Data-Dependent Generalization Bounds for Parameterized Quantum Models Under Noise

TL;DR

The paper tackles the challenge of understanding generalization in parameterized quantum machine learning under noise by deriving a data-dependent bound anchored in the quantum Fisher information (QFIM) geometry of the parameter space. It connects the parameter-space volume, training sample size, and noise effects through a bound of the form $R(\\theta) \\le \\hat{R}_N(\\theta) + \\frac{12\\sqrt{\\pi d} \, e^{C'/d}}{\\sqrt{N}} + 3\\sqrt{\\frac{\\log(2/\\delta)}{2N}}$, with $C' = \log V_\\Theta - \log V_d - \log m + d \\log L_f^p$, and introduces the concept of effective dimension $d_{eff}$ from QFIM eigenvalues to tighten bounds via local neighborhoods around the trained parameters. Numerical experiments on depolarizing-noise quantum circuits trained on Iris and MNIST-derived tasks show that local, QFIM-informed bounds better reflect observed generalization than global bounds, highlighting the practical value of geometry-based complexity control for NISQ-era QML. The work thereby provides a principled framework linking quantum state geometry, noise, and data to generalization performance, guiding robust quantum circuit design and training strategies in real hardware settings.

Abstract

Quantum machine learning offers a transformative approach to solving complex problems, but the inherent noise hinders its practical implementation in near-term quantum devices. This obstacle makes it difficult to understand the generalizability of quantum circuit models. Designing robust quantum machine learning models under noise requires a principled understanding of complexity and generalization, extending beyond classical capacity measures. This study investigates the generalization properties of parameterized quantum machine learning models under the influence of noise. We present a data-dependent generalization bound grounded in the quantum Fisher information matrix. We leverage statistical learning theory to relate the parameter space volumes and training sizes to estimate the generalization capability of the trained model. We provide a structured characterization of complexity in quantum models by integrating local parameter neighborhoods and effective dimensions defined through quantum Fisher information matrix eigenvalues. We also analyze the tightness of the bound and discuss the tradeoff between model expressiveness and generalization performance.

Figures (4)

• Figure 1: Rademacher Complexity $f(d,n)$ as a function of the number of samples $n$ and the dimensionality $d$. We have applied the $\log$ scaling on both axes for the third figure (second row) for better visualization.
• Figure 2: Two-qubit circuit with depolarizing noise channels for noise rate $p \in \{0.05,0.1,0.5\}$, parameterized single-qubit rotations, and controlled operations, preliminary CNOT gate. The Rot $(\cdot)$ is a single-qubit rotation gate with three Euler angles. $\theta_i,\theta_j,theta_k$ and $\theta_l$ are the parameters with distinct three Euler angles. The measurement is performed on the first qubit on a computation basis.
• Figure 3: Global and local bounds versus the generalization gap for the first two classes of the Iris dataset. The local bound more closely tracks the observed gap, indicating a tighter and more accurate estimate of generalization.
• Figure 4: Global and local bounds compared to the observed generalization gap for MNIST digits $\{0,1\}$. The local bound shows occasional spikes, possibly due to random parameter initializations, but remains closer to the empirical gap than the global bound.

Theorems & Definitions (11)

• Theorem 5.1
• Corollary 5.2
• proof
• Corollary 5.3
• proof
• Lemma 1.1: Covering Number and Volume
• proof
• Lemma 2.1: Empirical Rademacher Complexity Bound
• proof
• Theorem 2.2: Restatement of Theorem \ref{['th:gboundmain']}