Understanding Generalization in Quantum Machine Learning with Margins
Authors: Tak Hur
Daniel K. Park
Venue: ICML 2025 poster (2025)
Keywords: Quantum Machine Learning
Generalization
Margin
Primary area: theory->learning_theory
TL;DR: This paper establishes a margin-based generalization bound for QNNs, showing that margin distributions better predict generalization than parameter-based metrics.

Abstract: Understanding and improving generalization capabilities is crucial for both classical and quantum machine learning (QML). Recent studies have revealed shortcomings in current generalization theories, particularly those relying on uniform bounds, across both classical and quantum settings. In this work, we present a margin-based generalization bound for QML models, providing a more reliable framework for evaluating generalization. Our experimental studies on the quantum phase recognition dataset demonstrate that margin-based metrics are strong predictors of generalization performance, outperforming traditional metrics like parameter count. By connecting this margin-based metric to quantum information theory, we demonstrate how to enhance the generalization performance of QML through a classical-quantum hybrid approach when applied to classical data.

Understanding Generalization in Quantum Machine Learning with Margins
Tak Hur
Daniel K. Park
Abstract
quantities (Banchi et al., 2021; Caro et al., 2024). Despite these efforts, existing methods have shown limitations in fully capturing the behavior of QML models.
Understanding and improving generalization ca- pabilities is crucial for both classical and quantum machine learning (QML). Recent studies have revealed shortcomings in current generalization theories, particularly those relying on uniform bounds, across both classical and quantum set- tings. In this work, we present a margin- based generalization bound for QMI. models, providing more reliable framework for evaluating gener- alization. Our experimental studies the quan- tum phase recognition dataset demonstrate that margin-based metrics are strong predictors of generalization performance, outperforming tra- ditional metrics like parameter count. By connect- ing this margin- based metric to quantum infor- mation theory, we demonstrate how enhance the generalization performance of QML through a classical-quanture hybrid approach when applied to classical data.
The current understanding of generalization in QML pre- dominantly relies on uniform bounds, which hold uniformly across all hypotheses within a given class. However, in clas- sical deep learning, uniform bounds have faced significant criticism for their lack of practical relevance (Zhang et al., 2021; Nagarajan Kolter, 2019; Dziugaite Roy, 2017). In particular, Zhang al. (2021) demonstrated that mod- emn deep learning models can easily overfit random labels. Since uniform bounds apply equally to all hypotheses in the function family, this result indicates that these bounds fail to distinguish between models that generalize well and those that merely memorize data, highlighting their vacuity in deep learning. Building on this observation, Gil-Fuster et al. (2024) explored the use of parameterized quantum circuits, also known Quantum Neural Networks (QNNs), benchmark quantum dataset with randomized labels. Despite the small number of qubits, the QNNs were able to overfit the random labels, indicating that uniform bounds are equally ineffective the QMI. framework.
1. Introduction
In classical deep learning, the looseness of uniform bounds has prompted many researchers to shift their focus toward explanatory tools, such exploring their correlation with observed generalization. Notably, Bartlett al. (2017) pro- posed margin- baseen generalization bound for deep net- works and demonstrated that spectrally normalized margin distribution is strongly correlated with generalization per- formance. Subsequent research has consistently found that margin- metries are reliable predictors generaliza- tion performance (Neyshabur et al., 2018; Jiang et al., 2018; Dziugaite al., 2020).
Quantum machine learning (QML) presents exciting oppor- tunities to expand the horizons of machine learning beyond classical approaches. Generalization the ability to learn from examples and make accurate predictions on unseen data-is is a core component of intelligence and a critical fac- tor that quantifies the effectiveness of machine learning mod- els in real- world applications. Thus, fundamental challenge in QML, as in classical machine learning, is understand, characterize, and optimize generalization. Generalization in QMI. has been studied through various factors, such as the number of parameters (Caro et al., 2022), effective dimen- sion (Abbas et al., 2021a), quantum resource theory (Bu et al., 2021; 2022: 2023), and quantum cntematiionthhoortte
In this work, we demonstrate that margins are strong predic- tors of generalization performance within the QML frame- work as well. Our goal is to shift the focus away from the commonly emphasized parameter count, highlighting mar- gins as more effective tool for evaluating and controlling generalization. Moreover, this margin-based perspective provides a systematic pathway to improving the general- ization of QML models when applied to classical data, as it reveals that maximizing the separability of classes em- bedded in the quantum feature space is key to optimizing
Department of Statistics and Data Science, Yonsei Univer- sity, Scoul, Republic of Korea *Department of Applied Statis- tics, Yonsei University, Seoul, Republic of Korea. Correspon- (o0040000@@oosi.c...mmmm
dence to: Tak Hur <dkd.park park@ya yonke@ooo...cck.>...
Daniel K. Park
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Understanding Generalization in Quantum Machine Learning with Margins
performance.
theory (Neyshabur et al., 2018). More recently, Jiang et al. (2018) have underscored the importance of analyzing the margin distribution across training samples for accurately predicting the generalization gap.
The central contributions of this work are:
We establish margin-based based generalization bound for multiclass classification with QNNs by adapting the ap- proach of Bartlett et al. (2017), originally developed for classical neural networks, to the quantum domain. This involves interpreting quantum measurements Lips- chitz continuous nonlinear activations and extending matrix covering techniques to complex-valued spaces. The unitary constraints of quantum circuits and normal- ization of quantum states simplify the complexity of the original framework, enabling us to assess general- ization performance through the margin normalized by the spectral norms of the measurement operators. This margin-based bound yields tighter upper bound when the hypothesis achieves larger margins on the sample set, addressing the limitations of uniform bounds in capturing meaningful generalization behavior.
Concurrently, efforts to understand generalization in QML have often involved extending established techniques from classical learning theory. These include metrics such as parameter counts (Caro al., 2022), effective model di- mensions (Abbas et al., 2021b), and quantum information- theoretic measures (Caro et al., 2024). However, a dedicated extension of margin- QML domain has remained a significant gap. This paper directly addresses this by developing a margin margin-based frame- work for QMI. models, aiming to provide more effective predictions of their generalization capabilities.
generalization analysis to the
The process of data embedding is another critical aspect when applying QMI. models to classical data, with demon- strable impacts on approximation error (Schuld et al., 2021), optimization landscapes (Holmes et al., 2022), and on gen- cralization error (Caro et al., 2021). Specifically, Caro et al. (2021) investigated the role of quantum embedding on gen- eralization utilizing fat-shattering arguments. In Section 5, we introduce a novel approach to investigating the influ- ence of data encoding on generalization by leveraging our margin- based framework. We establish connection be- tween the margin mean and the trace distance, a key mea- sure of quantum state distinguishability. This perspective not only contributes to the theoretical understanding of how data embeddings affect generalization but also offers a prac- tical guide for enhancing the performance of QML models.
We experimentally demonstrate a strong correlation between generalization and margin distribution, even in scenarios with random labels, where traditional met- ries like parameter count ineffective. Furthermore, we compare margin-basee based metrics (e.g., lower quartile, median, and mean) against parameter-based metrics, consistently finding margins to be more reliable predic- tors of generalization in QNNs.
We further establish a connection between margins and quantum state discrimination, a core concept in quantum information theory. This insight underscores the pivotal role of quantum embeddings in both opti- mization and generalization. providing valuable guid- ance for designing effective QMI. models when applied to classical data. We demonstrate that Neural Quan- tum Embedding (NQE) (Hur et al., 2024), a classical- quantum hybrid approach optimized for high data dis- dinguishability, yields large margins, thereby enhances generalization performance.
3. Margin Bound for Quantum Neural Networks
3.1. Multiclass Classification with Quantum Neural Networks
Consider an unknown joint probability distribution D gov- erning the quantum state p and its corresponding label y. where E CN and E [k], representing an n-qubit (n = log, N). k-class classification task. QNNs employ a parameterized quantum eircuit U(),, along with a set of positive operator- -pllue measurements (POVMs) {E,) to perform classification. Here, U()) is a unitary operator parameterized by 0, satisfying UUT = ULU I, where Ut is the conjugate transpose of U. POVMs are a gener- alization of projective measurements in quantum mechan- ics, where {Ex) represents a set of positive semidefi- nite operators that sum to the identity. Specifically, the QNN maps quantum state to k-dimensional vector, he(p) (() ())) where each element represents the probability of the state being assigned to particular class.
2. Related Works
The concept of margin was popularized in machine learn- ing by the development of support vector machines (Cortes & Vapnik, 1995). Following this, the theoretical connec- tions between large margins and generalization bounds were explored for model classes such as two-layer neural net- works (Bartlett, 1996) and linear classifiers Shawe-Taalor et al., 1998), establishing margin analysis as a valuable tool for understanding generalization. In the context of modern deep learning, spectrally normalized margin bounds were introduced, derived from perspectives including covering number arguments (Bartlett al., 2017) and PAC- PAC-Bayesian
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Understanding Generalization in Quantum Machine Learning with Margins
Given m independent and identically distributed (i.i.d.) sam- ples S = {(pi, } the goal is find hypothesis h" (or optimal parameters 0*) that minimizes the true er- ror, R(h²) = ((po)>=[ (arg max, h² (p), * y)l, where 1 (xx = if x is true, and O otherwise. Since the true distribution D is unknown, we instead seek a hypothesis (from a hypothesis class H) with small empirical risk, R(h) = m Ex= 1(arg max, h(p:), * y.).
In this section, we present an analytic bound for R ((( 1)s) in terms of quantum channel components, thereby estab- lishing a margin-based generalization bound for QML. We focus on pure state inputs, I E CN,, which can be general- ized to mixed state inputs via vectorization.
Given a set of POVMs {Er we define the quantum
measurement functions as gi(z) x E,r and g(z) {al E 1= For pure state inputs, the function class be- comes F, = {(x. v) + L, (M((((())))) U Uq}.
For a given hypothesis h, the generalization gap is de- fined as the difference between the true and empirical risks, g(h) = R(h)- - R(h). A common approach to under- standing generalization is to upper bound g(h) using a complexity measure that depends on the hypothesis class, : U E Uo], where Uo denotes the space of unitaries accessible by QNNs. The complexity of H depends hyperparameters such as the quantum circuit architecture, the choice of unitary ansatz, the number of unitary layers, and the selection of POVMs.
To derive an analytic expression for the Rademacher com- plexity R(F)))ss) in terms of the quantum channel com- ponents, we employ covering number techniques, which discretize the function class with a finite set of representa- tive elements. The e-covering number of a set A, denoted as (A e, || II), represents the minimum cardinality of any subset B S A such that suPaE minse ||a bll S e. Obtaining a covering number bound for (Fr))s involves two steps: (1) utilizing the Lipschitz continuity of ly, M. and g. and (2) employing matrix covering techniques in the context of QNNs.
H = (p= - (Tr(UpU
3.2. Margin Generalization Bound
The concept of margin has been extensively explored since the early days of machine learning, offering theoretical foundations for support vector machines (Cortes & Vapnik, 1995). Recently, margin is adopted to understand general- ization in deep learning (Bartlett et al., 2017; Neyshabur al., 2018).
Let || ll. denote the spectral norm, and let || llp. denote element-wise (p, q) matrix norm. For any y and p 2 1, L, (M y)) is 2/7-Lipschitzz in the Ip norm (see Lemma A.3 of Bartlett et al. ()). Moreover, 9; is 2 | Exllo-Lipschizz as [><(=)]2 < 2 Eilla for all normalized quantum states
2. Consequently,
Margin generalization utilizes the ramp loss function L, R-
llo(u) - g(v)ll2 S 2
Σ || 0112,
if > y
showing that g is 2E-Lipschitz, where E = Ei E.1?
(x) = 1 x//7/
if0 S x S
1.
if 0.
Lipschitz continuity reduces the covering numbers of (Fy)ss to matrix covering. For a data matrix X E C xm where each column represents a quantum state, the Frobe- nius norm satisfies || vm due to normalization. Given a reference unitary Uret and bound b such that - U S b for all U E Uq we obtain
For multiclass tasks, we define the margin operator M as M(v, y)- = byy - maxisly Vi measuring the gap between the correct label's probability and the highest competing label.
To derive margin bound for QNNs, we utilize Rademacher complexity on the function class Fyy = {(p.y) + (M((()))))) h E H) (Mohri et al., 2018). Let o; be ii.d. Rademacher random variables, each taking values +1 with equal probability. For a sample S = {(r. 3:) 1m the sample Rademacher complexity is defined as m -E[ssupppee Σm (( 9.))]. Then, for any 8 >
N ((F))ss,,, e, -111 S SN IINNNNNNNN (IUX: uuu)
67
ll 11-112)
4E
<exp ((32mg)
E 222
In 1N422
=
The first inequality follows from Lipschitz scaling, while the second adapts matrix covering techniques to our quantum setting, extending Bartlett et al. (2017) and Zhang (2002).
and 7 > O, with probability at least dom draw of an i..dd sample S of size m, the following inequality holds for all h H:
over the ran-
Finally. we bound the Rademacher complexity in terms of the covering number using the Dudley's 's entropy inte- gral (Mohri et al., 2018; Shalev-Shwartz & Ben-David, 2014). Consequently, we have following margin bound for QNNs.
In(2/8) 2m
R(h) < (h) + 2 (F))ss 33/
(L)
where (h) represents the empirical margin loss i.e.
Theorem 3.1. Consider an n- QNNs consist of unitary U E Uq and POVMs (E.) for k-class classification. Let
R,,(h) = m-1 E (h((p)) S77 + max h(p)))))
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Understanding Generalization in Quantum Machine Learning with Margins
b be distance bound such that |IU 2.. b holds for any U E Uq. with Uret serving as the reference unitary matrix. Then, for any 8 > 0 and y > O, with probability at least 1 - 8 over the random draw of an i.i.d. sample S of size m, the following inequality holds for all h E H.
significant relevance in condensed matter physies (Sachdev, 1999; 2023; Broecker et al., 2017; Ehadi et al., 2021; Car- rasquilla & Melko, 2017).
Specifically,, we examine the generalized cluster Hamilto- nian defined as J2) = > شー= (Zz X J2 - (+1),, where X, and Z3 are the Pauli operators acting on site j. This Hamiltonian has tunable parameters J/1 and J/ that control interaction strengths. Depending on the values of these parameters, the ground state of the Hamilto- nian falls into one of four distinct phases: (1) ferromagnetic, (2) antiferromagnetic, (3) symmetry--rotectted topological (SPT), or (4) trivial. Consequently, determining the phase of a given ground state with unknown interaction parameters is framed as a four-class classification problem.
1n(1/5)
R(h) < (1)
n VTR
E.ll +
Unlike uniform generalization bounds, margin bound of- fers tighter upper bound when the hypothesis classifies the sample set S with larger margins. Thus, the margin distribu- tion, normalized by the spectral norms of POVMs, is critical for assessing generalization performance. For example, a left-skewed margin distribution, where many samples are classified with small margins, results in large upper bound, indicating poor generalization performance.
QCNNs can overfit the randomized QPR labels, revealing the limitations of uniform bounds in assessing generaliza- tion performance (Gil- et al., 2024). We argue that the margin bound offers a more accurate measure of generaliza- tion in QML models, evidenced by the strong correlation between margin distribution and test accuracy.
Furthermore, when the POVMs are projective measurements (as in many QMI models), the Lipschitz constant for quan- tum mcasurements simplifies to 2//.. In this case, the margin bound simplifies as follows:
Figure I presents the margin distributions of optimized 8- qubit QCNNs using whisskrrr plots, along with cor- responding test accuracies and generalization gaps, for mod- els with one, five, and nine QCNN layers. A right-skewed box plot suggests that the data are classified with larger margins, which is associated with a tighter generalization upper bound (i.e., a smaller right-hand side in the equation of Theorem 3.1).
Corollary 3.2. Under the conditions of Theorem 3.1, if the POVMs {E,}-= are projective measurements, where E.- E, and E, 0 for all i + j. then:
b
nk
1n(1/))
R(h) < R,, +
m
