Rademacher Complexity Bounds for Parameterized Quantum Circuits Generated by Pauli Strings

Abstract: In this study, we analyze the Rademacher complexity $ \mathcal{R}_{M} $ of a parameterized unitary whose generators are chosen from $ n $-qubit Pauli strings. Although generalization bounds for quantum machine learning models have been studied in several settings, explicit Rademacher-complexity bounds for parameterized unitaries generated by Pauli strings remain less transparent. We derive simple scaling bounds in terms of the number of parameters $ L $ and the number of training samples $ M $: $ \mathcal{O}(\frac{L^{\frac{3}{2}}}{\sqrt{M}}) $ for the full parameter domain and $ \mathcal{O}(\frac{L}{\sqrt{M}}) $ for a restricted parameter domain. Furthermore, we compare the obtained results with those for a classical linear model class and suggest a potential statistical-complexity advantage when the norms of both the input and the parameter in the classical model scale with the number of parameters. Numerical experiments provide qualitative evidence consistent with the predicted scaling.

• The paper establishes explicit Rademacher complexity bounds for parameterized quantum circuits using Pauli strings, showing O(L^(3/2)/√M) scaling for full parameter domains.
• The methodology employs Lipschitz analysis and covering number estimates, with numerical experiments validating the empirical scaling with the number of parameters L and samples M.
• The findings highlight a generalization advantage of these quantum models over classical linear models, providing actionable insights for circuit design and sample requirements.

Rademacher Complexity Bounds for Parameterized Quantum Circuits with Pauli String Generators

Overview

This paper provides a rigorous analysis of the Rademacher complexity for parameterized quantum circuits (PQCs) whose unitary gates are generated via $n$-qubit Pauli strings. This setting is canonical in variational quantum algorithms and quantum machine learning (QML), where optimizing over PQC parameters is central to model capacity and generalization. While prior studies quantified generalization bounds in terms of gate counts or circuit resources, explicit Rademacher complexity scalings for PQCs generated by Pauli strings, especially in terms of the number of parameters $L$ and number of training samples $M$, were unresolved. This work addresses that gap, establishing asymptotic bounds, contrasting quantum and classical model classes, and supplementing the theoretical analysis with numerical validation.

Theoretical Contributions

The principal technical result is a derivation of Rademacher complexity bounds for PQCs where both the generators and observables are $n$-qubit Pauli strings:

• For the full parameter domain, the Rademacher complexity $\mathcal{R}_M$ is shown to scale as $\mathcal{O}(L^{3/2} / \sqrt{M})$.
• For restricted parameter domains where the parameter space diameter is constant, the scaling improves to $\mathcal{O}(L / \sqrt{M})$.

The derivation utilizes a Lipschitz analysis specific to Pauli-string-generated unitaries. The key lemma establishes that the Lipschitz constant with respect to the circuit parameters scales as $\mathcal{O}(\sqrt{L})$. This directly affects covering number estimates, and via Dudley's entropy integral, leads to the stated complexity bounds.

A comparison to linear classical models reveals that, when the parameter and input norm constraints scale with the parameter count, classical Rademacher complexity deteriorates as $\mathcal{O}(p^2/\sqrt{M})$ (where $p$ is the number of parameters), which is worse asymptotically than the Pauli PQC regime in the high-parameter case. This suggests a statistical complexity advantage inherent to quantum models in specific norm-scaling regimes.

Empirical Validation

The numerical studies focus on three axes:

• Generalization gap scaling: The generalization gap, upper bounded by the Rademacher complexity, decreases with sample size $L$0 but increases with parameter count $L$1, consistent with the theoretical predictions.
• Sample-size dependence: Regression analysis of empirical Rademacher complexity (approximated by a random-search procedure) confirms the $L$2 scaling.
• Parameter-size dependence: Log-log regression yields slopes $L$3 1.5 for the loose parameter domain and $L$4 1 for the restricted domain, both broadly tracking the predicted $L$5 and $L$6 dependences (modulo finite-sample estimation error and limitations of the random-search technique).

The empirical results do not attain the theoretical exponents exactly, which is attributed to finite-sample effects, optimization landscape challenges, and limited resolution in parameter-space sampling.

Implications

The scaling of Rademacher complexity with $L$7 and $L$8 critically determines the generalization capability of quantum ML models. The results here imply that PQCs generated by Pauli strings are, a priori, unlikely to suffer from the same degree of capacity-induced overfitting as classical linear models under comparable parameter-norm scaling, especially in large-$L$9 regimes. However, the generalization bounds derived depend solely on function-class complexity and do not account for data distribution, parameter initialization, or practical training effects—factors known to influence empirical performance.

From a foundational standpoint, the analysis substantiates claims of statistical efficiency in QML architectures rooted in parameterized Pauli-string circuits, complementing empirical reports of quantum advantage in curated settings. These bounds also provide actionable guidelines for QML practitioners regarding circuit sizing and sample requirements.

Limitations and Future Directions

The current analysis is contingent on several simplifications: bounded observable/operator norms (each set to unity), expectation values framed within $M$0, and the focus on Pauli-string circuits with no hardware-induced noise. Numerical validation is limited by the random-search-based lower-bound approximation to the true Rademacher complexity and is feasible only for moderate $M$1.

Future research could extend this framework via:

• Optimization-based supremum solvers for more incisive empirical Rademacher complexity estimates,
• Investigation of data-dependent complexity measures (e.g., covering numbers or localized Rademacher complexity),
• Adapting the theoretical analysis to noisy/interacting circuits and non-Pauli observables,
• Exploring QML architectures with more expressive generator sets within the Lie algebra formalism for deeper complexity-theoretic insights.

Conclusion

This paper establishes scaling laws for Rademacher complexity of PQCs generated by $M$2-qubit Pauli strings. The bounds $M$3 (full domain) and $M$4 (restricted domain) are qualitatively corroborated by empirical approximation and suggest statistical-complexity favorability over certain classical classes. The results clarify a fundamental aspect of QML generalization and lay groundwork for deeper analyses of quantum statistical learning phenomena.