Empirical Characterization of Learning Geometry in Hybrid Quantum Forecasting Models

# Empirical Characterization of Learning Geometry in Hybrid Quantum Forecasting Models
Source: [https://arxiv.org/html/2608.19497](https://arxiv.org/html/2608.19497)
Sandra Leticia Juárez\-Osorio1, Jorge I\. Hernandez\-Martinez1, Jesus Ivan Ruiz\-Martinez1, Andres Mendez\-Vazquez1, Eduardo Rodriguez\-Tello2Affiliation:1Department of Computer Science, CINVESTAV Guadalajara, MexicoAffiliation:2Cinvestav Unidad Tamaulipas, MexicoAffiliation:Email:\{sandra\.juarez, jivan\.hernandez, jesus\.ruiz, andres\.mendez, ertello\}@cinvestav\.mx

###### Abstract

We characterize the learning dynamics of a compact hybrid quantum forecasting model through comparison with a structurally aligned classical baseline\. Using stationary harmonic\-mixture and nonstationary chirp benchmarks with controlled spectral complexity and data availability, we analyze empirical Neural Tangent Kernel dynamics through kernel\-target alignment, kernel drift, spectral concentration, and training loss\. The classical model exhibits stronger early target alignment, whereas the hybrid model generally develops a less concentrated kernel spectrum and smaller kernel drift\. Despite these distinct optimization geometries, both architectures attain similar held\-out performance across the evaluated regimes\. Notably, the hybrid model uses 125 trainable parameters compared with 281 for the classical baseline and reaches its validation\-selected checkpoint earlier in 15 of 18 frequency conditions\. A Fourier\-augmented classical baseline does not reproduce the observed training behavior, while a controlled re\-uploading ablation shows that repeated encoding systematically modifies both optimization and kernel geometry\. These results demonstrate that comparable generalization can emerge from substantially different learning trajectories and that individual NTK diagnostics do not provide monotonic predictors of validation convergence\. Rather than claiming a general quantum advantage, the study identifies architecture\-dependent learning behavior that is masked by endpoint accuracy alone\.

## IIntroduction

Neural networks are known to exhibit structured learning biases during optimization\. One of the most widely studied phenomena is spectral bias, where lower\-frequency or smoother components of a target function are typically learned earlier than higher\-frequency components\[[10](https://arxiv.org/html/2608.19497#bib.bib11)\]\. This behavior has been analyzed theoretically through the Neural Tangent Kernel \(NTK\) framework, where training dynamics can be interpreted through the eigenspectrum of an induced kernel\[[5](https://arxiv.org/html/2608.19497#bib.bib13)\]\. In this view, components associated with larger eigenvalues are learned more rapidly, while smaller\-eigenvalue directions converge more slowly\. As a result, kernel geometry provides a principled lens for understanding optimization speed, inductive bias, and generalization behavior\.

These ideas raise a natural question in quantum machine learning: Do hybrid quantum models induce learning dynamics that differ in a meaningful way from comparable classical architectures? Variational quantum circuits \(VQCs\) have been proposed as trainable nonlinear feature maps defined through quantum state evolution and parametrized unitary transformations, potentially capturing correlations differently from standard neural networks\[[3](https://arxiv.org/html/2608.19497#bib.bib2),[11](https://arxiv.org/html/2608.19497#bib.bib6)\]\. However, many comparisons between quantum and classical models focus primarily on final predictive accuracy, often leaving the underlying optimization geometry less explored\. Understanding learning dynamics is particularly relevant in forecasting tasks involving oscillatory or spectrally rich signals\. In such problems, differences in learning geometry may become visible not only through final error, but also through how quickly models learn distinct frequency components, how stable their kernels remain during training, and how concentrated their spectral representations become\.

In this work, we investigate these questions through a controlled empirical study comparing a compact hybrid quantum model against a classical baseline with matched structure on synthetic multivariate forecasting tasks with tunable spectral complexity\. We consider both stationary harmonic\-mixture signals and nonstationary chirp signals, while varying training set size to probe data limited regimes\.

Rather than focusing exclusively on final predictive performance, we analyze optimization behavior using empirical NTK diagnostics computed during training\. Specifically, we track kernel\-target alignment, kernel drift relative to initialization, top\-1 spectral mass, and learning efficiency through the area under the learning curve \(AULC\)\. Validation loss is monitored throughout training to identify when each model reaches its best validation\-selected solution, which is subsequently evaluated on test set\.

