Understanding Generalization in Quantum Machine Learning with Margins

TL;DR

This paper introduces a margin-based generalization bound for quantum neural networks (QNNs), addressing the shortcomings of uniform bounds in quantum machine learning. It adapts classical margin theory to the quantum setting, deriving a bound that depends on the sample margin and POVM norms, and validates it empirically on quantum phase recognition tasks using QCNNs. The results show that margin distributions strongly predict generalization better than parameter counts, and the work connects margins to quantum state discrimination via quantum embeddings, with Neural Quantum Embedding (NQE) achieving larger margins and better generalization. These insights guide the design of QML models and embeddings for improved generalization on both quantum and classical data.

Abstract

Understanding and improving generalization capabilities is crucial for both classical and quantum machine learning (QML). Recent studies have revealed shortcomings in current generalization theories, particularly those relying on uniform bounds, across both classical and quantum settings. In this work, we present a margin-based generalization bound for QML models, providing a more reliable framework for evaluating generalization. Our experimental studies on the quantum phase recognition (QPR) dataset demonstrate that margin-based metrics are strong predictors of generalization performance, outperforming traditional metrics like parameter count. By connecting this margin-based metric to quantum information theory, we demonstrate how to enhance the generalization performance of QML through a classical-quantum hybrid approach when applied to classical data.

Figures (6)

• Figure 1: A Tukey box-and-whisker plot depicting the margin distributions of optimized 8-qubit Quantum Convolutional Neural Networks (QCNNs). The results for QCNNs with one, five, and nine layers are displayed, along with their corresponding test accuracies indicated in the legend. QCNNs were trained for 4-class classification task aimed at quantum phase recognition (QPR). The experiment was performed with varying degrees of label noise: QPR dataset with pure labels (left), half randomly labelled dataset (middle), and full randomly labelled datasets (right). As the noise (randomization) level increases, the margin distributions tend to exhibit a more pronounced skew towards the left, indicating that a greater proportion of samples are classified with smaller margins. Notably, the margin distribution exhibits a strong positive correlation with test accuracy across all scenarios.
• Figure 2: Illustration of how the generalization gap, median of the margin distribution (a margin-based metric), and effective parameters with a $10^{-2}$ threshold (a parameter-based metric) vary with the number of layers, percentage of randomized labels, and the choice of variational ansätze.
• Figure 3: Comparative analysis of mutual information (solid) and Kendall rank correlation coefficients (shaded) between the generalization gap and various metrics. The first three columns represent margin-based metrics, while the last three columns represent parameter-based metrics.
• Figure 4: A Tukey box-and-whisker plot illustrating the margin distributions of optimized 8-qubit Quantum Convolutional Neural Networks (QCNNs). The plot shows results for QCNNs using fixed quantum embedding (left), trainable quantum embedding (middle), and neural quantum embedding (right). The QCNNs were trained on a binary classification task using the MNIST (bottom), Fashion-MNIST (middle), and Kuzushiji-MNIST (top) datasets. In addition to the margin distributions, the mean of the margins is indicated by a black cross, and the trace distance between ensemble quantum states is marked by a red circle.
• Figure 5: Reproduction of margin distribution (Figure \ref{['fig:margin_qpr']}) using different variational ansätze.

Theorems & Definitions (5)

• Theorem 2.1
• Corollary 2.1.1
• Lemma A.1
• Lemma A.2
• Lemma A.3