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arXiv:2606.20183

Effective Dimension Governs Generalization in Quantum Kernel Vision Models

Jian Xu, Delu Zeng, John Paisley, Qibin Zhao

Published Jun 18, 2026• 0 citations
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Abstract

Recent quantum vision models-quantum vision transformers and quantum convolutional networks-report two striking but unexplained empirical phenomena: (i) ansatze with more, or more uniformly distributed, entanglement generalize better, and (ii) injecting quantum noise can improve test accuracy rather than degrade it. These observations are currently treated as curiosities, discovered by grid search and explained, if at all, by hand. We show that both are manifestations of a single, measurable quantity: the \emph{effective dimension} $d_{\rm eff}$ of the (noise-shaped) quantum feature kernel. Working primarily with quantum-kernel vision models-a quantum feature map read out by a kernel classifier-we give a spectral account in which entanglement structure and quantum noise are two knobs that move $d_{\rm eff}$; in an overfitting regime, contracting $d_{\rm eff}$ acts as ridge-like regularization. We analyze the mechanism: an \emph{exact} decomposition of the depolarized kernel $K_p=(1-p)^2K+\tfrac{p(2-p)}{D}\mathbf{1}\mathbf{1}^\top$ with $d_{\rm eff}(K_p)\to1$, a contraction result (and its boundary) for amplitude damping, a kernel-machine capacity bound, and a capacity/alignment risk decomposition; the monotone contraction operative in our entangled experiments is verified empirically, not proven in general. Along the one-parameter depolarizing family the collapse is instead exact by construction; we use it only to confirm the kernel decomposition to machine precision and at up to $12$ qubits, not as evidence for $d_{\rm eff}$. Amplitude damping contracts $d_{\rm eff}$ and lifts test accuracy by up to $+13\%$ along an inverted-U sweet spot; the effect's sign flips between the over- and under-fitting regimes; noise injection matches an explicit spectral-filtering frontier. Our results organize two reported anecdotes into a single measurable principle for designing quantum-vision models.