Our main contributions are summarized as follows:

- •We introduce an empirical NTK framework for comparing hybrid quantum and classical learning dynamics under controlled spectral complexity and data availability\.
- •We identify persistent architecture\-dependent differences in kernel alignment, drift, and spectral concentration across stationary and nonstationary forecasting benchmarks\.
- •We show that the 125\-parameter hybrid model attains held\-out performance within a few percent of a 281\-parameter classical baseline and reaches its validation\-selected checkpoint earlier in most evaluated conditions\.
- •Through Fourier\-feature and re\-uploading studies, we show that explicit periodic features alone do not reproduce the observed training dynamics and that individual NTK diagnostics are not monotonic predictors of convergence\.

The remainder of this paper is organized as follows: Section II reviews the relevant background, Section III describes the methodology and experimental setup, Section IV presents the results, and Section V concludes the paper\.

## IIBackground and Related Work

The optimization dynamics of neural networks are strongly shaped by the frequency structure of the target function, often learning simpler functions before more complex ones\. Rahaman et al\.\[[10](https://arxiv.org/html/2608.19497#bib.bib11)\]showed that networks trained with gradient\-based methods preferentially fit low\-frequency components of a target before capturing higher\-frequency details, a phenomenon known asspectral bias\. Their Fourier\-based analysis further showed that low\-frequency components are more robust to parameter perturbations, highlighting an asymmetry in how neural networks learn and represent different spectral components\.

The Neural Tangent Kernel \(NTK\) framework provides a principled view of these learning dynamics\[[5](https://arxiv.org/html/2608.19497#bib.bib13)\]\. In the infinite\-width limit, the evolution of a neural network under gradient descent can be described as kernel gradient flow in function space, governed by a deterministic kernel that remains effectively constant during training\. Decomposing the dynamics along the kernel eigenfunctions gives

ft=∑i\(1−e−λi​t\)​⟨y,ϕi⟩​ϕi,f\_\{t\}=\\sum\_\{i\}\\left\(1\-e^\{\-\\lambda\_\{i\}t\}\\right\)\\langle y,\\phi\_\{i\}\\rangle\\phi\_\{i\},\(1\)whereλi\\lambda\_\{i\}andϕi\\phi\_\{i\}denote the eigenvalues and eigenfunctions of the kernel\. Components associated with larger eigenvalues are therefore learned more rapidly, whereas directions associated with smaller eigenvalues converge more slowly\. The NTK eigenspectrum thus provides a direct connection between kernel geometry and optimization dynamics\.

The spectral properties of the NTK also characterize its inductive bias\. Bietti and Mairal\[[1](https://arxiv.org/html/2608.19497#bib.bib14)\]analyzed the reproducing kernel Hilbert space induced by the NTK through spherical harmonics and showed that eigenvalues decay with increasingly oscillatory components\. Consequently, smoother functions are preferentially represented, while more complex components receive progressively smaller spectral weights\. Their analysis also identifies a trade\-off between smoothness and approximation capacity, since slower spectral decay allows richer function classes at the cost of reduced regularity\.

These theoretical predictions are supported by empirical studies\. Cao et al\.\[[2](https://arxiv.org/html/2608.19497#bib.bib12)\]tracked the optimization of functions constructed from spherical harmonics and observed that lower\-degree components are learned earlier than higher\-degree components, consistently with the spectral dynamics predicted by the NTK framework\. Together, these results motivate the use of kernel eigenspectra and their evolution during training as empirical diagnostics of learning dynamics and inductive bias\.

### II\-AQuantum Computing and Variational Quantum Circuits

Quantum computing exploits properties such as superposition and entanglement to process information through quantum state evolution\[[7](https://arxiv.org/html/2608.19497#bib.bib1),[9](https://arxiv.org/html/2608.19497#bib.bib4)\]\. Beyond quantum algorithms for tasks such as factoring and search\[[13](https://arxiv.org/html/2608.19497#bib.bib8),[4](https://arxiv.org/html/2608.19497#bib.bib9)\], these properties have motivated the development of quantum machine learning models that use quantum circuits as trainable computational components, particularly in hybrid quantum–classical settings\.

Variational Quantum Circuits \(VQCs\), also called parametrized quantum circuits, are among the most widely studied models for near\-term quantum machine learning\[[3](https://arxiv.org/html/2608.19497#bib.bib2),[11](https://arxiv.org/html/2608.19497#bib.bib6)\]\. A VQC combines classical data encoding, layers of parametrized unitary gates, and observable measurements\. Its trainable circuit parameters are typically optimized by a classical optimizer, forming a hybrid learning loop\. Because VQCs can be constructed with relatively shallow circuits, they are commonly investigated in the context of noisy intermediate\-scale quantum \(NISQ\) devices\[[3](https://arxiv.org/html/2608.19497#bib.bib2)\]\.