This shows that, for QNNs with projective measurements, the margin bound is independent of the choice of measure- ment operators. Consequently, when comparing generaliza- tion performance across models with projective measure- ments, it suffices to evaluate the margin distribution alone, as normalization by the spectral norm is unnecessary due to its uniformity across models.
Across all layer configurations, increasing label randomiza- tion leads to decreased test accuracy. Concurrently, label randomization shifts the margin distributions significantly to the left. This leftward shift in the margin distributions results in a larger upper bound on the true error, as indicated in Theorem 3.1, showing that the margin bound effectively captures the generalization behavior under randomized la- bels.
Details on the derivation of margin bound for QNNs and its extension mixed input quantum states are provided in Appendix A.I.
Additionally, when labels are not completely randomized, QCNNs with deeper layers tend to achieve higher test accu- racy. This suggests that deeper QCNNs, due to increased expressibility, can identify hypotheses closer to the optimal one. As the test accuracy increases, we observe a corre- sponding rightward shift in the margin distributions, further highlighting that margin distributions are reliable indicators of generalization performance in the QML framework.
4. Experimental Results
4.1. Margin Distribution and Generalization Performance
In this section, we empirically demonstrate a strong corre- lation between margin distribution and the generalization performance of QML models. Building on the framework presented by Gil-Fuster et al. (2024), we revisit the Quantum Phase Recognition (QPR) with randomized labels, utilizing Quantum Convolutional Neural Networks (QCNN) (Cong et al., 2019; Hur et al., 2022). QPR is a classification task aimed at identifying quantum phases of matter, a problem of
Detailed experimental procedures and additional results using different variational ansätze, including more fine- grained analyses of noise levels and layer counts, are pro- vided in Appendix A.3, Figures 20 and 21.
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Understanding Generalization in Quantum Machine Learning with Margins
Noise 0%
Noise 50%
Noise 100%
Test (Gen)
Test (Gen)
Test (Gen)
0.81 (0.06)
0.43 (0.24)
0.25 (0.33)
11-
(0.10)
1L-
(0,28)
1L-
(0.39)
0.85 (0.10)
(0.30)
(0.41)
Number of Layers
5L-
0.75 1.00
0.25
0.50
0.75
0.00
0.25
0.50
Margin Distribution
Figure 1. Tukey box-and-whisker plot depicting the margin distributions of optimized 8-qubit Quantum Convolutional Neural Networks (QCNNs). The results for QCNNs with one, five, and nine layers are displayed, along with their corresponding test accuracies and generalization gaps indicated in the legend. QCNNs were trained for 4-class classification task aimed quantum phase recognition (QPR). The experiment was performed with varying degrees of lahel noise: QPR dataset with pure labels (left), half randomly labelled dataset (middle), and full randomly labelled datasets (right). As the noise (randomization) level increases, the margin distributions tend exhibit more pronounced skew towards the left, indicating that a greater proportion of samples classified with smaller margins. Notably, the margin distribution exhibits a strong positive correlation with test accuracy across all scenarios.
4.2. Predicting Generalization Gap: Parameters Margins
ered effective.
Figure depicts how the generalization gap, median margin, and effective parameters (with 10-2 threshold) change in response variations the number of QCNN layers, the percentage of randomized labels, and the choice of varia- tional ansatz. Since margins are inversely correlated with generalization gap (as shown in Theorem 3.1), the inverse of the median margin is plotted instead. Additionally, fol- lowing the results of Caro et al. (2022), we plot the square root of number of effective parameters.
In our earlier analyses, we identified a strong correlation be- tween margin distributions and generalization performance. Here, we demonstrate the effectiveness of margin- based metrics in estimating the generalization gap, highlighting their advantages over traditional uniform bounds.
Caro et al. (2022) showed that the generalization gap in QML models can be estimated based on the number of train- able parameters. Specifically, they proved that, in the worst case, the generalization gap scales with the square root of the parameter count. Furthermore, when only subset of parameters undergoes substantial change during training, the generalization gap scales with the square root of the number of effective parameters that undergo significant up- dates. Despite being uniform bounds, the parameter-baue based generalization approach has become a standard method for understanding generalization behavior in QML models.
Figure 2(a) shows how the generalization gap, inverse me- dian margin, and the square root of effective parameters vary with the number of QCNN layers- 1, 3, 5, 7, and 9, The generalization gap reaches its peak at five layers before slightly decreasing. While the effective parameters increase monotonically with the number of layers, the median margin effectively captures the peak five layers.
Similarly, in Figure 2(b), the margin effectively captures the rising generalization gap the percentage of randomized labels increases, while the number of effective parameters show an inverse trend.
To illustrate the effectiveness of margins estimating the generalization gap. compare three margin- based met- rics with three parameter-basee metrics. For margin- based metrics, we analyze the lower quartile, median, and mean values of the margin distribution. For parameter-basee met- rics, we examine both the total number of parameters and count of effective parameters that undergo substantial change during optimization. Specifically, we define effec- tive parameters using two thresholds, 10 and 10-2, which represent the minimum change in parameter (the rotation angle of parameterized quantum gates) for it to be consid-
In Figure 2(c), we analyze three variational ansätze: QCNN, QCNN with shared parameters (Cong et al., 2019; Hur et al., 2022), and Strongly Entangling Layers (Bergholm al., 2020), arranged in decreasing order of expressibility. In this case, the QCNN with shared parameters constrains the local parameterized unitaries within the convolutional layers to share identical parameter values. The margin correctly reflects the rising generalization gap as expressibility de-
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Understanding Generalization in Quantum Machine Learning with Margins
2,55
3.50
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35.330
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-.-- <eet. 130
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0.0
0.5
0ChN
OCHN shared) Variatonal Ansatz
SEL
Number of Layers
Percentage fandomized Labels
(a)
(b)
(c)
Figure2 2. Illustration of how the generalization gap, median of the margin distribution (a margin-based metric), and effective parameters with a 10-2 threshold (a parameter-based metric) vary with the number of layers, percentage of randomized labels, and the choice variational ansätze.
creases, while the number of effective parameters shows an inverse trend.
- H(glu). Here, H()) represents the entropy of g- quantifying the uncertainty of the generalization gap, while H (g|و) is the conditional entropy of g given μ. indicating the remaining uncertainty about the generalization gap given μ. Thus, higher mutual information I (g: μ4) suggests that μ provides more information about g, reducing the uncertainty associated with
-
In summary, the margin reliably captures variations in the generalization gap across different hyperparameters, whereas effective parameters fail to do so and sometimes show the opposite trend. Note that the median margin is (Q1) to avoid infinite inverse margins when QI is zero, a scenario that occasion- ally arises when the labels are fully randomized and the model employs an inexpressive variational ansatz. Effec- tive parameters are shown instead of the total number of parameters, as the latter remains constant despite changes in randomized label percentages, unlike the generalization gap. mean and QI versus total and effective parameters- without withouu label noise are provided in Appendix A.3, Figure 22.
used instead of the lower quartile
In contrast, the Kendall rank correlation coefficient T mea- sures the strength and direction of the association between two variables. For pairs of generalization gap and metric values, (A)))) and (o(A2), p(22)), we expect the metric to accurately capture the ranking of generalization performance. Specifically, if g(11) < g(>2). then μ(\1) should also be less than (x)2). This implies that the metric i effectively predicts relative generalization, with lower values corresponding to better generalization (lower g). Con- sider lists of generalization gaps and corresponding met- rics, denoted as G [91, In] and M [Ma, - [un). where each g; and μ4 corresponds to g(A,) and p(x,). re- spectively. The Kendall rank correlation coefficient between G and M is given by
Additional comparisons-marge
Thus far, we have compared how margins and the number of parameters capture generalization by varying one hyper- parameter at time while keeping the others fixed. Now, we present more comprehensive comparison with all hyper- parameters varied simultaneously. More specifically, for a given set of hyperparameters X, we examine the correlation between the generalization gap g\)) and the correspond- ing metric p()). Here, as before, the metrics can be either margin- or parameter-bon based. To assess correlation, we analyzed two different evaluation methods: mutual in- formation and Kendall rank correlation coefficient. Both methods (or their variants) have been used to assess gen- eralization capacity in classical deep learning (Jiang et al.; 2020; Dziugaite et al., 2020).
1 TG.M 2n(n 1)
Σ
[1 + sgn(g) 95) sgn(s: (5))
This coefficient ranges from 0 to 1, where 7 = indicates perfect agreement between the rankings of g and fa (all pairs are concordant). and r = 0 indicates perfect disagreement (all pairs are discordant).
Figure 3 shows the mutual information (solid) and Kendall rank correlation coefficient (shaded) between the general- ization gap and various metrics. In both evaluation methods, margin- based metrics demonstrate stronger correlations than barsee metrics, indicating that margins are more effective in evaluating the generalization performance of QMI models.
In mutual information analysis, we treat the hyperparameter A as random vector, with g()) and μ(() as functions of this vector. Mutual information, denoted by 14); is P(g.a)
defined as I (g: 11) = E log P()P(()) which simplifies to
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Understanding Generalization in Quantum Machine Learning with Margins
Mutual Information
0.8
Kendall's Tau
0.7
Correlation Values
0.6
0.5
0.4
0.3
0.2
0.1
0.0
margin (Q1)
margin (median)
margin (mean)
params
eff. params (10 1)
eff. params (10-2)
Figure 3. Comparative analysis of mutual information (solid) and Kendall rank correlation coefficients (shaded) between the generalization gap and various metrics. The first three columns represent margin- based metries, while the three columns represent parameter- metrics.
Consistent with previous analyses, we used inverse margin values measure correlation, larger margins correspond smaller generalization gaps. Further details this exper- iment can be found in Appendix A.2.
Here, / Ei represents the quantum states corresponding to the ensemble of each class. The proba- bilities for each class denoted by p N and Dir(,.) denotes the trace distance.
To further demonstrate the robustness of our results, we present additional experimental results that reproduce the main findings (Figures 1-3). These experiments were con- ducted on supplementary datasets the Transverse Field Ising Model and the XXZ Heisenberg spin chain-with 8 and 10 qubits. These results detailed in Appendix A.3.
Given sample dataset S, the quantum embedding circuit determines Drr(= pr , A quantum embedding that produces large initial trace distance results in a smaller loss, and vice versa. This because quantum channels are completely positive and trace- pace-preserving (CPTP) maps, which cannot increase the distinguishability between quan- tum states. Formally, for quantum channel A and dis- tance metric D (such as trace distance or infidelity), the contractive property D(A(+t), A,A())) < D(pt, ,-) ap- plies to any quantum states pt and p This indicates that the distance between two quantum states (representing data from different classes) cannot increase as quantum opera- tions applied, emphasizing the importance of starting with a large initial trace distance.
5. Margins and Quantum Embedding
Up to this point, we have focused on scenarios involving the classification n-qubit quantum data using QML models. Another important application of QML the classification of classical data. To work with classical data in quan- tum framework, the d-dimensional data, E Rd, must be mapped to an n-qubit quantum state, represented by density matrix p(z) E x2" This process, known as quantum embedding. typically involves applying a quan- tum embedding circuit Uemb to the ground state, resulting in p(x) Uemb (x)) (I0) (0)) @r (x).
To address this limitation, Lloyd et al. (2020) introduced trainable quantum embedding (TQE), where a parameter- ized quantum circuit is optimized to maximize the sep- aration between data classes Hilbert space. Unlike a fixed quantum embedding, the L-layer TQE incorporates additional parameterized unitary circuits, V(Ø), to con- struct Uye (2, Φ) l (()... The quantum state associated with data x and parameter ò is given by p(x, = (xr, (10) (0) ( UUppex Φ). The trainable parameters o optimized enhance the distinguishabil- ity between quantum states corresponding to different data classes. This concept was later extended to quantum kernel frameworks by Hubregtsen et al. (2022).
Quantum embedding is crucial to the overall performance of QML models, it establishes the lower bound of the training loss (Lloyd et al., 2020; Hur et al., 2024). For exam- ple, in a binary classification with a dataset of N samples, S {., - J {x +1 the training loss with respect to a linear loss function, denoted by Ls. is lower bounded as
Ls 2 2
Dulp p ppppppppp'' -p****).
Building on this, Hur al. (2024) proposed Neural Quan-
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Understanding Generalization in Quantum Machine Learning with Margins
Fixed Quantum Embedding
Trainable Quantum Embedding
Neural Quantum Embedding
Test (Gen)
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0.76 (0.01)
0.31 (0.00)
0.95 (0.00)
0.77 (0.02)
FASHION
MNI ST
Margin Distribution
Margin Distribution
Margin Distribution
Figure 4. Tukey box-and-whisken plot illustrating the margin distributions of optimized 8-quhit Quantum Convolutional Neural Networks (QCNNs). The plot shows results for QCNNs using fixed quantum embedding (left), trainable quantum embedding (middle), and neural quantum embedding (right). The QCNNs were trained on a binary classification task using the MNIST (bottom), Rishion-MNIS (middle), and Kuzushiji-MNDI MNNTTT (top) datascts. In addition to the margin distributions, the mean of the margins is indicated by black cross, and trace distance between ensemble quantum states marked by a red circle.
tum Embedding (NQE), which leverages classical neural networks to optimize quantum embeddings. Their study highlighted the inherent limitations of TQE maximizing trace distance and empirically demonstrated that NQE can effectively identify quantum embeddings with large initial trace distance, outperforming the capabilities of TQE.
embeddings with large trace distance, thereby increasing the upper bound of the margin mean. This margin- baseed perspective on generalization enables systematic optimiza- tion of the generalization performance of QML models by selecting quantum embeddings with large trace distances.
Figure 4 presents the classification of MNIST (LeCun et al., 2010), Fashion-MNIST (Xiao et al., 2017), and Cetushiji-MNISSI (Clanuwat et al., 2018) datasets using 8 8-qubit QCNN with various quantum embedding schemes. The comparison highlights how the initial trace distance af- fects the margin distribution and the model's generalization performance. Three quantum embedding schemes are em- ployed to vary the initial trace distance: the fixed quantum embedding (77. Feature Map introduced by Havlícek et al. (2019)). TQE, and NQE.
While both TQE and NQE are theoretically designed to im- prove the lower bound of sample training loss, they have also been shown to significantly enhance test accuracy, thereby improving generalization performance. Although a posi- tive relationship between large initial trace distance and improved generalization has been observed empirically, theoretical explanation for this relationship was previously missing. Margin generalization offers this missing theoreti- cal basis, as the trace distance serves as an upper bound on margin mean.
As anticipated from previous research, we observed a pro- gressively increasing trace distance (indicated by a red dot) in the order of fixed quantum embedding, TQE, and NQE, This increase in trace distance corresponds to higher margin mean values (indicated by black cross). Larger margin means, coupled with right-skewed margin distributions, are associated with improved test accuracies, aligning with the principles of margin- based generalization. In summary, using an optimal quantum embedding with large initial trace distance not only reduces sample training loss but also results in larger margins and enhanced generalization. Experimental details are provided in Appendix A.2.
In binary classification, the margin mean simplifies to 1, where U denotes the
N Σ, Tr(Up((;)((tE,,) optimized unitary of QNN after the training. By defining Ei = Ut E+ the margin mean can be reformulated as,
2 N+
N Σ [T((((7)EL)) 1=1
Σ
p =
N
= 2T pt F "TTppT>>> ) -1
< Dielptpp poopptt.p> pp.pppppp -ppppp)) p)),
(2)
where the final inequality becomes equality if and only if (Eii} corresponds to the Helstrom measurement (see Ap- pendix A.4). In other words, TQE and NQE enhance the model's 's generalization capability by identifying quantum
6. Discussion
In this work, we proposed a margin understand generalization in QML models. We began by
baaeed- based framework to bet-
8

