteration variance.The paper describes λ = 0.1 as providing modest exploratory benefits with minimal noise.Source: 2) Hyperparameter Selection · p. 2
  • The qubit count determines how many parameters receive sinusoidal updates; using four limits quantum effects to a subset and reduces simulation overhead.Larger values extend exploration at increased computational cost.Source: 2) Hyperparameter Selection · p. 2
  • The optimizer retains Adam moments (β1 = 0.9, β2 = 0.999, ϵ = 1 × 10−8) and uses depth-2 Ry and Rz circuits to balance nonlinearity with simulation time.The paper cautions that very high λ can introduce excessive variance and destabilize training.Source: 2) Hyperparameter Selection · p. 2

B. Quantum Transformer Architecture

Quantum attention augments standard scaled dot-product attention with a quantum-circuit contribution. The circuit uses Hadamard gates, learned rotations, and entanglement operations.

B. Quantum Transformer Architecture

  • Quantum Attention augments standard scaled dot-product attention with a quantum circuit contribution, Φ(Q, K, C).The contribution is defined using amplitudes from circuit C operating on attention scores.Source: B. Quantum Transformer Architecture · pp. 2–3
  • For input sequences X ∈ R^(b×s×d), the quantum attention computation uses the circuit contribution in its attention calculation.The prose identifies ψ_k as the k-th circuit amplitude and h as the attention-head index.Source: B. Quantum Transformer Architecture · pp. 2–3
  • Circuit C applies Hadamard gates, learned Ry and Rz rotations, then CNOT entanglement between adjacent qubits.The CNOT operations are denoted U_entangle; the rotations use learned parameters θ_i and ϕ_i.Source: B. Quantum Transformer Architecture · p. 3

IV. IMPLEMENTATION AND EXPERIMENTAL RESULTS

The implementation extends Adam with quantum-inspired perturbations and evaluates the optimizer on text classification and LLM fine-tuning. It reports improved classification accuracy and convergence, alongside lower GSM8K fine-tuning loss than AdamW.

A. Experimental Setup

  • The PyTorch and Qiskit implementation extends Adam with quantum-inspired perturbations and evaluates Superpositional Gradient Descent on synthetic text classification and LLM fine-tuning tasks.The evaluated transformer used 64-dimensional embeddings, 4 attention heads, and 2 layers, with a 4-qubit circuit; λ = 0.5 was a key hyperparameter.Source: A. Experimental Setup · p. 3, B. Results · p. 3

B. Results

  • 93.8% ± 0.7% final accuracy, 2.3 percentage points above standard Adam; 90% accuracy arrived in 4.6 versus 7.4 epochs, a 37.8% reduction in training time.The reported result is for Superpositional Gradient Descent with λ = 0.5; Fig. 1 shows the text-classification learning curves.Source: Figure 1 · p. 3, B. Results · p. 3
  • On GSM8K fine-tuning, both configurations outperform AdamW in convergence speed and final loss; λ = 0.5 has the most favorable trajectory, indicating enhanced training stability.The evaluation uses Llama-3.2-1B-Instruct.Source: B. Results · p. 3
  • 4.11% and 4.16% lower mean loss than AdamW after one epoch for λ = 0.1 and λ = 0.5, respectively; the marginal gain suggests diminishing returns.Table I summarizes mean fine-tuning loss on GSM8K after one epoch.Source: Table I · p. 3, B. Results · p. 3

C. Computational Efficiency

Superpositional Gradient Descent reduces total time to reach 90% accuracy despite spending more time per epoch than Adam. On GSM8K fine-tuning, it also achieves lower loss and more stable convergence.

C. Computational Efficiency

  • Quantum circuit simulation accounts for approximately 35% more time per epoch than Adam.Source: C. Computational Efficiency · p. 3
  • GSM8K fine-tuning loss curves show lower loss and more stable convergence with Superpositional Gradient Descent.The comparison is shown in Figure 2.Source: Figure 2 · p. 4
  • 16% lower total time to reach 90% accuracy offsets Superpositional Gradient Descent’s higher per-epoch cost.The optimizer requires approximately 35% more time per epoch than Adam because of quantum circuit simulation.Source: C. Computational Efficiency · pp. 3–4

V. CONCLUSION

The conclusion presents Superpositional Gradient Descent as a quantum-inspired framework that improves classical gradient-based optimization. Its reported benefits are faster convergence and better final model performance, while scaling and real-processor implementations remain future-work targets.

V. CONCLUSION

  • Superpositional Gradient Descent enhances classical gradient-based optimization by incorporating quantum superposition principles.Source: §V · p. 4
  • The experiments report improvements in convergence speed and final model performance across various tasks.Source: §V · p. 4
  • Quantum-inspired perturbations may help escape poor local minima by exploring multiple parameter configurations simultaneously.Source: §V · p. 4
  • Future work targets scaling to larger models, more sophisticated circuit designs, and implementations for real quantum processors.Source: §V · p. 4

APPENDIX

The appendix outlines implementation and experimental setup for text classification and LLM fine-tuning. It describes a depth-2 quantum circuit for text classification and mixed-precision training with a warmup-and-decay schedule for fine-tuning.

APPENDIX

  • The appendix provides implementation and experimental setup details for text classification and fine-tuning.Source: APPENDIX · p. 5

A. Text Classification Setup

  • Text classification uses PyTorch and Qiskit, with its model architecture and training configuration listed in Table II.The quantum circuit applies Ry and Rz rotation gates after Hadamard gates, then uses adjacent-qubit CNOT gates; its depth is 2.Source: A. Text Classification Setup · p. 5, Table II · p. 5

B. LLM Fine-tuning Setup

  • LLM fine-tuning uses GSM8K with Llama-3.2-1B-Instruct, with configuration details listed in Table III.Mixed-precision FP16 reduces memory use and accelerates training; warmup followed by linear decay helps stabilize training.Source: B. LLM Fine-tuning Setup · p. 5, Table III · p. 5

C. Implementation Details

The implementation combines a parameterized quantum circuit, a quantum-inspired optimizer, and a quantum-enhanced transformer with separate training scripts. Circuit simulation uses Qiskit statevector and GPU-accelerated Aer simulators.

C. Implementation Details

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