Classical data must first be mapped to a quantum state through an encoding operation\. Common strategies include amplitude and angle encoding\[[12](https://arxiv.org/html/2608.19497#bib.bib3)\]\. In angle encoding, classical input values parameterize single\-qubit rotations, introducing data\-dependent amplitudes or relative phases according to the rotation axis and state preparation\. The encoded state is subsequently transformed by trainable quantum gates and entangling operations, and observable measurements produce classical outputs that can be used as learned features or predictions\.

The representational properties of a VQC depend on its encoding strategy, circuit depth, parametrization, and entangling topology\[[11](https://arxiv.org/html/2608.19497#bib.bib6),[14](https://arxiv.org/html/2608.19497#bib.bib5)\]\. Data re\-uploading extends this construction by repeatedly encoding the input at multiple circuit layers, rather than introducing it only once\. This repeated encoding can increase the class of input\-dependent functions represented by the circuit\[[8](https://arxiv.org/html/2608.19497#bib.bib7)\]\. In this work, we employ rotation encoding with data re\-uploading; the specific architecture is defined in Section III\-B\.

Here, VQCs are studied as compact nonlinear feature maps whose learning dynamics can be characterized through empirical NTK analysis and compared with closely matched classical baselines\. Our objective is to identify differences in learning geometry rather than to assume a universal quantum advantage\.

## IIIMethodology

### III\-AProblem setting

We study forecasting tasks on synthetic multivariate signals with controlled spectral complexity\. Given an input sequence

X∈ℝL×F,X\\in\\mathbb\{R\}^\{L\\times F\},whereLLis the sequence length andFFis the number of features, the goal is to predict the next value of a target variable at time stepL\+1L\+1\. Our analysis focuses on comparing how a compact hybrid quantum model and a classical baseline with comparable structure learn under increasingly challenging spectral regimes\.

### III\-BModels under comparison

#### III\-B1Hybrid quantum model

The hybrid quantum model consists of a classical input projection, a variational quantum circuit \(VQC\), and a linear readout head\. The input sequence is first projected to a latent dimension equal to the number of qubits\. Only the last projected time step is used by the quantum block and is scaled as:

xq=πtanh\(X~L,:\)∈ℝQ,x\_\{q\}=\\pi\\tanh\\left\(\\widetilde\{X\}\_\{L,:\}\\right\)\\in\\mathbb\{R\}^\{Q\},\(2\)whereQQdenotes the number of qubits\.

Before data encoding, each qubit is prepared in an equal superposition, yielding the initial state

\|Ψ0⟩=H⊗Q​\|0⟩⊗Q=\|\+⟩⊗Q\.\|\\Psi\_\{0\}\\rangle=H^\{\\otimes Q\}\|0\\rangle^\{\\otimes Q\}=\|\+\\rangle^\{\\otimes Q\}\.\(3\)This preparation ensures that the subsequentRZR\_\{Z\}encoding introduces input\-dependent relative phases from the first encoding operation\.

The VQC containsnlayersn\_\{\\mathrm\{layers\}\}data re\-uploading layers,

U\(l\)​\(xq\)=Uent​\[⨂i=1QRot⁡\(ϕl,i,ϑl,i,ωl,i\)\]​\[⨂i=1QRZ​\(sl​xq,i\)\],U^\{\(l\)\}\(x\_\{q\}\)=U\_\{\\mathrm\{ent\}\}\\left\[\\bigotimes\_\{i=1\}^\{Q\}\\operatorname\{Rot\}\(\\phi\_\{l,i\},\\vartheta\_\{l,i\},\\omega\_\{l,i\}\)\\right\]\\left\[\\bigotimes\_\{i=1\}^\{Q\}R\_\{Z\}\(s\_\{l\}x\_\{q,i\}\)\\right\],\(4\)wheresls\_\{l\}is a trainable re\-uploading scale andUentU\_\{\\mathrm\{ent\}\}applies CNOT gates in a ring topology\. The input representation is re\-encoded at every circuit layer\. The final state is

\|Ψ\(xq\)⟩=U\(nlayers\)\(xq\)⋯U\(1\)\(xq\)\|Ψ0⟩\.\|\\Psi\(x\_\{q\}\)\\rangle=U^\{\(n\_\{\\mathrm\{layers\}\}\)\}\(x\_\{q\}\)\\cdots U^\{\(1\)\}\(x\_\{q\}\)\|\\Psi\_\{0\}\\rangle\.\(5\)One Pauli\-ZZexpectation value is measured per qubit,

zi=⟨Ψ\(xq\)\|Zi\|Ψ\(xq\)⟩,i=1,…,Q,z\_\{i\}=\\langle\\Psi\(x\_\{q\}\)\|Z\_\{i\}\|\\Psi\(x\_\{q\}\)\\rangle,\\qquad i=1,\\ldots,Q,\(6\)and the resulting feature vector is passed to a linear head to produce the scalar forecast\.