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Understanding Generalization in Quantum Machine Learning with Margins
addressing the limitations of uniform generalization bounds, which have proven ineffective when models overfit noisy labels in both classical and quantum settings. Drawing inspi- ration from margin-based approaches in deep learning, we developed a margin generalization bound for QNNs, offer- ing tighter and more meaningful estimates of generalization performance.
ity, assessing how sensitive A is to changes in the training data (Bousquet & Elisseeff, 2002; Neyshabur al., 2017; Dziugaite & Roy, 2017). Another promising approach ex- plores the relationship between generalization and optimiza- fion speed, such as the number of iterations required to reach a certain training loss (Hardt et al., 2016). Extending these lines of research to QML could provide deeper insights into the generalization behavior of quantum models.
Through extensive empirical studies, we demonstrated that margin-basee based metrics are highly predictive of generaliza- tion performance in QML models. Our experiments with QCNNs on the QPR dataset revealed a strong correlation be- tween margin distributions and generalization. Furthermore, we compared margin- based metrics against parameter parameter-based metrics for predicting the generalization gap. In all settings, margin-based metrics proved be better predictors of the generalization gap, showing higher mutual information and Kendall rank correlation coefficients.
A Acknowledgments
This work was supported by Institute for Information & com- munications Technology Promotion (ITTP) grant funded by the Korea government (No. 2019-0-00003, Research and Development of Core technologies for Programming. Run- ning, Implementing and Validating of Fault-Tolerant Quan- tum Computing System), the Yonsei University Research Fund of 2024 (2024-22-0147) the National Research Foun- dation of Korea (2023M3K5A (AAS5A119441111 and the Ministry of Trade, Industry, and Energy (MOTIE), Korea, under the Industrial Innovation Infrastructure Development Project (Project No. R0-2004-004666933.
Additionally, we explored the connection between margin- based generalization and quantum state discrimination, a fundamental concept in quantum information theory. We showed that quantum embeddings with a large trace distance lead to larger margins, which in turn improve generalization. By comparing different quantum embedding methods, we experimentally demonstrated that embeddings with higher trace distances consistently result in better generalization performance.
Impact Statement
This paper presents work whose goal is to advance the field of Quantum Machine Learning. There are many potential societal consequences of our work, none of which we feel must specifically highlighted here.
Overall, our results suggest that margin- metrics pro- vide more reliable and interpretable framework for under- standing generalization in QML models. This framework not only offers theoretical insights but also practical guid- ance for designing QML architectures that generalize well.
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Understanding Generalization in Quantum Machine Learning with Margins
A. Appendix
A.1. Derivation of Margin Bound and Generalization to Mixed States
In Section 3, we present a theoretical generalization bound for Quantum Neural Networks (QNNs) using margin- based approaches. This bound quantifies how well QNN model trained on given dataset will perform on new, unseen samples. In this appendix, expand on the derivation of this bound, offering additional insights into the steps and calculations involved.
To derive the upper bound, we apply Rademacher complexity to the hypothesis classes of QNNs that incorporate margin loss. We proceed by utilizing covering number arguments, which enable further refinement of the upper bound on sample Rademacher complexity. Specifically, we employ Dudley's Entropy Integral, a technique that converts the bound on Rademacher complexity into a function covering numbers.
Our derivation relies on three established lemmas, summarized below without proofs. For further details, proofs of Lemma A.1 and Lemma A.3 can be found in Shalev-Shwards & Ben-David (2014) and Mohri et al. (2018), while proof of Lemma A.2 can be found in Zhang (2002) and Bartlett et al. (2017).
Lemma A.1. (Rademacher Complexity) Let F be class of functions where each f(2) takes values within an interval [O, 1}. Then, for any 5 > O, with probability at least 1 - 8 over the random draw of an i.i.d. sumple S of size m, the following inequality holds for all f E F:
m +
In(2/0)
sup E [[(()]]
<
Σ
+ +3
fer
m
2m
Lemma A.I provides upper bound the expectation of any function within F using sample Rademacher complexity (Fis))
Lemma A.2. (Maurey's Sparsification Lemma) Consider Hilbert space H. with norm || and set of vectors == =VV,,..., V.] where each V, E H. Define scaled convex combination U = a conv(V), where a is positive real constant, and cVnv(V) denotes the convex hull ofV. V. For any integer ki there exists a vector (k1,...,ku, E 24 such that Ee_ ki k, and
a S
U
K
max 1IV11
(4)
k
Lemma A.2 provides an error bound for approximating a vector using a linear combination of discrete coefficients.
Lemma A.3. (Dudley's 's Entropy Integral) Given a set U C [-1,++1]" that contains the origin, the Rademacher complexity RUU) can be bounded by:
4a m
12 mm
R(U) S inf a>0
+
Ja
(5)
m
(12) to bound the Rademacher complexity through integral over scales 3, with a acting as an adjustable parameter to tighten the bound.
Lemma A.3 leverages covering numbers NN B| ||
Applying Lemma A.1 to the function class F, = (p. + (M((()), v)) h E H}. we establish a bound on the expected margin loss in terms of the empirical sample. With probability at least - 8, for all h E H. we have:
m Σ (M))))))) 29{(()))ss) + 3/
In(2/8)
E (M((())))))
(6)
2m
Since the ramp loss is lower bounded by the indicator function, 1(x 0) < (z),, the expected loss is directly related to the misclassification probability: E (h(p),, S max h(p))) < (M((p))))) Similarly, because (x) S 1(x & 7),
the sample loss is bounded above by the empirical margin loss: m ". (Mb((p))) y:)) I m " (((())) S
y + maxizy, hpp)))) Substituting these relationships into the bound from Equation yields Equation
12

