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 · Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany · Fraunhofer Heinrich Hertz Institute, 10587 Berlin, Germany · Helmholtz-Zentrum Berlin für Materialien und Energie, 14109 Berlin, Germany
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 ϕj 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 Wj,λ of size mλ×mλ , while the repetition count dλ 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 βP(T)M(∑j=1L∥Wj∥F2)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 ∥Wj∥F2 replaced by the dimensionally rescaled value suggested by the alternative convention ∥W~j∥F2 , while βPTM is kept unchanged. The complexity term becomes significantly smaller, yet the correlation with the observed generalization error remains r=0.46 .
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 O~(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.
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
A central challenge in quantum machine learning is understanding the scaling behavior of parameterized quantum circuits (PQCs). In particular, it remains unclear how their performance on unseen data changes as the number of trainable parameters increases. Prior works have derived formal generalization guarantees for quantum models, but it is well-known that many such results do not fully characterize generalization behavior in practice. In this work, we show that gradient-based PQCs can exhibit improved performance on unseen data as model size increases, displaying the phenomenon of double descent. This contrasts with the traditional view that larger models lead to degraded generalization. We provide analytical results rigorously underpinning this behavior by leveraging add-one-in perturbation techniques and spectral properties of random matrices. We support these results with numerical experiments on re-uploading PQCs across several data sets and training set sizes, consistently observing the predicted double descent behavior. While other obstacles on the path toward practical quantum machine learning remain, our finding that deeper parameterized quantum circuits do not necessarily exhibit degraded performance provides reasons for cautious optimism.
Marie Kempkes, Elies Gil-Fuster, Carlos Bravo-Prieto +5
Leiden University, Niels Bohrweg 1, 2333 CA Leiden, Netherlands · Volkswagen Group Innovation, Berliner Ring 2, 38440 Wolfsburg, Germany · Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany +4
Parameterized quantum circuits (PQCs) are increasingly used as policies and value functions in quantum reinforcement learning, yet it remains unclear when and why quantum policies generalize. We give a PAC-Bayesian account in which generalization is governed not by the raw number of circuit parameters, but by the effective dimension of the Fisher geometry induced by the circuit. This quantity is inflated by entanglement, making entangling connectivity an independent axis of complexity.In controlled experiments that fix the number of trainable rotations and vary only entanglement, we find that circuits with larger Fisher effective dimension exhibit larger train-test gaps, while parameter count is a weak predictor. The resulting bound acts primarily as a ranking certificate: it correctly orders circuits with identical parameter count, which parameter-counting bounds cannot do. We validate this mechanism across supervised classification, quantum contextual bandits, and value-function generalization, where entangled circuits consistently generalize worse than non-entangled circuits of equal parameter count, with gaps shrinking as sample size increases.Our strongest evidence comes from low-variance decision models, including single-observable classifiers, value heads, and one-step policies. In end-to-end multi-step policy learning, entanglement effects remain statistically significant but high return variance leaves the full ordering only partially resolved. Partial-correlation analysis shows that Fisher effective dimension screens off entangling pattern, and controls for training accuracy, readout, and optimizer rule out major optimization confounders. The effect also persists on an IBM Heron quantum processor under real noise. Overall, our results reframe quantum policy design around an entanglement--generalization trade-off rather than expressivity alone.
Jian Xu, Delu Zeng, John Paisley +1
1RIKEN iTHEMS · 2RIKEN AIP · 3South China University of Technology +1