#### III\-B2Classical baseline

To make the comparison as controlled as possible, we designed a classical baseline with the same structure of the hybrid quantum model\. As in the quantum case, the input sequence is first projected to a latent dimension of sizeQQand scaled withπ​tanh⁡\(⋅\)\\pi\\tanh\(\\cdot\)\. Again, only the last time step is used for prediction\.

The reuploading strategy of the quantum model is introduced in the classical part as well\. To this end, lety0∈ℝQy\_\{0\}\\in\\mathbb\{R\}^\{Q\}denote the projected last time step\. The classical latent state is initialized ash\(0\)=y0h^\{\(0\)\}=y\_\{0\}, and then updated overnlayersn\_\{\\text\{layers\}\}stages according to

h\(l\+1\)=W\(l\)​\(π​tanh⁡\(h\(l\)\+y0\)\),h^\{\(l\+1\)\}=W^\{\(l\)\}\\left\(\\pi\\tanh\\left\(h^\{\(l\)\}\+y\_\{0\}\\right\)\\right\),where eachW\(l\)W^\{\(l\)\}is a trainable linear mixing layer without bias\. In this way, the original input representationy0y\_\{0\}is re\-injected at every layer, providing a classical analogue of the repeated data encoding performed by the quantum circuit\. The number of classical layers is matched to the number of quantum reuploading layers\.

Finally, the resulting latent representation is passed through a linear layer to generate the prediction\. This construction does not make the two models identical, but it provides a closer comparison by ensuring that both architectures repeatedly mix the same latent input across multiple layers before the final readout\.

### III\-CTrainable parameters

Both models use a latent dimension of eight and four trainable re\-uploading stages\. The resulting hybrid quantum and classical models contain 125 and 281 trainable parameters, respectively\. Thus, the comparison matches latent dimension and depth, but not parameter count; the classical baseline has approximately2\.25×2\.25\\timesmore trainable parameters\.

### III\-DSynthetic datasets

#### III\-D1Harmonic\-mixture dataset

Our first benchmark consists of multivariate signals built from a sinusoidal component and a higher\-frequency harmonic\. For each sample and feature, amplitudes and phases are drawn uniformly at random :

A∼𝒰⁡\(0\.5,1\.5\),ϕ∼𝒰⁡\(0,2​π\)\.A\\sim\\mathcal\{U\}\(0\.5,1\.5\),\\qquad\\phi\\sim\\mathcal\{U\}\(0,2\\pi\)\.Given a base frequencyff, the signal is defined as

x⁡\(t\)=A​sin⁡\(2​π​f​t\+ϕ\)\+0\.5​A​sin⁡\(2​π​\(3​f\)​t\+1\.5​ϕ\)\+ϵ⁡\(t\),x\(t\)=A\\sin\(2\\pi f t \+ \\phi\)\+0\.5A\\sin\(2\\pi\(3f\)t \+ 1\.5\\phi\)\+\\epsilon\(t\),whereϵ⁡\(t\)∼𝒩⁡\(0,σ2\),σ=0\.2\.\\epsilon\(t\)\\sim\\mathcal\{N\}\(0,\\sigma^\{2\}\),\\qquad\\sigma=0\.2\.The time grid is

t=\{0,1L\+1,2L\+1,…,LL\+1\},t=\\left\\\{0,\\frac\{1\}\{L\+1\},\\frac\{2\}\{L\+1\},\\dots,\\frac\{L\}\{L\+1\}\\right\\\},implemented as a uniform partition of\[0,1\)\[0,1\)withL\+1L\+1points\. The model input consists of the firstLLpoints of the signal across all features, while the target is the next\-step value of the first feature:X=x1:L,y=xL\+1X=x\_\{1:L\},\\qquad y=x\_\{L\+1\}\.