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Understanding Generalization in Quantum Machine Learning with Margins
Equation I includes the Rademacher complexity term R((FF)ss) which analyze using covering number techniques. Specifically, are interested in the 6-covering number of (F=))s with respect the 22 norm, denoted by (V((F)) ll-
For pure input states, the function class can be expressed as F, - {(z, v) - (.M(((U))) v)) U E Uq}. where g(z) = {x E and x E CN. As shown in the main text, L, (M(g()) 3)) is 4E/r-Lipschitz with respect to the l norm, where E = E. Eallo-
For any L-LLpsshitz function f over a set S, we have the following covering number bound: el,( I]). Applying this property, we obtain
U U IN (1 l1a) In (IUX: I
Uq}. 4E" 1E-1--11).
where X is N x m complex valued matrix whose i-th column represents the quantum state
The next step is to bound the covering number for the set {UX U E Ug). Since the Frobenius norm is invariant under conjugate transposition, we can write N(UUX : E Uoh,e,1 |l l12) = N E U2) e, 1122).
Define Y E Cm*N by normalizing each column of Xt to have unit no-m: Y., = _ l2. Also, let a E NxN be a scaling matrix where each element in the i-th row is set to || |X|,|l2- Then, we can express XtUt as ((a OU) = YB, where C denotes the Hadamard product (element-wise multiplication) of matrices.
Consider the set V (V VaN= [gYe,et gg {+1, 1, +i, -i, [V], k E [N]]] For any matrix B, can express YB as
YB= EBjYYeje |B conv(V),
j.k
where ||]|_ is defined as Ej (|Re( (B5x)l )1 (B)))))).
For distance bound b such that ||7||2,1 S b for all U Ug (which can be generalized to ||U Uretll2,1 for arbitrary reference matrix Uzet). ||||| can be bounded by
B S V
Σ j.k
BBnk s V21 S by2m,
where the first inequality follows from the vector norm inequality, the second from Holder's inequality, and the final inequality from the fact that ||a||2,>0 ||||| |||
vm.
For any > 0, by setting k = (2mb² and applying Lemma A.2, we conclude that there exists a vector kANz) E 24
with Σ; b; = such that
xtut
||B|
Σ
K
<
|BI
max
m c
(8)
k
k
i
2
Consequently, the set C [BI11 Σ KV
2mb
serves as an c-cover for (XtUt U e U2} with respect to
k
2mp?
normm. Since |C| < 2
follows that
2mb²
In INUX Uoheel (12)
In
2
(9)
2
Combining this with Equation 7, we obtain
32mbb E2
In (nFa)sss
112) <
In4N2.
(10)
222
13

