A PAC-Bayesian approach to generalization for quantum models

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 (4)

• 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 \ref{['th:pac-bayes-PM', 'th:pac-bayes-PTM']} 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.
• Figure F.1: Hybrid evaluation of the PAC-Bayes bound on the QCNN experiment. Density corresponds to the same models as in \ref{['fig:numerics_results']}(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 $r = 0.46$.

Theorems & Definitions (46)

• Lemma 1: PAC-Bayes bound for randomized predictors McAllester2003
• Lemma 2: PAC-Bayes margin bound perturbation_bound
• Theorem 3: PAC-Bayes generalization bounds for quantum models -- PM framework (Informal)
• Theorem 4: PAC-Bayes generalization bound for quantum models -- PTM framework (Informal)
• proof : Proof sketch
• Corollary 5: PAC-Bayes generalization bounds for data re-uploading quantum models
• Theorem 6: PAC-Bayes generalization bound for equivariant quantum models (Informal)
• Lemma 7: Perturbation bound for quantum models -- PM framework
• Lemma 8: Perturbation bound for quantum models -- PTM framework
• Lemma 9: Perturbation bound for equivariant quantum models