#### III\-D2Chirp dataset

To test whether the observed behavior persists beyond stationary harmonic signals, we also consider nonstationary chirp signals\. For each feature, amplitudes and phases are sampled from the same distributions as above\. The base chirp is constructed from a linearly varying instantaneous frequency fromf0f\_\{0\}tof1f\_\{1\}, whose integrated phase is

Φbase​\(t\)=2​π​\(f0​t\+12​\(f1−f0\)​t2\)\.\\Phi\_\{\\text\{base\}\}\(t\)=2\\pi\\left\(f\_\{0\}t\+\\frac\{1\}\{2\}\(f\_\{1\}\-f\_\{0\}\)t^\{2\}\\right\)\.A harmonic chirp component is added using a corresponding sweep from3​f03f\_\{0\}to3​f13f\_\{1\}:

Φharm​\(t\)=2​π​\(3​f0​t\+12​\(3​f1−3​f0\)​t2\)\.\\Phi\_\{\\text\{harm\}\}\(t\)=2\\pi\\left\(3f\_\{0\}t\+\\frac\{1\}\{2\}\(3f\_\{1\}\-3f\_\{0\}\)t^\{2\}\\right\)\.The resulting signal is

x⁡\(t\)=A​sin⁡\(Φbase​\(t\)\+ϕ\)\+wh​A​sin⁡\(Φharm​\(t\)\+1\.5​ϕ\)\+ϵ⁡\(t\),x\(t\)=A\\sin\(\\Phi\_\{\\text\{base\}\}\(t\) \+ \\phi\)\+w\_\{h\}A\\sin\(\\Phi\_\{\\text\{harm\}\}\(t\) \+ 1\.5\\phi\)\+\\epsilon\(t\),wherewh=0\.5w\_\{h\}=0\.5is the harmonic weight andϵ⁡\(t\)\\epsilon\(t\)is Gaussian noise with standard deviationσ=0\.2\\sigma=0\.2\. As in the harmonic dataset, the target is the next\-step value of the first feature\.

#### III\-D3Experimental regimes

For the harmonic\-mixture benchmark, we varied the base frequency overf∈\{1,2,4,8,12,16\}f\\in\\\{1,2,4,8,12,16\\\}to generate tasks with increasing oscillatory complexity\. For the chirp benchmark, we fixed the initial frequency tof0=2f\_\{0\}=2and varied the final frequencyf1=\{8,12,16\}f\_\{1\}=\\\{8,12,16\\\}\. Both benchmarks were evaluated with 200 and 50 training samples to examine the effect of data\-limited regimes\.

Both architectures use a latent dimension of eight and four re\-uploading stages\. The hybrid model therefore employs eight qubits and four rotation–entanglement layers, while the classical baseline uses four corresponding mixing layers\. Both models were trained using the Adam optimizer\[[6](https://arxiv.org/html/2608.19497#bib.bib10)\]with mean squared error \(MSE\) loss\. To reduce optimizer mismatch between architectures, learning rates were selected independently by minimizing the mean AULC over 20 random seeds across\{0\.001,0\.003,0\.01,0\.03,0\.1,0\.3\}\\\{0\.001,0\.003,0\.01,0\.03,0\.1,0\.3\\\}, yielding0\.10\.1for the hybrid quantum model and0\.030\.03for the classical baseline\. For each condition, fixed validation and test sets of 500 and 1000 samples were shared across models and runs\. The 20 random seeds vary only parameter initialization\. Validation MSE was evaluated at every epoch, and the checkpoint with the lowest validation loss was selected independently for each model and seed\. Test MSE was then evaluated at the selected checkpoint\. Training loss, AULC, and NTK diagnostics are computed on the training set\.

### III\-EEmpirical NTK analysis

To characterize differences in learning beyond final predictive accuracy, we analyze training dynamics through the empirical NTK\. For each model and selected training epoch, we compute the kernel induced by the Jacobian of the model outputs with respect to trainable parameters on the training set\. This gives a local representation of how samples are related by the current parameterization of the model\.

Our goal is not to derive new NTK theory, but to use NTK\-based diagnostics as a principled empirical framework for comparing the learning geometry of the quantum and classical models\.

From the empirical kernel, we compute the following quantities\.

##### Kernel\-target alignment\.

We quantify how strongly the kernel supports the target direction through

λeff​\(K,y\)=y⊤​K​yTr⁡\(K\)\+ε\.\\lambda\_\{\\mathrm\{eff\}\}\(K,y\)=\\frac\{y^\{\\top\}Ky\}\{\\mathrm\{Tr\}\(K\)\+\\varepsilon\}\.This metric measures how well the target projects onto the dominant directions of the kernel\. Larger values indicate that the kernel is better aligned with the target, which is generally associated with faster early\-stage learning\.

