Q-Newton: Hybrid Quantum-Classical Scheduling for Accelerating Neural Network Training with Newton's Gradient Descent

TL;DR

This paper tackles the heavy computational cost of second-order optimization in neural networks by introducing Q-Newton, a hybrid quantum-classical scheduler that dynamically routes Hessian inversions between quantum and classical solvers based on real-time conditioning and sparsity. It combines Hessian estimation, pruning, and adaptive regularization to make Hessian inversions more quantum-friendly, enabling selective offloading to quantum linear solvers (QLSAs) where advantageous. Empirical results across DNNs, BERT, and GPT demonstrate up to $90\%$ reductions in training time compared to classical Newton's GD without sacrificing accuracy, with Hessians progressively becoming more favorable for quantum acceleration during training. This work highlights a practical quantum-classical co-design pathway for accelerating ML training and lays groundwork for applying hybrid strategies to other matrix-intensive ML operations, illustrating how quantum advantages can be realized in the near term through selective offloading rather than wholesale replacement of classical methods.

Abstract

Optimization techniques in deep learning are predominantly led by first-order gradient methodologies, such as SGD. However, neural network training can greatly benefit from the rapid convergence characteristics of second-order optimization. Newton's GD stands out in this category, by rescaling the gradient using the inverse Hessian. Nevertheless, one of its major bottlenecks is matrix inversion, which is notably time-consuming in $O(N^3)$ time with weak scalability. Matrix inversion can be translated into solving a series of linear equations. Given that quantum linear solver algorithms (QLSAs), leveraging the principles of quantum superposition and entanglement, can operate within a $\text{polylog}(N)$ time frame, they present a promising approach with exponential acceleration. Specifically, one of the most recent QLSAs demonstrates a complexity scaling of $O(d\cdotκ\log(N\cdotκ/ε))$, depending on: {size~$N$, condition number~$κ$, error tolerance~$ε$, quantum oracle sparsity~$d$} of the matrix. However, this also implies that their potential exponential advantage may be hindered by certain properties (i.e. $κ$ and $d$). We propose Q-Newton, a hybrid quantum-classical scheduler for accelerating neural network training with Newton's GD. Q-Newton utilizes a streamlined scheduling module that coordinates between quantum and classical linear solvers, by estimating & reducing $κ$ and constructing $d$ for the quantum solver. Our evaluation showcases the potential for Q-Newton to significantly reduce the total training time compared to commonly used optimizers like SGD. We hypothesize a future scenario where the gate time of quantum machines is reduced, possibly realized by attoseconds physics. Our evaluation establishes an ambitious and promising target for the evolution of quantum computing.

Figures (5)

• Figure 1: Overview. We propose the hybrid quantum-classical solver of Newton's gradient descent as Q-Newton, a general neural network training framework. Starting from Newton's gradient descent problem, we develop solutions via both classical and quantum solvers, integrating them through our hybrid approach. We evaluate Q-Newton along five dimensions: convergence improvements over first-order methods, comparative analysis between pure quantum/classical and hybrid implementations, Hessian matrix dynamics throughout training, computational efficiency across different neural architectures, and the effects of Hessian pruning and regularization techniques. Through these investigations, Q-Newton demonstrates significant training acceleration capabilities by dynamically leveraging the strengths of quantum and classical computation for matrix inversion operations.
• Figure 2: Performance evaluation and ablation study of Q-Newton. (a) Overview of neural network models used for evaluation, showing architecture dimensions, layer counts, and parameter sizes. (b) Impact of quantum oracle sparsity on model accuracy, demonstrating minimal accuracy drop ($<0.5\%)$ with up to $65\%$ sparsification. (c) Comparison of computational costs between quantum and classical solvers across training steps, showing the dynamic scheduling decisions of Q-Newton with quantum advantage emerging after step $4$. (d) Relationship between regularization coefficient, model accuracy (red line), and Hessian condition number (blue bars), highlighting the optimal regularization range for maintaining accuracy while improving matrix conditioning. (e) Convergence comparison between Newton's gradient descent and SGD on BERT and GPT pre-training tasks, demonstrating $56\%$ and $66\%$ training step reduction, respectively. (f) Runtime analysis of quantum solver with varying sparsity levels against classical solver (dashed line) as condition number increases, showing crossover points where quantum advantage emerges. (g) Comparative accuracy evaluation of Q-Newton against alternative pruning strategies across different sparsity levels, showing Q-Newton's superior preservation of model performance at high sparsity levels.
• Figure 3: Distribution of $p\%$ quantum sparsity for Hessian matrix at training steps $0$ (up) and $10$ (down). $p\%$-sparsity represents the minimum number of elements in a row summing to $p\%$ of its total absolute magnitude. Data from DNN training on MNIST (Hessian size: $12544\times 12544$), showing Hessian magnitude distribution evolution. We evaluate the $p\%$ across $\{50\%, 75\%, 90\%, 95\%\}$.
• Figure 4: Overview of Q-Newton. Both purely classical and quantum Newton's GD are unscalable and impractical. Purely classical Newton's GD calculates the Hessian inversion via LU decomposition etc., with a time complexity of $\mathcal{O}(N^3)$. Quantum Newton's GD requires substantial time when the Hessian is either ill-conditioned or non-sparse. Hybrid classical-quantum Newton's GD adaptively schedules between classical and quantum inversion solvers, and incorporates pruning and regularization techniques, making it scalable and practical.
• Figure 5: Schematic of the QLSA circuit flowing from left to right.