Efficient and Accurate Estimation of Lipschitz Constants for Hybrid Quantum-Classical Decision Models

TL;DR

This work tackles the challenge of controlling sensitivity in hybrid quantum-classical decision models by developing a unified semidefinite programming framework to bound the Lipschitz constant $K^*$ across both classical and quantum components. It extends classical LipSDP techniques to quantum variational circuits and derives a convex program for the quantum part, then couples these into a joint optimization to bound the overall network Lipschitz constant in hybrid architectures. The approach yields tight, computationally efficient guarantees that support robustness, generalization, and fairness in quantum-enhanced decisions, demonstrated through Iris dataset experiments with Lipschitz-regularized training and PGD. The framework provides a principled, scalable path for robust quantum-classical learning, with potential extensions to deeper models and dynamic Lipschitz-constrained training regimes.

Abstract

In this paper, we propose a novel framework for efficiently and accurately estimating Lipschitz constants in hybrid quantum-classical decision models. Our approach integrates classical neural network with quantum variational circuits to address critical issues in learning theory such as fairness verification, robust training, and generalization. By a unified convex optimization formulation, we extend existing classical methods to capture the interplay between classical and quantum layers. This integrated strategy not only provide a tight bound on the Lipschitz constant but also improves computational efficiency with respect to the previous methods.

Figures (3)

• Figure 1: Evolution of the Lipschitz constant during training epochs, comparing three loss metrics ($\ell_1$, $\ell_2$, and $\ell_{\infty}$). The left plot illustrates training on the original Iris dataset, demonstrating divergent behavior dependent upon the selected norm, indicative of varying sensitivities and stability of convergence. The right plot shows training outcomes when labels are uniformly altered to a single class, highlighting the model's inherent stabilization toward minimal sensitivity across epochs. Regularization parameter is set to zero, isolating the direct impact of the chosen metric.
• Figure 2: Analysis of the Lipschitz constant as a function of the regularization parameter, The left plot shows a pronounced decrease in the Lipschitz constant with increasing regularization, illustrating the theoretical relationship with regularization. Different norms exhibit similar asymptotic behaviors, converging towards minimal sensitivity values at higher regularization parameters. The right plot shows training outcomes when labels are uniformly altered to a single class with Lipschitz regularization, highlighting the effect of regularization on generalization and convergence rate of quantum models.
• Figure 3: Relationship between Lipschitz constant and model accuracy across training epochs, comparing Projected Gradient Descent, Lipschitz Regularization-based training, and a Naive training method. Notably, PGD consistently maintains stable robustness and high accuracy, whereas naive training fluctuates significantly.

Theorems & Definitions (9)

• definition 1: Neural Network
• definition 2: Quantum Variational Circuit
• definition 3: Hybrid Quantum-Classical Neural Network
• proposition 1: LipSDP fazlyab2019efficient
• lemma 1: Lemma 1, guan2022fairness
• theorem 1
• proof
• theorem 2
• proof