Balancing Expressivity and Learnability in Quantum Kernel Bandit Optimization

Abstract: We investigate Gaussian process (GP) bandit optimization with quantum kernels, assuming the mean reward function lies in the reproducing kernel Hilbert space (RKHS) induced by the quantum kernel. This setting is motivated by NISQ-era tasks such as quantum control, state preparation and variational quantum algorithms. While quantum kernels can offer a `quantum advantage' via domain-specific inductive biases, naïvely using full, high-dimensional kernels increases model complexity and information gain, leading to higher cumulative regret and poor learnability. To address this, we propose projected quantum kernels and classical kernel approximation techniques that reduce feature dimensionality while preserving key quantum properties. Using these approximate kernels, we develop misspecified GP bandit algorithms and derive regret bounds that characterize the trade-off between approximation error and information gain. The regret bounds provide principled guidance for selecting the optimal model complexity. Empirically, our methods outperform full quantum kernels in sample efficiency, while substantially reducing computational overhead, enabling scalable GP optimization for quantum-native applications.

• The paper introduces a quantum kernel bandit framework that balances expressivity and learnability by approximating complex fidelity kernels with reduced surrogate models.
• It employs methodologies like projected quantum kernels, random Fourier features, and P-greedy Newton basis to optimize GP bandit performance and manage cumulative regret.
• The work provides theoretical regret bounds and empirical validation, offering guidelines for optimal parameter selection in scalable quantum and classical kernelized bandit settings.

Summary of "Balancing Expressivity and Learnability in Quantum Kernel Bandit Optimization" (2607.01080)

Quantum Kernel Bandit Framework and Motivation

The paper addresses sequential optimization in noisy intermediate-scale quantum (NISQ) scenarios using Gaussian process (GP) bandit algorithms with quantum fidelity kernels, motivating applications such as quantum control, state preparation, and variational quantum algorithms (VQAs). The reward function is assumed to reside in the reproducing kernel Hilbert space (RKHS) induced by a quantum kernel, typically encoding domain-specific structure via quantum feature maps. While full quantum kernels are maximally expressive, they incur exponential feature dimension scaling with qubit count, resulting in excessive information gain and poor sample efficiency. Additionally, kernel value concentration requires an exponentially growing number of quantum measurements for accurate estimation.

Learnability Barriers and Model Approximation

The exponential scaling of information gain and kernel concentration with qubit number $n$ is formally identified as a fundamental learnability barrier. Specifically, the maximum information gain for an $n$-qubit quantum kernel scales as $O(4^n \log T)$ in iteration count $T$, producing cumulative regret bounds that increase with kernel complexity. The authors propose reducing this complexity through either quantum subsystem projection or classical kernel approximation techniques. The kernel approximation framework includes projected quantum kernels (LPQKs), random Fourier feature (RFF) methods, and Newton basis expansions (P-greedy).

Approximate Quantum Kernel Methods

Projected Quantum Kernels (LPQKs):

LPQKs operate by tracing out subsystems to reduce the dimensionality of quantum feature space. Summing projected kernels over multiple subsystems provides expressive surrogate models, while limiting Pauli-weight projections curtails complexity. Theoretical analysis connects Pauli-weight structure in quantum observables to both misspecification error and information gain, showing that observables with low-weight or rapidly decaying high-weight Pauli tails benefit most from projection.

Random Fourier Features (RFF):

For shift-invariant quantum kernels, RFF approximates the kernel via sampled Fourier frequencies derived from circuit generator eigenvalue gaps. The misspecification error decays as $O(1/\sqrt{D})$, where $D$ is the feature dimension, while information gain is suppressed relative to the full quantum kernel. Structured sampling strategies (e.g., prioritizing dominant frequencies) can further accelerate convergence. The theoretical regret bounds specify optimal $D$ as a function of $T$ (e.g., $D = \sqrt{T}$) to minimize total regret.

P-greedy (Newton Basis):

P-greedy iteratively selects kernel basis functions maximizing the power function criterion, yielding low-rank approximations tailored to the underlying spectral properties. The misspecification error is governed by the Kolmogorov width of the quantum RKHS's spectral tail, which decays rapidly for physical systems with localized observables or Gaussian profiles.

Regret Trade-off Analysis

Regret bounds are derived for misspecified GP bandit algorithms, quantifying the interplay between reduction in information gain and induced kernel misspecification. The cumulative regret is shown to be minimized at an optimal model complexity, with closed-form guidance for parameter settings (e.g., subsystem size $b$ in LPQK or RFF dimension $n$0). For observables with $n$1-locality, projected kernels achieve polynomial scaling of regret and complexity, compared to the exponential scaling of the full kernel. For physically motivated settings with rapid spectral decay, both LPQK and P-greedy can achieve near-optimal sample efficiency without sacrificing expressivity.

Empirical validation across synthetic and real quantum tasks—including phase classification and VQE optimization—demonstrates that appropriately chosen approximate kernels consistently outperform the full quantum kernel in both sample efficiency and computational overhead. The characteristic "U-shaped" regret curves confirm the theoretical trade-off, and the approximation parameter selection aligns with theoretical predictions.

Computational Implications and Scalability

The approximation-based algorithms enable significant reductions in both computational and quantum measurement cost. Full GP-UCB with fidelity quantum kernels is $n$2 in time per $n$3 rounds, whereas surrogate models with $n$4-dimensional feature maps are $n$5, with $n$6. For large qubit systems, feature reduction is essential for tractable optimization, especially when quantum hardware imposes practical limits on circuit evaluation and measurement repetitions.

Theoretical and Practical Implications

This research provides actionable guidelines for balancing quantum kernel expressivity and learnability in NISQ-era optimization. By exploiting observable structure or spectral decay, practitioners can tailor kernel surrogate models and bandit algorithms to physical constraints and task-specific requirements. The misspecified kernel bandit framework is broadly applicable, enabling principled trade-off analysis for both quantum and classical kernelized bandit settings. Theoretical regret bounds allow for model complexity selection that optimally balances underfitting and overfitting, with direct implications for quantum control experiment design and scalable optimization of variational circuits.

Future Directions

Key limitations include conservativeness of regret bounds, possible pessimism in misspecification error rates due to problem-dependent spectral properties, and the requirement of tractable Fourier or Newton expansions. Realistic noise models and hardware-dependent measurement costs warrant further study. Extending the framework to infinite-dimensional kernels, non-Gaussian noise, and more general reward landscapes may further broaden applicability. Measurement-aware regret and runtime analysis, as well as robust dimension selection strategies grounded in empirical kernel structure, comprise promising directions for future work.

Conclusion

The paper establishes a principled framework for scalable, sample-efficient GP bandit optimization using quantum kernel approximations, with theoretical regret guarantees and empirical validation across quantum-native tasks. The balance of expressivity and learnability, rooted in quantum observable structure and kernel spectral decay, sets a foundation for practical optimization in both quantum and high-dimensional classical bandit contexts.