##### Top\-1 spectral mass\.

We compute the fraction of spectral mass carried by the leading eigenvalue:

m1​\(K\)=λmax∑jλj\.m\_\{1\}\(K\)=\\frac\{\\lambda\_\{\\max\}\}\{\\sum\_\{j\}\\lambda\_\{j\}\}\.This metric captures how concentrated the kernel spectrum is around its dominant mode\. A larger top\-1 mass indicates that the kernel is dominated by a single principal direction, whereas smaller values reflect a more distributed spectral structure\.

##### Kernel drift\.

To measure how much the kernel changes during training relative to initialization, we compute

drift⁡\(Kt,K0\)=‖Kt−K0‖‖K0‖\.\\mathrm\{drift\}\(K\_\{t\},K\_\{0\}\)=\\frac\{\\\|K\_\{t\}\-K\_\{0\}\\\|\}\{\\\|K\_\{0\}\\\|\}\.This quantity measures the relative deformation of the kernel geometry during optimization\. Larger values indicate that the model departs more strongly from its initial kernel, suggesting a greater degree of kernel reconfiguration during training\. Smaller values indicate that the model remains closer to its initialization, which is consistent with a more stable or near\-frozen kernel regime\.

Experiments were implemented in PennyLane \[16\] using default\.qubit, the JAX\-JIT interface, backpropagation, and vectorized circuit evaluation, and executed on an RTX 3080 GPU with a Ryzen 7 5700X CPU\.

## IVResults

In the results below, we first examine four training diagnostics: kernel\-target alignment, kernel drift, top\-1 spectral mass, and training loss\. Together, these quantities characterize target alignment, kernel reconfiguration, spectral concentration, and in\-sample optimization\. Validation\-selected checkpoints and held\-out test performance are analyzed separately\. In all figures, curves and shaded regions show the mean±\\pmone standard deviation over 20 initialization seeds\.

![Refer to caption](https://arxiv.org/html/2608.19497v1/ntk_main_metrics_200.png)Fig\. 1:NTK Dynamics for the Harmonic\-mixture dataset with 200 training samples\.Figures[1](https://arxiv.org/html/2608.19497#S4.F1)and[2](https://arxiv.org/html/2608.19497#S4.F2)show the harmonic\-mixture results with 200 and 50 training samples, respectively\. Across both regimes, the classical baseline exhibits stronger early kernel\-target alignment and larger kernel drift, whereas the hybrid model remains closer to its initial kernel and develops a less concentrated spectrum\. The training\-loss trajectories also differ across spectral regimes, with the separation becoming more pronounced in the 50\-sample setting\. These results reveal persistent differences in the optimization geometry whose relation to validation convergence is examined below\.

![Refer to caption](https://arxiv.org/html/2608.19497v1/ntk_main_metrics_50.png)Fig\. 2:NTK Dynamics for the Harmonic\-mixture dataset with 50 training samples\.![Refer to caption](https://arxiv.org/html/2608.19497v1/ntk_main_metrics_200chirp.png)Fig\. 3:NTK Dynamics for the Chirp dataset with 200 training samples\.Figures[3](https://arxiv.org/html/2608.19497#S4.F3)and[4](https://arxiv.org/html/2608.19497#S4.F4)extend the analysis to nonstationary chirp signals\. The classical baseline again develops stronger kernel\-target alignment and abrupt early kernel drift, whereas the hybrid model evolves more gradually and generally maintains a less concentrated spectrum\. The difference in training\-loss trajectories is particularly visible with 50 samples, where the classical model degrades more strongly relative to the 200\-sample regime\. Thus, the characteristic kernel\-level differences persist beyond stationary harmonic signals and remain visible under limited training data\.

![Refer to caption](https://arxiv.org/html/2608.19497v1/ntk_main_metrics_50chirp.png)Fig\. 4:NTK Dynamics for the Chirp dataset with 50 training samples\.Table[I](https://arxiv.org/html/2608.19497#S4.T1)examines the frequency dependence of the learning dynamics in the 50\-sample harmonic\-mixture regime\. The hybrid model reaches its validation\-selected checkpoint earlier at five of the six evaluated frequencies, with the largest epoch reduction atf=12f=12\(58\.73%58\.73\\%\)\. Significant paired differences in checkpoint epoch are observed forf=1,2,4,8,f=1,2,4,8,and1212\. However, this behavior does not directly track training AULC\. At lower frequencies, earlier validation selection occurs despite a higher cumulative training loss, whereas atf=12f=12both metrics favor the hybrid model\. Atf=16f=16, the AULC advantage persists although the classical model reaches its selected checkpoint slightly earlier\. Taken together, these results reveal a decoupling between cumulative training efficiency and validation\-selected convergence: lower training AULC does not systematically imply earlier attainment of the best validation solution\.