---

Understanding Generalization in Quantum Machine Learning with Margins
Next, we bound the Rademacher complexity term using covering numbers via Dudley's Entropy Integral. Applying Lemma A.2 yields,
4a vm +
32mb2 72
vm
R (((()))s)) I inf
22n
4N
(11)
as0
m
a
Setting
18 In 4N 4N
32mbb F
we find
m
12 R((Fg) s) S _2 2 m
32mb2 E
bE
(12)
This yields the margin-based generalization bound (Theorem 3.1) in terms of the QNN components.
The margin bound for QNNs can be extended to mixed quantum states via vectorization. Given a mixed quantum state Eig Pij ji) (jl. we consider its vectorized form as a pure state: (p)) = En, Psy (t) @ (j). In this framework, the measurement function d: (corresponding to the POVM element F.) for the input mixed state given by gi(p) Tr(E.p) = (pEs)))). Since ( = || the function gi || I-Liischiiz Therefore, following the derivation in the main text, the combined measurement function g is E E-pischhi, where E VV E
For the mixed input state, the function class F, is defined as F, ((,,)) + (,,0)))+MMMM(C( (MM(((U)U)))) v)) Using the property [UpUi)) = (U & p))) we extend the margin bound from Theorem 3.1 to mixed input states. The extension involves a slight adjustment: the distance bound b now applies to U @ U rather than U, and = =v2 E 1 E is evaluated using the Frobenius norm, unlike the spectral norm used for pure input states.
A.2. Experimental Details
QUANTUM PHASE RECOGNITION: MARGIN DISTRIBUTION
In Section 4.1, 8-qubit quantum convolutional neural networks (QCNNs) were trained to address the quantum phase recognition (QPR) task using a generalized cluster Hamiltonian. The QCNNs employed nearest-neighbor two-qubit parameterized quantum circuits (PQCs). For this experiment, we used the architecture proposed in Hur et al. (2022), where the two-qubit POCs follow the structure outlined in Figure of the reference.
The model was trained 20 data points, evenly split across four classes. A small training sample was deliberately chosen to explore the overfitting regime, where labels were intentionally randomized with noise, following a methodology similar to that of Gil-Fuster et al. (2024). The test accuracy was measured on 1,000 test samples-far exceeding the size of the training set-to provide a robust estimate of true accuracy. The model was trained using Adam optimizer, with a learning rate of 0.001 and bull-batch updates (Kinga et al., 2015). The model was trained for up to 5,000 iterations, with early stopping triggered based on a convergence interval of 500 iterations. Specifically, the training halted when the difference between the average loss over two consecutive intervals became smaller than the standard deviation of the most recent interval. To ensure reliable margin distribution boxplots given the small training set, the results were averaged over 15 experimental repetitions, each using different training sample.
QUANTUM PHASE RECOGNITION: MARGIN VS PARAMTER
In Section 4.2, we compare parameter-based metrics against margin- metrics for predicting the generalization gap. The experiment was conducted under the same conditions as described above, except for the convergence interval, which was set 300 iterations.
For the mutual information calculation, the empirical joint probability distribution used for the generalization gap g and metrics approximate the unknown true distribution. For Kendall rank correlation coefficient calculation, random tie-breakers were used when equal generalization gap metrics occurred.
Figure 2 compares the median the margin distribution with the effective number of parameters (threshold 10-2) in predicting the generalization gap. The experiments vary a single hyperparameter- either the number of layers, noise level, or variational ansatz while keeping the others fixed seven layers, no noise, and QCNN.
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Understanding Generalization in Quantum Machine Learning with Margins
MARGINS AND QUANTUM EMBEDDING
In Section 5, we explore the influence of quantum embeddings and initial trace distances on margin and generalization performance when applying QML models to classical data. To maintain binary classification setting, we focused on the first two classes from the MNIST, fashion-MNIST, and KuzushijiMMNIST datasets. Unlike previous experiments, did not examine the overfitting regime under label noise, choosing instead to utilize the full training and test datasets. Principal component analysis (PCA) was applied to reduce the dimensionality of the data to match the number of qubits. The experimental setup remained the same as before, except for using a batch size of 16 instead of full-batch training.
We compared three quantum embedding methods, each resulting in distinct initial trace distances. For the fixed embedding, used the ZZ Feature Map with three repeated layers. The ZZ Feature Map consists of blocks of Hadamard gates, single-qubit Pauli-Z. rotations, and two-qubit Pauli-77. rotations described in Havlícek al. (2019).
For the trainable quantum embedding, we introduced trainable quantum layers along with the ZZ Feature Map. The trainable quantum layers incorporate parameterized single-qubit Pauli-Y rotations, exp(-i -igY,): and nearest-neighbor two-qubit YY rotations, exp(i (Y
Neural Quantum Embedding (NQE) utilizes classical neural network, (x; w), where z represents the data and w the network parameters. NQE maps classical data into quantum states p(x) = eumb (a (0) [0|@ru (or w)). Here, the ZZ Feature Map was also employed as the quantum embedding circuit. The neural network was fully-connected ReLU network with layer dimensions of [8, 16, 32, 32, 16, 8], where each element represents the number of nodes in that layer.
A.3. Additional Experimental Results
This section details supplementary experimental investigations, where we replicated the core experiments presented in Figures 1-3 of the main text. These replications involved larger 10-qubit systems and more diverse set of Hamiltonians to further validate the robustness of our methodology. In addition to the generalized cluster Hamiltonian (detailed in the main text), our analysis includes the one-dimensional (ID) XXZ-Heisenberg spin chain and the ID Transverse Field Ising Model (TFIM).
XXZ--eisenberg Spin Chain
The Hamiltonian for this model is defined as:
N Σ
((J,A) J
X1X 8,Y41 + AZiZit1
(13)
In our simulations, we set the coupling strength J = 1. The anisotropy parameter A was sampled uniformly from the
combined intervals [-2, 11) U (1, (1, This configuration leads a ground state that either ferromagnetic (when A -1)
antiferromagnetic (when A 1). QMI. models were subsequently trained to perform binary classification between these two distinct phases.
Transverse Field Ising Model (TFIM)
The TFIM Hamiltonian is given by:
N
N
H(J = -J Σ ZiZe+1 + A Σ X;
(14)
For dataset construction, both the coupling J and the transverse field strength A were sampled from uniform distributions. Our investigation focused on the ordered phase, characterized by | < 1. The coupling J was varied within the range
[-1, 1]. allowing the system to exhibit either a ferromagnetic (J > 0) or an antiferromagnetic (J < 0) phase. A QML model was then trained for the binary classification task of distinguishing these phases.
15

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Understanding Generalization in Quantum Machine Learning with Margins
10-QUBIT GENERALIZED CLUSTER HAMILTONIAN
Noise 0%
Noise 50%
Noise 100%
Test (Gen)
Test (Gen)
Test (Gen)
0.85 (0.07)
0.40 (0.29)
0.26 (0.33)
11-
0.89 (0.07)
1L-
0.42 (0.34)
0.25 (0.41)
0.88 (0.08)
0.41 (0.40)
0.25 (0.43)
Number of Layers
SL-
5L-
5L-
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
Margin Distribution
Figure 5. Reproduction of Figure with 10-qubit generalized cluster Hamiltonian.
gen- $46
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marghn/02)
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Figure 6. Reproduction of Figure 2 with 10-qubit generalized cluster Hamiltonian.
0.7
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Figure Reproduction of Figure 3 for 10-qubit generalized cluster Hamiltonian.
16

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Understanding Generalization in Quantum Machine Learning with Margins
8- 8-QUBIT XXZ-HEISENBERG SPIN CHAIN
Noise 0%
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Figure Reproduction Figure with 8-qubit XXZ-Ileisenberg spin chain.
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im sharest
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Figure 9, Reproduction of Figure 2 with 8-qubit XXZ-Heiseneeeeee spin chain.
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Figure 10, Reproduction of Figure 3 with 8-qubit XXZ-Heisenberg spin chain.
17

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Understanding Generalization in Quantum Machine Learning with Margins
10-QUBIT XXZ-HEISENBERG SPIN CHAIN
Noise 0%
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Noise 100%
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Figure 11. Reproduction of Figure with 10-qubit XXZ-Heisenberg spin chain.
gaf, gAp
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QCWN sharod)
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Figure 12. Reproduction of Figure 2 with 10-qubit XXZ-Heiieenbegg spin chain.
Mutual Information
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Figure 13, Reproduction of Figure 3 with 10-qubit XXZ-Heisenberg spin chain.
18

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Understanding Generalization in Quantum Machine Learning with Margins
8-QUBIT TRANSVERSE FIELD ISING MODEL
Noise 0%
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Figure 14. Reproduction of Figure 1 with 8-qubit Transverse Field Ising Model.
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(MN Pimmee DD
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oen qep
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off,, params 11gst
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GCHN (parim shared]
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Figure 15. Reproduction of Figure 2 with 8-qubit Transverse Field Ising Model.
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Figure 16. Reproduction of Figure 3 with 8-qubit Transverse Field Ising Model.
19

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Understanding Generalization in Quantum Machine Learning with Margins
10-QUBIT TRANSVERSE FIELD ISING MODEL
Noise 0%
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Noise 100%
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H|
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Figure 17. Reproduction Figure with 10-qubit Transverse Field Ising Model.
0.04A
gon. gap
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mangin122)
paraers (10
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Figure 18. Reproduction of Figure 2 with 10-qubit Transverse Field Ising Model.
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Figure 19, Reproduction of Figure with 10-qubit Transverse Field Ising Model.
20

---

Understanding Generalization in Quantum Machine Learning with Margins
MARGIN DISTRIBUTION PLOTS
Corruption 0%
Corruption 50%
Corruption 100%
Accuracy
Accuracy _
Accuracy
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1
1L
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(a) QCNN with shared parameters
Corruption 0%
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Acouracy
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(b) Strongly Entangling Layers
Figure 20, Reproduction of margin distribution (Figure 1) using diffcrent variational ansätze.
Figure 21. Fine-grained margin distribution plots.
Figure presents results for QCNNs, where the two-qubit PQCs are permitted to have distinct parameter values within the convolutional layer. Figure 20, we provide additional experimental results using different variational ansätze: QCNN with shared parameters and Strongly Entangling Layers. Strongly Entangling Layers consist of repeated single-qubit rotations and two-qubit CNOT gates, provided as built-in feature by PennyL Lane (Bergholm et al., 2020). Consistent with the findings from the main text, we observe a decline in test accuracy as the noise level rises, accompanied by left-skewed margin distribution. Additionally, Figure 21 provides plot with more fine-grained experimental results. These details were excluded from the main text purely for reason of display conciseness. The figure show performance across corruption levels of 0%, 25%, 50%, 75%, 100% as well as for 1,3, 5, 7, numbers of layers.
21