DOI
10.48550/arxiv.2606.20183

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Quantum kernel methods are the most concrete near-term proposal for quantum machine learning on classical data, yet published benchmarks disagree on whether they outperform tuned classical kernels, and the disagreement cannot be resolved from the literature because the comparisons are not protocol-controlled. We argue that the obstacle is one of measurement, not theory: the usual setup fixes the quantum feature map's data scaling while grid-searching the classical kernel's hyperparameters, recomputes nothing per fold, and reports cost in simulator time. We therefore treat the comparison itself as the object of study and impose three controls. The bandwidth is tuned on the same grid budget as the classical γγγ, the kernel is recomputed inside every fold from fold-fitted preprocessing, and cost is counted in hardware circuit evaluations. On top of these we run a factorial design of 324 cells in which dimensionality, training size, class separability and data structure are explicit covariates, together with four standard datasets, a shots-versus-gate-noise ablation, and a cost sweep. The dominant effect we measure is not quantum: tuning the feature map's bandwidth is worth 0.120 accuracy on synthetic data and 0.339 on real data, exceeding every quantum-classical difference we observe. Tuned quantum kernels are then statistically indistinguishable from a tuned RBF on all four real datasets. The advantage on synthetic data is confined almost entirely to data generated by the feature map itself, and it shrinks with dimensionality (-0.0045 per qubit) while growing with sample size. Finite sampling costs an order of magnitude more accuracy than gate and readout noise, and a fidelity kernel needs 5×1075\times10^{7}5×107 circuits at n = 10,000. Protocol variance is thus sufficient to explain the field's contradictory conclusions.
Two recent results have reshaped quantum Gaussian processes (QGPs). On the one hand, \citet{lowe2025assessing} rule out the exponential speedups claimed by HHL-based QGP regression in the typical, well-conditioned regime; on the other, an independent line of work shows that highly expressive quantum kernels suffer posterior pathologies that break Bayesian optimization. We show that these seemingly unrelated phenomena are governed by the same quantity: the normalized spectral entropy S(K)/log⁡nS(K)/\log nS(K)/logn of the kernel Gram matrix. We prove a Cauchy--Schwarz tail bound on Nyström approximation error, a finite-sample variance-contraction identity in terms of Bach's degrees of freedom dσ(K)d_σ(K)dσ​(K), and a characterization of the \emph{target-dependent} optimal entropy via the intrinsic dimension of the target in the kernel eigenbasis. Empirically, the diagnostic is kernel-agnostic: hardware-efficient, matchgate, IQP \emph{and} RBF/Matérn/RFF/deep-kernel families all collapse onto identical S/log⁡nS/\log nS/logn curves on dequantization, ECE, and variance-contraction panels. The NLL sweet spot lives at high entropy for smooth targets and at low entropy for band-limited quantum-data targets. The diagnostic transfers from simulator to IBM Heron hardware with median absolute error 3.2%3.2\%3.2% and mean 5.2%5.2\%5.2% in S/log⁡nS/\log nS/logn across 242424 configurations at nq=4n_q = 4nq​=4, with matchgate and IQP within 5%5\%5% mean and a single HE configuration returning a 30%30\%30% outlier that drops to 0.5%0.5\%0.5% on rerun (attributed to calibration drift); the same diagnostic transfers to a second Heron backend (mean error 2.7%2.7\%2.7%) and to a nq=6n_q = 6nq​=6 scale-up on the original backend (mean error 1.7%1.7\%1.7%). No error mitigation is applied throughout.
Abstract Quantum machine-learning models are useful only if classical computers cannot cheaply imitate them. We test this directly for quantum Kolmogorov--Arnold networks (QKANs)---neural networks whose tunable ``activation functions'' are small quantum circuits---by asking whether each model can be contracted as an efficient tensor network (TN). Our task-aware yardstick is the bond dimension a classical surrogate needs to copy the model's output at the accuracy the model itself reaches. We find a clear ladder. The amplitude-encoded Chebyshev QKAN uses only logarithmically many qubits and is exactly classical. The variational QKAN, built from all-to-all controlled rotations, looks hard at random settings but generalizes only weakly once trained (held-out R2 ⁣≤ ⁣0.27R^2\!\le\!0.27R2≤0.27) and is copied at that attained accuracy by a low-bond-dimension surrogate (n ⁣≤ ⁣10n\!\le\!10n≤10); at fixed strict tolerance the required bond dimension grows near-exponentially. Depolarizing noise erases it (purity collapse) near p ⁣≈ ⁣0.03p\!\approx\!0.03p≈0.03--0.050.050.05 at the sizes we reach. Every generic attempt to force hardness---a global measurement or an interaction-based (IQP) feature map---instead destroys trainability through exponentially small gradients (a ``barren plateau''). The escape is a discrete-logarithm feature map: its kernel variance is flat from 444 to 141414 qubits while a generic IQP kernel concentrates exponentially, and a support-vector machine with it learns a classically hard concept to ≥99%\ge99\%≥99% accuracy while generic and classical kernels stay at chance; wired into the KAN architecture, it fits discrete-log targets to R2 ⁣≈ ⁣1R^2\!\approx\!1R2≈1. Hardness placed in number theory stays classically hard, shot-efficient, and learnable. The work provides a measurement-based reproducibility criterion and a design rule: seek advantage in structured, provably hard encodings rather than in deep entanglement.
Kernel methods compare inputs through feature maps. Quantum kernels follow the same principle: input data are encoded into quantum states, which define quantum feature representations in Hilbert spaces. Kernel values are then obtained by estimating inner products between these states using suitable quantum circuit measurements. As a result, quantum kernels may be intractable to compute classically while remaining efficiently computable on quantum hardware, potentially leading to a quantum advantage. However, designing effective quantum kernels remains a major challenge. Many quantum kernels, such as the fidelity kernel, suffer from exponential concentration. This results in near-identity kernel matrices that fail to capture meaningful data correlations and lead to overfitting and poor generalization. In this paper, we propose a novel strategy for constructing quantum kernels that achieve good generalization performance, drawing inspiration from benign overfitting in classical machine learning. We introduce the concept of Local-Global quantum kernels, which combine two components: a local quantum kernel based on measurements of small subsystems, and a global quantum kernel derived from full-system measurements. To support the effectiveness of the proposed construction, we show theoretically and empirically that Local-Global quantum kernels exhibit benign overfitting.
Quantum Machine Learning (QML) has recently emerged as a highly promising research frontier. Within this domain, Quantum Neural Networks (QNNs),characterized by Variational Quantum Circuits (VQCs) at their core and featuring layers of quantum gates optimized by classical algorithms, have garnered significant attention. However, a rigorous and exhaustive evaluation of their practical performance remains largely incomplete. In this study, we conduct a comprehensive comparative analysis of three prominent hybrid classical-quantum architectures: Quantum Convolutional Neural Networks (QCNN), Quantum Recurrent Neural Networks (QRNN), and Quantum Vision Transformers (QViT), focusing on the critical dimensions of generalization, accuracy, and robustness. Our findings provide novel insights that address previous evaluative gaps. Notably, while these models exhibit exceptional performance on low-feature datasets such as MNIST, their learning efficacy degrades significantly when transitioned to high-feature datasets. Furthermore, convolutional-based models like QCNN appear less effective on high-dimensional data than other machine learning architectures. Additionally, while all models are susceptible to adversarial noise, traditional architectures, such as recurrent and convolutional networks, demonstrate superior resilience. Conversely, in the presence of quantum noise, the transformer-based architecture proves its strength by maintaining high robustness against measurement noise, channel noise, and finite-shot effects, whereas other architectures suffer marked performance declines. These results provide a granular perspective on the current state of the field and underscore the critical importance of tailoring model selection to the constraints of contemporary Noisy Intermediate-Scale Quantum (NISQ) environments.
We aim to rigorously address how local noise affects classical simulability of quantum dynamics benchmarked by tensor-network methods. Using operator entanglement entropy (OEE), we prove the following: (1) For single-qubit depolarizing noise on arbitrary circuits, tensor networks with poly(n)\mathrm{poly}(n)poly(n) bond dimension suffice for fixed absolute Hilbert-Schmidt error after $\order{1}$ depth, while relative error demands $\order{\log n}$ depth; and this bound is optimal. (2) For single-qubit depolarizing noise on 1D local circuits, the existence of whole-trajectory error-bounded matrix product operator (MPO) of poly(n)\mathrm{poly}(n)poly(n) bond dimension at all depths. (3) For general single-qubit noise in 1D brickwall circuits, random two-design gates with contraction coefficient c<1/3c<1/3c<1/3 yield an $\order{1}$ OEE plateau with probability 1−Te−Ω(n)1-Te^{-Ω(n)}1−Te−Ω(n), while arbitrary gates with c<1/48c<1/48c<1/48 give $\order{\log n}$ OEE in the worst case. (4) In higher dimensions, these bounds yield uniform-in-depth poly(n)\mathrm{poly}(n)poly(n) average boundary-bond dimensions for projected entangled pair operators~(PEPO) across every cut -- under depolarizing noise at either absolute or relative accuracy, and under general noise with strong contraction at absolute accuracy. Our results establish a rigorous connection between certain noise models, circuit types, and their classical simulability.