A PAC-Bayesian approach to generalization for quantum models

Pablo Rodriguez-Grasa; Matthias C. Caro; Jens Eisert; Elies Gil-Fuster; Franz J. Schreiber; Carlos Bravo-Prieto

quant-phMar 24, 2026

A PAC-Bayesian approach to generalization for quantum models

Authors: Pablo Rodriguez-Grasa, Matthias C. Caro, Jens Eisert, Elies Gil-Fuster, Franz J. Schreiber, Carlos Bravo-Prieto

Organizations: Department of Physical Chemistry, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain · EHU Quantum Center, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain · TECNALIA, Basque Research and Technology Alliance (BRTA), 48160 Derio, Spain · Department of Computer Science, University of Warwick, UK ·

Abstract

Generalization is a central concept in machine learning theory, yet for quantum models, it is predominantly analyzed through uniform bounds that depend on a model's overall capacity rather than the specific function learned. These capacity-based uniform bounds are often too loose and entirely insensitive to the actual training and learning process. Previous theoretical guarantees have failed to provide non-uniform, data-dependent bounds that reflect the specific properties of the learned solution rather than the worst-case behavior of the entire hypothesis class. To address this limitation, we derive the first PAC-Bayesian generalization bounds for a broad class of quantum models by analyzing layered circuits composed of general quantum channels, which include dissipative operations such as mid-circuit measurements and feedforward. Through a channel perturbation analysis, we establish non-uniform bounds that depend on the norms of learned parameter matrices; we extend these results to symmetry-constrained equivariant quantum models; and we validate our theoretical framework with numerical experiments. This work provides actionable model design insights and establishes a foundational tool for a more nuanced understanding of generalization in quantum machine learning.

Figures & tables

Figure 1: Visualization of the proposed PAC-Bayesian framework for QML. The framework unifies a broad spectrum of quantum architectures, including (a) standard unitary circuits, (b) mid-circuit measurement-and-feedforward architectures, and (c) dissipative channels. (d) Under the PAC-Bayesian lens, this methodology produces operationally interpretable and solution-dependent bounds that serve two main purposes: (e) providing theoretical guidance for model design, and (f) establishing a more nuanced understanding of generalization in quantum models.

Figure 2: Architecture and parameterization of equivariant quantum models. (a) Schematic of a multi-layer equivariant quantum model. The model consists of a sequence of quantum channels $\phi_{j}$ and unitary representations $R^{(j)}$ that define the symmetry constraints at each layer. The implementation concludes with an observable $O$ yielding an invariant prediction. (b) The block-diagonal structure of a channel’s Choi operator arising from the isotypic decomposition. By virtue of Schur’s Lemma, the parameters decouple into independent blocks acting on the multiplicity spaces. Here, the colored squares represent the learnable parameter matrices $W_{j,\lambda}$ of size $m_{\lambda}\times m_{\lambda}$ , while the repetition count $d_{\lambda}$ corresponds to the dimension of the associated irreducible representation.

Figure 3: Correlation between complexity term and generalization gap. Generalization gap as a function of the complexity term $\beta_{\text{P(T)M}}({\sum_{j=1}^{L}\left\lVert W_{j}\right\rVert_{F}^{2}})^{1/2}$ derived in Theorems 3 and 4 for the (a) dynamic PQC and (b) QCNN. The density plots represent the outcomes of $1400$ individual and independent trained models, yielding a positive Pearson correlation coefficients of $r=0.26$ and $r=0.46$ for the dynamic PQC and QCNN architectures, respectively. Overall, we observe that models with smaller complexity term, and therefore lower norms, tend to exhibit smaller generalization gaps.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure F.1: Hybrid evaluation of the PAC-Bayes bound on the QCNN experiment. Density corresponds to the same models as in Figure 3 (b) in the main text, but with each $\|W_{j}\|_{F}^{2}$ replaced by the dimensionally rescaled value suggested by the alternative convention $\|\tilde{W}_{j}\|_{F}^{2}$ , while $\beta_{\mathrm{PTM}}$ is kept unchanged. The complexity term becomes significantly smaller, yet the correlation with the observed generalization error remains $r=0.46$ .

Generalization Bounds

Generalization bounds quantify how training performance can differ from population or test performance as sample size and model complexity change.

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Oct 17, 2026J/K move · Enter open · S save

Margin-Based Generalization for Quantum Machine Learning with Classical Data Encoding

Ju-Young Ryu, Tak Hur, Daniel K. Park

Korea Institute of Science and Technology Information, Republic of Korea · Department of Statistics and Data Science, Yonsei University, Republic of Korea · Department of Applied Mathematics, University of Waterloo, Canada+2

Quantum Machine Learning Generalization Bounds Quantum Data Encoding

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Understanding generalization is essential for assessing the practical viability of quantum machine learning. In many applications, quantum models operate on classical data that must first be encoded into quantum states, making the encoding part of the effective hypothesis class. We develop a margin-based generalization framework for this setting, covering both fixed and trainable quantum data encodings. For fixed encodings, the existing margin bound transfers unchanged. For trainable encodings, we derive a bound using a two-scale covering argument that combines the original margin-dependent term with an additive encoding-capacity term of order

\tilde O(\sqrt{L/m})

, where

L

is the number of trainable encoding parameters and

m

is the training-sample size. Across 19,440 simulated training runs spanning various image datasets, circuit architectures, and optimization settings, the inverse margin performs best among five baseline and two bound-derived complexity measures under conditional-independence and 90th-percentile sign-error criteria, within bootstrap uncertainty. These results provide a unified margin-based view of generalization in quantum machine learning with classical data.

Abstract

Generalization Bounds

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