---

Understanding Generalization in Quantum Machine Learning with Margins
MARGIN VS PARAMETERS
0.22-
morgnida
get EAp
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Tparame
26
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11
vef. puroms SUpeS
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Number of Layers
Mumseraa Layers
(a) Lower quartile of margin versus parameter-based metries for predicting the generalization gap.
0.300
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(b) Margin mean versus parameter brasee metrics for predicting the generalization gap.
Figure 22. Comparison of the lower quartile (a) and the mean (b) of the margin against parameter-based metrics for predicting the generalization gap. The experiments use QCNNs (left) and QCNNs without parameter sharing (right) as the variational ansatz.
Figure 22 (a) and (b) compare parameter-based metrics with the lower quartile and mean of the margin, respectively, Consistent with the results in the main text, the margin- based metrics more effectively capture the generalization gap across all settings.
A.4. Margin mean and Quantum State Discrimination
In Section 5, established connection between quantum state discrimination and margin. Specifically, showed that the margin mean is upper bounded by the trace distance between quantum state ensembles, making large initial trace distance essential for achieving strong generalization. Here, we provide the detailed derivations.
Equation demonstrated that margin mean can be expressed as:
j = 2Tr(pp xt E + 2Tr(p p E21) 1.
(15)
Recall that {Ei}) forms set of POVMs, meaning Ei + E = I. Utilizing this property, rewrite:
Tr(p (+EE) Tr(p" p E2 = + Tr (p+pp= - pppp) Fin)
= =p+ Tl (pp p+ - pp }{
1
- lr (pt p- p p- (Ei -E")),
(16)
2
where the final equality averages the first two.
22

---

Understanding Generalization in Quantum Machine Learning with Margins
We are particularly interested in the maximum of Tr(p*p=EE p² sition p+pp -p Σ (xab) where X real due to Hermitian nature of the density matrix. The maxi- mum occurs when Et1 are Helstrom measurements, which are projects onto the positive and negative subspaces, i.e. E Epesso03 (15)) and E = Ettt.co (1)().)..
T Consider the spectral decompo-
Thus, we have:
p+ max Tr(p* Tr(ppp=E TrppTpTT p-E_ Ei
1
+ Σ Ac
2 2
1 + 2
pu
(17)
2
Substituting this into Equation 15, we obtain p S (u p" P p ) where equality holds when Eii are the Helstrom measurements.
23