TABLE I:Frequency\-dependent learning dynamics for the 50\-sample harmonic\-mixture benchmark\. Positive epoch reduction indicates earlier attainment of the validation\-selected checkpoint by the hybrid model; positive AULC gain indicates lower training AULC\.pEp\_\{E\}andpAp\_\{A\}correspond to paired Wilcoxon signed\-rank tests over 20 initialization seeds\.FreqEpoch reduction\(%\)pEp\_\{E\}Train AULC gain\(%\)pAp\_\{A\}111\.870\.0048\-9\.440\.001229\.200\.0185\-12\.40<<0\.0001426\.800\.0197\-5\.350\.0353824\.360\.00316\-0\.350\.84691258\.730\.00098\.84<<0\.000116\-6\.060\.09905\.72<<0\.0001Table[II](https://arxiv.org/html/2608.19497#S4.T2)summarizes validation\-selected convergence and held\-out performance across benchmarks\. The hybrid model reaches its selected checkpoint earlier in five of six harmonic conditions and in two of three evaluated chirp conditions\. In contrast, test gains are small and do not reveal a consistent held\-out advantage, with aggregated test differences remaining within3\.6%3\.6\\%\. This parity is obtained despite the hybrid model using approximately55%55\\%fewer trainable parameters\. Thus, comparable held\-out performance is reached through distinct optimization trajectories, with the hybrid model more frequently attaining its validation\-selected solution earlier\.

TABLE II:Validation\-selected convergence and held\-out performance across forecasting benchmarks\. Values are averaged across evaluated frequencies and 20 initialization seeds\. Positive test gain indicates lower test MSE for the hybrid model; epoch reduction indicates earlier attainment of the validation\-selected checkpoint\. The last column indicates in how many frequency the quantum model achieved the checkpoint earlierDatasetNNTest gain \(%\\%\)Epoch reductionQ earlierHarmonic200\-2\.214\.725/6Harmonic50\-1\.075\.435/6Chirp2001\.4011\.533/3Chirp50\-3\.583\.122/3To assess whether the observed performance differences can be explained solely by the spectral structure introduced by quantum angle encoding, we considered an additional classical baseline augmented with explicit Fourier features\. The model preserves the input projection, data re\-uploading structure, number of mixing layers, and linear readout of the original classical baseline\. However, before each mixing layer, the latent representation is expanded using sinusoidal featuresϕ⁡\(z\)=\[z,\{s​i​n​\(b​z\),c​o​s​\(b​z\)\}b∈B\]\\phi\(z\)=\[z,\\\{sin\(bz\),cos\(bz\)\\\}\_\{b\\in B\}\]over a fixed set of frequency bandsB=\(1,2,4,8,16\)B=\(1,2,4,8,16\), together with the raw latent features\. The resulting feature vector is normalized by the square root of its dimension and projected back to theQQ\-dimensional latent space\. This baseline provides the classical model with an explicit periodic representation while retaining the overall re\-uploading architecture used in the original comparison\.

The Fourier\-augmented classical baseline does not eliminate the optimization gap observed on the harmonic\-mixture benchmark\. Across the complete frequency sweep, the hybrid model achieves7\.637\.63–20\.31%20\.31\\%lower final training loss and18\.7018\.70–45\.90%45\.90\\%lower training AULC\. The final\-loss differences are significant at all six frequencies \(p≤0\.0194p\\leq 0\.0194\), while the AULC differences satisfyp<0\.0001p<0\.0001throughout\. These results indicate that explicit sinusoidal features alone are insufficient to explain the observed differences in training dynamics\.

To isolate the contribution of repeated data encoding, we vary the number of re\-uploading operations while keeping the four trainable rotation–entanglement layers fixed\. Figure[5](https://arxiv.org/html/2608.19497#S4.F5)shows that increasing the number of re\-uploads consistently reduces final loss across the three representative frequency regimes\. This improvement is accompanied by larger kernel drift and, particularly forf=8f=8andf=12f=12, lower top\-1 spectral mass\. Kernel\-target alignment does not increase with additional re\-uploading and instead remains stable or decreases\. These results show that repeated encoding affects both predictive performance and the learned kernel geometry\. They also indicate that stronger global alignment or smaller kernel drift, when considered individually, are not sufficient indicators of lower final error\.