Reviews:
[{"review_id": "ZRKAoGwL8j", "reviewer": "Reviewer_tcJi", "summary": "This paper provides generalization error upper bounds for parameterized quantum neural networks using arguments from Bartlett et al. (2017) 's construction.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer tcJi for constructive feedback.\n\n\nWe understand the concern that our theoretical contribution could appear incremental, given the established history of margin-based bounds in classical ML. However, our work is the first to systematically extend this theoretical framework to QNNs. This extension is neither straightforward nor trivial. The translation of classical techniques to the quantum setting requires addressing unique mathematical and physical challenges that arise from quantum information processing, and in doing so, our work makes the following nontrivial technical advances:\n\n1. **Lipschitz Continuity of POVMs** (Sec.2): Unlike classical neural networks, QNNs employ POVMs. Handling the Lipschitz continuity of quantum measurements required specialized analysis unique to quantum states, involving spectral norms of POVM operators.\n\n2. **Covering Number of Complex Unitary Matrices** (A.1): Classical margin bounds rely on covering numbers of real-valued neural networks. By contrast, QNNs require derivations for complex-valued unitary matrices constrained by quantum mechanics. This differentiates our theoretical approach from existing classical frameworks.\n\n3. **Normalization properties of quantum states** (A.1): Quantum states possess normalization constraints that substantially simplify and alter complexity arguments. We exploit this property to derive tighter bounds, a simplification with no direct classical counterpart.\n\n4. **Extension to mixed input states via vectorization** (A.1): Our margin bound is further generalized to mixed quantum states utilizing vectorization arguments. This step is fundamentally quantum-specific, with no classical counterpart.\n\nNotably, our work addresses key open problems raised by two recent publications in Nature Communications ([13:4919, 2022](https://www.nature.com/articles/s41467-022-32550-3) and [15:2277, 2024](https://www.nature.com/articles/s41467-024-45882-z)), which represent the state of the art in QML generalization theory. The first introduces a generalization bound based on Rademacher complexity, while the second demonstrates the vacuity of such uniform bounds through extensive randomization experiments, concluding that existing theoretical tools are insufficient to explain generalization in QML. \n\nOur work goes beyond these foundational studies by introducing a margin-based framework that more accurately predicts generalization behavior in QNNs. In addition, our analysis applies to arbitrary quantum states—pure or mixed—whereas the prior works provide theoretical guarantees only for pure states. Rather than being a routine extension of classical results, our contribution directly addresses the theoretical gap identified in the second Nature Communications paper and significantly advances the understanding of generalization in QML beyond what is provided in the first.\n\nMoreover, our work establishes a previously unexplored theoretical connection between margin bounds and quantum state discrimination. This provides not only conceptual insight but also practical guidance for optimizing quantum data embedding. Specifically, by directly linking margins to trace distances, our theory offers a principled route for improving generalization performance when applying QNNs to classical datasets. This perspective also enables us to systematically optimize the quantum feature maps introduced in [Nature 567, 209–212 (2019)](https://www.nature.com/articles/s41586-019-0980-2), which proposed using quantum-enhanced feature spaces for supervised learning. We experimentally demonstrate that our margin-based approach significantly improves classification accuracy compared to the method in the Nature paper (Fig. 4). Thus, our work not only advances the theoretical foundations of QML, but also delivers a concrete performance gain over one of its most recognized experimental baselines.\n\nAdditionally, our experimental section provides novel and extensive evidence showing that margin-based metrics outperform standard parameter-count metrics across multiple challenging scenarios, including randomized labels, varying network depths and ansatz, and different embedding methods. In response to Reviewer 2’s suggestion, we have added additional experiments using the [Transverse Field Ising Model](https://tinyurl.com/47mpp43c) and [XXZ-Heisenberg Model](https://tinyurl.com/3r598nmm), further validating the robustness and practical relevance of our margin-based framework.\n\nWe believe these clarifications demonstrate the depth and originality of our contributions, which go beyond a trivial adaptation of classical techniques. By addressing open questions in QML with quantum-specific analysis and expanded empirical evidence, we believe our work offers timely and meaningful progress. We are grateful for the reviewer’s thoughtful feedback and hope this response helps convey the full value and relevance of our submission.", "claims_and_evidence": "I find all the claims in the paper to be reasonable. What helps this work is that there is a long line of work in deriving generalization error bounds based on margins starting from Bartlett (1996) [cited in the paper] and Bartlett et al. (1998) [1]. This also constitutes a drawback of the paper, in my opinion - the theoretical contributions of this paper are a very straightforward generalization of existing techniques. I invite the authors to correct me on this point if I have missed any non-trivialities in A.1 that they feel should be highlighted as an important contribution.\n\n[1]  Peter Bartlett. Yoav Freund. Wee Sun Lee. Robert E. Schapire. \"Boosting the margin: a new explanation for the effectiveness of voting methods.\" Ann. Statist. 26 (5) 1651 - 1686, October 1998. https://doi.org/10.1214/aos/1024691352.", "methods_and_evaluation_criteria": "I find the proposed evaluation criteria to be acceptable. However, since I am not an experimentalist, I will defer to the judgment of other reviewers.", "theoretical_claims": "I have gone over the theoretical parts of this paper in detail (and I am already familiar with the existing papers in the literature) and I don't see any obvious issues with the theoretical claims of this paper.", "experimental_designs_or_analyses": "No issues.", "relation_to_broader_scientific_literature": "This paper makes a nice addition to the literature on the generalization bounds of parameterized QNNs.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "This paper should only be accepted at a premier ML conference like ICML if the experimental section is worthy of being highlighted, since the theoretical contributions are incremental at best in my opinion. This is not meant as a targeted criticism of the author's efforts - my issue is that margin bounds have long since shown to be loose for generalization bounds (especially for overparameterized circuits).\n\n---------------------------------------------\nPost Rebuttal\n---\nI am still unconvinced on the second point. However I am raising my score to reflect that I no longer stand by the \"incremental advance\" part.", "other_comments_or_suggestions": "See the above comments. I am willing to raise my score after the rebuttal period based on discussions with other reviewers on the novelty in the experimental section. As it stands, despite the paper being well written, its contributions are incremental (merely quantizing existing analysis does not constitute a good contribution in my opinion), I cannot justify recommending acceptance.", "questions_for_authors": "See Claims section.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ly6ceavLbR", "reviewer": "Reviewer_uadb", "summary": "The manuscript describes a theoretical and experimental analysis of quantum machine learning models, with the focus on generalization bounds. The authors build on prior quantum machine learning results indicating vacuity of bounds based on parameter count or other measures of complexity of the hypothesis space. They adapt bounds based on margin distribution from classical to quantum ML, and show that it explains generalization gap more accurately than parameter-based metrics.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer uadb for thorough evaluation and insightful suggestions.\n\n---\n### 1) **Experimental Scope and Additional Results**\nWe acknowledge the reviewer's concern regarding the breadth of empirical evidence. To address this directly, we conducted extensive additional experiments on two canonical quantum many-body benchmarks: **Transverse Field Ising Model** and **XXZ-Heisenberg Model**.\n\nThese additions go beyond the original QPR dataset and validate that our margin-based framework generalizes well across distinct data distributions. All main-text experiments were systematically repeated on these datasets. Here are the anonymous link for additional experimental results: [TFIM](https://tinyurl.com/47mpp43c) and [XXZ-Heisenberg](https://tinyurl.com/3r598nmm).\n\nWe consistently observe that margins strongly correlate with generalization gaps across multiple datasets and experimental conditions, significantly strengthening the relevance and robustness of our margin bounds for QNNs.\n\n---\n### 2) **Unified Theorem Incorporating Quantum Data Embedding**:\nWith [Neural Quantum Embedding](https://tinyurl.com/4nyhdv5k), the quantum classifier for classical data is constructed through a two-step training procedure: first optimizing only the embedding circuit, and subsequently training the quantum neural network while keeping the embedding circuit fixed. Developing a fully unified theory would require separately deriving generalization bounds for the embedding optimization stage—which fundamentally differs from the classification task—and then integrating these results into a bound for overall classification performance. Such an analysis is inherently non-trivial and is currently beyond the scope of our work. \n\nNote that our existing margin-based bound implicitly captures the embedding circuit parameters, as these parameters directly influence the margin distribution of the trained model. Nevertheless, we agree that developing unified theoretical bounds explicitly incorporating embedding circuit properties is an interesting direction for future research. \n\nMoreover, many quantum machine learning applications involve quantum data directly, with no classical-to-quantum embedding required. In these cases, explicitly incorporating embedding circuit properties is not relevant.\n\n---\n### 3) **Margin and Generalization Gap Analysis in Figures**:\n\nWe confirm that Figures 1 and 4 initially emphasized test accuracies for clarity. However, margin distributions also strongly correlate with generalization gaps in these experiments. We have updated these figures to include both test accuracy and generalization gaps on this [anonymous link](https://tinyurl.com/4xtxt2a7).\n\nAs predicted by our theory, we observe in Figure 1 (QPR) that generalization gaps consistently decrease as margin distributions shift rightward. For Figure 4 (MNISTs), we see the same trend explicitly for the MNIST and Fashion-MNIST. For the KMNIST, we observe a slight discrepancy, where the gap marginally increases.\n\nThis minor deviation can be explained by the significantly larger number of samples (~12,000) relative to system complexity (8-qubit QCNN). Our margin bound scales as $1/\\sqrt{m}$, where $m$ is the sample size. Thus, when the number of samples greatly exceeds system complexity, generalization gaps become naturally small, and margin effects appear less pronounced. Indeed, the gaps observed for the MNIST-based dataset are substantially smaller compared to those with the QPR dataset, which includes only 20 samples following the experimental setup of [Caro et al.](https://tinyurl.com/mrsrrjk4) and [Gil-Fuster et al.](https://tinyurl.com/4kjnvb44). Therefore, this subtle deviation aligns with our theory rather than contradicting it. We will explicitly clarify this nuance and present both test accuracies and generalization gaps clearly in the revised manuscript.\n\nIn addition, we would like to clarify that generalization gap analysis was already presented in the main manuscript through Figures 2 and 3. These figures present the relationship between margin-based metrics and generalization gap across various architectural choices and data corruption levels, supporting our theoretical claims. This is also why we originally placed Figure 6, which provides supplementary generalization gap analysis comparing parameter-based metrics with the lower quartile and mean of the margin for the QPR problem, in the appendix.\n\nLastly, our work substantially broadens the experimental scope compared to prior state-of-the-art studies, such as [Caro et al.](https://tinyurl.com/mrsrrjk4) and [Gil-Fuster et al.](https://tinyurl.com/4kjnvb44). While these focus on a single task involving quantum data, our experiments span multiple QML architectures and a range of both classical and quantum datasets. This broader scope enables a more comprehensive assessment of the practical relevance of margin-based generalization theory and deepens our understanding of generalization in QML.", "claims_and_evidence": "The two main claims are:\n\n1) Generalization in QML theoretically depends on margin. Here, the authors adapt existing margin-based generalization bound to take into account the characteristics of quantum models. The evidence, in the form of proof of the bound, is sound. The assumptions relating to the quantum nature of the model architecture (model is a parameterized unitary followed by measurement, projective or more generally, POVM) are also sound. \n\n2) The bound from 1) has practical relevance. Here, the authors perform an experimental analysis evaluating to what extent inverse margin aligns with generalization gap. The experimental evidence is convincing (inverse margin aligns well, qualitatively in Fig 2 and quantitatively in Fig 3) with generalization gap for varying number of layers in the model, varying internal architecture, and varying randomization of the dataset). The evidence is, however, only limited to one dataset.\n\nThe two results above are focused on quantum model working on quantum input data. The paper also extends the margin-based generalization analysis to classical data – quantum model setup, and provides a margin-based explanation of the previously observed relationship between the choice of how classical data is embedded into quantum state and generalization. The theoretical link between mean margin and quantities previously shown to lower-bound loss is sound, and the authors show experimental evidence for the link between margin and test accuracy. The experimental limited (three MNIST-based datasets, one model QCNN). The presentation in the main manuscript is in terms of test accuracy, without showing training set accuracy, which does not directly support the claims that are related to generalization gap. Some limited exploration of generalization gap is provided in the Appendix (A.2, Fig. 6), but lacks details in the description (e.g. which MNIST dataset is used in Fig. 6?).", "methods_and_evaluation_criteria": "Empirical results are somewhat limited in scope, and (in section 4) not perfectly aligned with theory (see above for details).", "theoretical_claims": "Correctness of proofs in Appendix has not been checked in detail.", "experimental_designs_or_analyses": "The overall choice of experimental setup (models and dataset in Sect. 3 & 4) is sound, though limited in breadth.", "relation_to_broader_scientific_literature": "The paper extends prior findings on generalization bounds for quantum data-quantum model, and prior findings about quantities that affect how quantum embeddings related to effectiveness of classical data – quantum model; in both cases they extend the prior findings by introducing margin as an explanatory variable. This mirrors earlier developments in classical machine learning.", "essential_references_not_discussed": "None noted.", "other_strengths_and_weaknesses": "In terms of strengths, the paper extends our understanding of generalization in quantum machine learning. \nOne weakness of the work is the limited nature of the experimental evidence. Another weakness is the separate treatment of classical data scenario (Sec. 4) from the earlier parts (Sect. 2). The paper would be stronger if it could present a unifying theoretical framework that incorporates properties of the embedding circuit as parameters in the bounds, in the same manner as different options for measurements (properties of E) are incorporated.", "other_comments_or_suggestions": "The authors should consider moving the generalization gap-focused results into the main manuscript (Fig. 6), providing more detailed description.", "questions_for_authors": "None.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "t50s4RkHPB", "reviewer": "Reviewer_rwMH", "summary": "This paper establish a margin-based generalization bound for multiclass classification with Quantum Neural Networks, adapting techniques from classical neural networks to the quantum domain. This approach interprets quantum measurements as nonlinear activations and extends matrix covering techniques to complex-valued spaces. Through experiments on quantum phase recognition datasets, they demonstrate that margin-based metrics strongly correlate with generalization performance, even when traditional metrics like parameter count fail. Also they conduct experiments on three quantum embedding methods and showing that Neural Quantum Embedding (NQE), a classical-quantum hybrid approach, enhances generalization by yielding larger margins through increased data distinguishability.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer rwMH for thoughtful evaluation.