![Refer to caption](https://arxiv.org/html/2608.19497v1/reuploading_sweep_summary.png)Fig\. 5:Effect of the number of data re\-uploading operations on final loss and empirical NTK metrics for the harmonic\-mixture benchmark with 50 training samples\. Circuit depth is fixed to four rotation–entanglement layers\.### IV\-ADiscussion

The central result of this study is not a systematic difference in endpoint generalization, but the observation that comparable held\-out performance can arise from substantially different learning geometries\. The 125\-parameter hybrid model remains within a few percent of the 281\-parameter classical baseline across all evaluated regimes, while more frequently reaching its validation\-selected solution in fewer optimization epochs\. At the same time, the architectures exhibit persistent differences in kernel\-target alignment, kernel drift, and spectral concentration\. Across the experimental settings, the empirical NTK diagnostics reveal a recurring distinction between the learning dynamics of the hybrid quantum and classical models\. The classical baseline generally exhibits stronger early kernel\-target alignment and faster initial loss reduction, whereas the hybrid model tends to remain closer to its initial kernel and develop a less concentrated spectral structure\. These patterns remain qualitatively visible for both stationary harmonic\-mixture and nonstationary chirp signals\.

Importantly, the hybrid model preserves broadly comparable held\-out performance despite using substantially fewer trainable parameters and following a systematically different optimization trajectory\. The frequency analysis reveals a decoupling between cumulative training efficiency and validation\-selected convergence: the hybrid model frequently reaches its best validation checkpoint earlier even in regimes where its training AULC is higher\. Moreover, stronger early kernel\-target alignment in the classical model is not systematically associated with earlier validation convergence\. Together, these observations indicate that neither cumulative training loss nor a single NTK diagnostic provides a monotonic metric for how rapidly the best validation solution emerges\.

The Fourier\-feature comparison and re\-uploading ablation further support this interpretation\. The persistence of the optimization gap under explicit Fourier augmentation shows that the observed learning dynamics are not reproduced by periodic feature expansion alone\. Moreover, increasing the number of quantum data re\-uploads reduces final training loss despite increasing kernel drift and without strengthening kernel\-target alignment\. This second form of decoupling reinforces the observation that individual kernel diagnostics cannot be interpreted as monotonic indicators of optimization quality\. Rather, learning behavior appears to emerge from the joint evolution of kernel alignment, spectral organization, and kernel reconfiguration\.

Instead, the interaction between repeated data encoding and trainable quantum transformations appears to contribute to the resulting kernel geometry\. However, the present experiments do not isolate a uniquely quantum mechanism, as the hybrid architecture combines angle encoding, trainable rotations, and entangling operations\. Dedicated ablations of these components remain necessary\.

The observed results also cannot be explained by a larger quantum model: the hybrid and classical architectures contain 125 and 281 trainable parameters, respectively\. Nevertheless, parameter count is not a direct measure of equivalent representational capacity across different architectures; the comparison should therefore be interpreted as structurally aligned rather than parameter matched\.

Overall, empirical NTK analysis provides a useful perspective for characterizing hybrid quantum models beyond endpoint accuracy\. Our results identify reproducible differences in optimization geometry and convergence epoch, despite presenting comparable testing metrics\. Because the quantum experiments use exact expectation values on a noiseless state\-vector simulator, whether these kernel\-level patterns persist under finite\-shot and hardware noise remains an open question\. Future work should investigate their architectural origin and robustness under realistic quantum noise and more diverse classical baselines\.

## VConclusion

In this work, empirical NTK diagnostics reveal reproducible differences between the learning geometries of a compact hybrid quantum forecasting model and a structurally aligned classical baseline\. The classical model generally exhibits stronger early kernel\-target alignment and faster initial training\-loss reduction, whereas the hybrid model tends to develop a less concentrated kernel spectrum and undergo smaller kernel drift\.

These differences are accompanied by a non\-monotonic relationship between optimization efficiency and validation\-selected convergence\. Thus, earlier validation convergence is not systematically predicted by training AULC or stronger kernel\-target alignment\.

A Fourier\-augmented classical baseline does not eliminate the observed in\-sample optimization gap, while the re\-uploading ablation shows that lower training loss can emerge despite increased kernel drift and without stronger alignment\. Together, these findings show that individual NTK diagnostics do not provide monotonic relations for learning behavior across the studied architectures\. Rather than establishing a general quantum advantage, our results identify systematically different optimization trajectories that reach comparable held\-out performance and motivate a more joint interpretation of kernel alignment, spectral organization, and kernel evolution\.

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