\n\n---\n### 1) **Experiments on small qubit systems**\nWe agree that our experiments were conducted on relatively small quantum systems (8-qubit QCNNs). While it is possible to increase the number of qubits by one or two, we chose 8 qubits as a practical design decision. First, powers of two are a natural choice for QCNNs due to their hierarchical structure. Second, we do not expect a small increase in the number of qubits to significantly affect the generalization behavior.\n\nWhile extending the numerical studies to much larger systems (16 or 32 qubits) would be of interest, classical simulation becomes exponentially expensive with the number of qubits. For example, training a single 8-qubit QCNN requires ~**0.5hours** on state-of-the-art commercial CPUs. Our experiments were performed across extensive hyperparameters: [1,3,5,7,9] for QCNN layers, [0, 0.5, 1] for label noise, [QCNN, QCNN_shared, SEL] for variational ansatz. Each setting was repeated **15 times** to control variability, resulting in a total time of ~**300 hours**. Increasing the number of qubits by one would double simulation time, quickly becoming computationally impractical.\n\nNonetheless, our theoretical results are valid beyond 8-qubits. Our margin bound scales linearly with the number of qubits, suggesting that larger systems would exhibit analogous behaviors, provided the sample grows proportionally. Thus, while explicit numerical validation at larger scales is limited by classical computational constraints, our theory remains scalable.\n\nMoreover, [prior work](https://tinyurl.com/4kjnvb44) has shown that QNNs can efficiently memorize polynomially increasing training data, implying uniform bounds can remain vacuous even in larger quantum systems. Thus, the generalization challenges we address are fundamental to QML, not artifacts of small system size. This further motivates non-uniform measures, such as our proposed margin bound, which remain meaningful regardless of scale. \n\n---\n### 2) **Computational costs for calculating margins vs. parameter counts**\nThe margin for a single data point $(\\rho, y)$ is defined as $h(\\rho)_y - \\max\\_{j \\neq y} h(\\rho)_j$. In principle, evaluating margin distributions for an $m$-sample dataset requires $O(m)$ quantum circuit executions. However, practically, the predictions $h(\\rho)$ are computed as part of the final step of model training. Therefore, calculating margin distributions **requires no additional computational costs** beyond what is already used during model training. We will explicitly mention this practical consideration in our revised manuscript.\n\n---\n### 3) **Lack of analysis regarding real quantum hardware noise**\nOur work offers a new perspective on how quantum noise affects generalization by connecting margin bounds to quantum state discrimination. Quantum noise, which are contractive CPTP maps ($\\Lambda$), reduces trace distance $D(\\Lambda(\\rho_1),\\Lambda(\\rho_2)) \\leq D(\\rho_1,\\rho_2)$. In Section 4, we explicitly show that the margin mean is upper-bounded by the trace distance. Consequently, quantum noise shrinks trace distance and shifts margin distributions leftward, resulting in decreased generalization performance. \n\nThis connection is not captured by previous generalization bounds in QML and, represents a meaningful advancement in the theoretical understanding of generalization. The margin-based generalization framework naturally incorporates quantum noise effects, making it uniquely insightful. We agree that deeper experimental investigation of noise effects would be a valuable next step, and we believe our work provides a strong theoretical foundation for such studies.\n\n---\n### 4) **Small training set (20 samples) and statistical significance**\n\nWe recognize that our QPR experiments employed relatively small training datasets (20 samples). This experimental choice was deliberate and aligned precisely with previously established works ([Caro et al.](https://tinyurl.com/mrsrrjk4) and [Gil-Fuster et al.](https://tinyurl.com/4kjnvb44)), which served as direct references. Specifically, Gil-Fuster et al. experimentally demonstrated that QCNNs could overfit randomized labels with small datasets, questioning the effectiveness of uniform generalization bounds presented in Caro et al. Since a core contribution of our work is to propose margin bounds as a tight, non-uniform alternative to these uniform bounds, we intentionally retained similar experimental setups for direct comparison. \n\nFurthermore, to bolster empirical validation, our manuscript also presents additional extensive experiments on classical datasets (MNIST, Fashion-MNIST, and Kuzushiji-MNIST) in Section 4, each consisting of approximately **12,000 samples**. In this large-sample regime, we consistently observed strong correlations between margin metrics and generalization, confirming the statistical significance and robustness of our claims beyond the small-sample scenario.", "claims_and_evidence": "The claims made in the paper are generally well-supported by evidence through both theoretical development and experimental validation: The margin-based generalization bound is mathematically derived with clear steps, extending established techniques from classical to quantum settings. Experimental evidence strongly supports the predictive power of margin-based metrics for generalization. These evidences include usage of multiple datasets, testing various models and challenging scenarios like randomized labels", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria are appropriate for studying generalization in quantum machine learning. They are using margin-based approach is sensible given its success in classical deep learning and the demonstrated limitations of uniform bounds. Also the theoretical framework acknowledges quantum-specific properties like POVM measurements. For the evaluation, they test on both quantum data and classical data to show broader applicability. Comparing against parameter-based metrics establishes relative improvement over existing approaches.", "theoretical_claims": "There are no significant issues in the theoretical proofs to the best of my knowledge.", "experimental_designs_or_analyses": "The experimental designs and analyses in the paper are generally sound: Multiple experimental setups testing different aspects of the theory (generalization gap prediction, margin distribution analysis, quantum embedding comparison), Appropriate statistical measures (mutual information, Kendall rank correlation) to quantify relationships, Controls for variability by averaging results over 15 repetitions with different training samples, Systematic exploration of hyperparameters (layers, noise levels, embedding strategies)", "relation_to_broader_scientific_literature": "This paper brings margin-based framework to the quantum machine learning area, continuing the trend of non-uniform generalization bounds that better predict real performance. Also it links fundamental quantum information theory (trace distance, Helstrom measurements) to machine learning performance, connecting QML to established quantum information concepts.", "essential_references_not_discussed": "No to the best of my knowledge.", "other_strengths_and_weaknesses": "Strengths: It successfully bridges classical margin theory with quantum information theory, creating a unified framework for understanding QML generalization. It provides actionable guidance for designing better quantum embeddings based on trace distance maximization. It tests across multiple datasets, model architectures, and hyperparameters. Mathematical derivations are well-structured and the experimental results are clearly presented.\n\nWeaknesses: Experiments on small qubit systems (8 qubits) may not fully represent challenges of larger quantum systems. No discussion of computational costs for calculating margins versus parameter counts. Lacks analysis of how real quantum hardware noise might affect the margin-based approach.", "other_comments_or_suggestions": "It would be good to include a diagram illustrating the conceptual relationship between quantum embeddings, trace distance, and margins. Figure labels in Figure 2 are somewhat difficult to understand.", "questions_for_authors": "How would your margin-based approach scale to larger quantum systems (beyond 8 qubits)? Do you anticipate any theoretical or practical challenges?\nWhat is the computational overhead of calculating margin-based metrics compared to parameter-based metrics?\nHow might real quantum hardware noise affect your margin-based approach to generalization?\nYour small training set (20 samples) for QPR experiments raises questions about statistical significance. Have you verified these results hold with larger datasets?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "KPsh4nbrjq", "reviewer": "Reviewer_5bgA", "summary": "The authors address generalization in quantum machine learning by introducing a margin-based framework. The authors critique traditional uniform generalization bounds, which have been shown to be ineffective in both classical and quantum settings, and propose margin-based generalization bounds as a more reliable alternative. They extend classical margin-based theory to quantum neural networks by leveraging Lipschitz continuity and matrix covering techniques. Their experiments on quantum phase recognition datasets demonstrate a strong correlation between margin distribution and generalization performance, surpassing traditional metrics like parameter count. They also establish a connection between margins and quantum state discrimination, highlighting how maximizing the separability of quantum embeddings improves generalization. Their findings suggest that margin-based methods provide better theoretical insights and practical guidance for designing QML models with improved generalization capabilities.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank Reviewer 5bgA for insightful comments. Below, we address each point directly and explain how we plan to incorporate your suggestions.\n\n---\n### 1) **Limited experiments; broader testing is needed for statistically robust validation**\nTo strengthen empirical validation, we have now conducted extensive additional experiments on two canonical quantum many-body datasets: **Transverse Field Ising Model** and **XXZ-Heisenberg chain**. We repeated all main-text experiments on these datasets; results are summarized clearly at the following anonymous links: [TFIM](https://tinyurl.com/47mpp43c) and [XXZ-Heisenberg](https://tinyurl.com/3r598nmm).\nFor both TFIM and XXZ datasets, the margin consistently correlate with generalization performance. This strongly reinforces our central claims regarding the effectiveness of margin bounds for QNNs. These additional datasets significantly broaden our experimental scope, providing comprehensive validation of margin-based generalization theory.\n\nWe appreciate the reviewer highlighting the coarse discretization of our initial experimental results (originally QCNN layers = $\\{1,5,9\\}$). To comprehensively validate our claims, we now provide fine-grained experimental results covering $\\{1,3,5,7,9\\}$ layers. We also extend these detailed margin distribution analyses to the TFIM and XXZ-Heisenberg model. Results are summarized [at this anonymous link](https://tinyurl.com/5n6j7us3). We will explicitly incorporate these detailed results into the appendix of our revised manuscript.\n\n---\n### 2) **Generalization gap results are mixed, especially in the appendix**\nWe acknowledge the reviewer’s observation regarding mixed generalization gap results presented in the appendix. However, even within these initial results, margin-based metrics (both Q1 and mean) still consistently outperform all parameter-based metrics.\n\nTo further address this concern, we conducted additional experiments using the TFIM and XXZ-Heisenberg datasets, available [at this anonymous link](https://tinyurl.com/mrx8943h). These new results demonstrate highly stable and consistent correlations between margin-based metrics (Q1 and mean) and the generalization gap, without mixed outcomes. This further supports our central thesis that margin-based metrics offer stable and robust predictors of generalization in QNNs.\n\n---\n### 3) **Lack of comparison with methods beyond parameter-based metrics**\nWe appreciate the reviewer highlighting this point. Compared to classical machine learning, Quantum Machine Learning (QML) is a much newer field, and currently has relatively few established methods for evaluating generalization. While several recent works have proposed bounds from different theoretical perspectives, most ultimately fall within the class of parameter-based or uniform generalization bounds.\n\nFor instance:\n\n- [Banchi et al.](https://tinyurl.com/yjx386x9) derive information-theoretic bounds based on the mutual information between the training data and the parametric quantum states. The analysis is based on Rademacher complexity, reflecting limitations shared with other uniform bounds, such as that proposed by [Caro et al](https://tinyurl.com/mrsrrjk4).\n\n- [Abbas et al.](https://tinyurl.com/mw2ua99y) propose using the effective dimension, defined via the Fisher information matrix, to capture model complexity. This approach captures model expressivity through parameter sensitivity and thus also remains fundamentally parameter-based.\n\nGiven this, we chose to compare against more recent work by [Caro et al.](https://tinyurl.com/mrsrrjk4) as a representative of this broader class of approaches. Their bounds are constructed using covering numbers and Rademacher complexity, directly in terms of trainable parameters and model capacity. This makes their work both relevant and representative of the prevailing generalization bounds in QML to date. \n\nMoreover, the recent critical analysis by [Gil-Fuster et al.](https://tinyurl.com/4kjnvb44) directly points to inherent shortcomings of parameter-based uniform bounds, motivating us to benchmark against parameter-based methods explicitly.\nWe will clearly outline these justifications in our revised manuscript, contextualizing our choice of comparisons and the current state of generalization theory in QML.\n\n---\n### **Additional Points**\n\n**Related Work Section:**  We agree and thank the reviewer for this suggestion. We will include a concise **Related Work** section in the revised manuscript, situating our contributions within recent quantum generalization literature.\n\n**Proofs in the Main Manuscript:**  To maintain readability, detailed proofs were provided in Appendix A.1. We will refer readers to these proofs from the main text, clearly guiding navigation without compromising readability.\n\n---\nWe greatly appreciate Reviewer 5bgA’s valuable suggestions and believe these adjustments significantly enhance the clarity, robustness, and overall quality of our manuscript.", "claims_and_evidence": "The claims made in the submission are supported by clear and convincing evidence", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria make sense for the problem, but they can be improved with more datasets.", "theoretical_claims": "No proofs in the main manuscript.", "experimental_designs_or_analyses": "The optimized 8-qubit QCNNs and the generalization gap analysis are well-designed experiments. However, the discretization of the experiments is somewhat coarse, which may impact the precision of the results.", "relation_to_broader_scientific_literature": "The authors relate to the previous margin-based generalization. The paper is missing a related works section.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strengths:\n\n1. The method is well-motivated, addressing the limited understanding of generalization in quantum machine learning. The connection between quantum state discrimination and generalization is a novel and valuable contribution to QML theory.\n\n2. The paper is well-written, and the results are engaging and clearly presented.\n\n3. The method outperforms the compared approaches, particularly in the comparative analysis. The experiments show a strong correlation between margin distribution and generalization performance, supporting the theoretical claims.\n\n4. The application of QML to classical data using quantum embeddings is not novel but is a useful and relevant demonstration.\n\nWeaknesses:\n\n1. Limited experiments, broader testing across different parameters is needed for statistically robust validation.\n\n2. Generalization gap results are mixed, especially in the appendix.\n\n3. Lacks comparison with other  methods, relying mainly on comparison with effective parameter analysis.", "other_comments_or_suggestions": "1. Add a relevant work section", "questions_for_authors": "1, How would the method of margins compare to other methods in terms of robustness, or how generalizable is this approach.\n\n## Update after Rebuttal\n\nThe additional results and clarifications strengthen the massage of the already interesting work. However, the new experiments are similar to the existing ones and the sections need to be improved to match the style of ICML better, e.g. clear Relevant works and limitations are missing. Additionally, I am skeptical about the significance of the results, partly due to the limited experimentation, in regard to a venue like ICML.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper extends previous results in classical learning theory on margin-based generalization to the quantum setting. It provides new generalization bounds for quantum neural networks (also known as parameterized quantum circuits), contributing to the theoretical understanding of generalization in quantum machine learning (QML). It also provides an experimental analysis to support the theoretical results. All the reviewers agree that this is a solid paper with valuable theoretical fundings. Concerns were raised regarding the experimental evaluation. The authors in their rebuttal provided additional experimental results to address these concerns. While further evaluation on devices with a larger number of qubits would strengthen the experimental part, this paper is a nice contribution to the field of QML and is worthy of publication. I recommend that the authors take into account all the reviewer’s comments to enhance the experimental results and the presentation of the paper (e.g., clear relevant work section).
Decision: Accept (poster)