Predictive Performance of Deep Quantum Data Re-uploading Models
Authors: Xin Wang
Hanxiao Tao
Rebing Wu
Venue: ICML 2025 poster (2025)
Keywords: Quantum machine learning
Data re-uploading
Predictive performance
Primary area: general_machine_learning->supervised_learning
TL;DR: We investigate the predictive performance of data re-uploading models with deep encoding layers and demonstrate that their performance degrades to near-random guessing as the number of encoding layers increases.

Abstract: Quantum machine learning models incorporating data re-uploading circuits have garnered significant attention due to their exceptional expressivity and trainability. However, their ability to generate accurate predictions on unseen data, referred to as the predictive performance, remains insufficiently investigated. This study reveals a fundamental limitation in predictive performance when deep encoding layers are employed within the data re-uploading model. Concretely, we theoretically demonstrate that when processing high-dimensional data with limited-qubit data re-uploading models, their predictive performance progressively degenerates to near random-guessing levels as the number of encoding layers increases. In this context, the repeated data uploading cannot mitigate the performance degradation. These findings are validated through experiments on both synthetic linearly separable datasets and real-world datasets. Our results demonstrate that when processing high-dimensional data, the quantum data re-uploading models should be designed with wider circuit architectures rather than deeper and narrower ones.

Predictive Performance of Deep Quantum Data Re-uploading Models
Xin Wang' Han-Xiao Tao
Re-Bing Wu
Abstract
D-dimersional datases
Quantum machine learning models incorporat- ing data re-uploading circuits have garnered sig- nificant attention due to their exceptional ex- pressivity and trainability. However, their abil- ity to generate accurate predictions on unseen data, referred to as the predictive performance, remains insufficiently investigated. This study reveals fundamental limitation in predictive per- formance when deep encoding layers are em- ployed within the data re-uploading model. Con- cretely, we theoretically demonstrate that when processing high-dimensional data with limited- qubit data re-uploading models, their predictive performance progressively degenerates near random- grndom-guessing levels as the number of encod- ing layers increases. In this context, the re- peated data uploading cannot mitigate the perfor- mance degradation. These findings are validated through experiments on both synthetic linearly separable datasets and real-world datasets. Our results demonstrate that when processing high- dimensional data, the quantum data re-uploading models should be designed with wider circuit ar- chitectures rather than deeper and narrower ones.
Encoding
LaD
Training
11]
Predicting
Shallow
10
Deep Eacoding
Eneoding
[1]
Figure I. For D-dimensional lincarly separable data, data re- uploading encodes the data into quantum eircuits and trains the model effectively. However, during prediction. data re-uploading with shallow encoding layers maintains prediction results close to the training outcomes, while deep encoding layers lead to predic- tions that approach random guessing.
processing quantum data (Huang et al., 2021b; 2022), current experiments on classical data are primarily re- stricted to simple, low-dimensional datasets (Schuld et al., 2020; Hubregtsen et al., 2022; Pérez-Salinas et al., 2020; Huang et al., 2021a; Xu et al., 2024). The potential for achieving a quantum advantage in the processing of prac- tical high-ddmeesiioal classical data remains to be demon- strated (Schuld & Killoran, 2022).
1. Introduction
Quantum Machine Learning (QML) (Biamonte et al., 2017) has emerged as promising field that integrates machine learning with quantum computing, offering potential compu- tational advantages (Ristè et al., 2017; Huang et al., 2021b; Zhong et al., 2024). In recent years, quantum machine learning models based on Parameterized Quantum Circuits (PQCs) (Benedetti et al., 2019) have garnered significant attention (Yan et al., 2023; Meyer et al., 2023; Zhao al., 2024). Although QMI. demonstrates advantages in
Encoding classical data into quantum states presents sig- nificant challenge in QML (Lloyd et al., 2020; Wiebe, 2020; Weigold et al., 2021; Rath & Date, 2024). Currently, two primary encoding paradigms exist in supervised quantum machine learning: the secodding-variationa paradigm and the data re-uploading paradigm (Jerbi et al., 2023). The encoding- -vriitiigaaiiaiinnnal paradigm encodes data into quantum states using methods such as basis encoding, angle encoding, or amplitude encoding, followed by training with parame- terized quantum circuits (Schuld & Petruccione, 2018). 8). In contrast, data re-uploading interleaves trainable parameter- ized gates between encoding gates, enabling more flexible data processing and enhancing model expressivity (Pérez- Salinas et al., 2020).
'Department of Automation, Tsinghua Univer- sity, Beijing, China. <rbwu@ esinghua.cdu.cn>.
Correspondence to: ReBing Wu
Proceedings of the 4rnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Predictive Performance of Deep Quantum Data Re-uploading Models
The crooding-variationa paradigm suffers from several trainability issues, including encoding-induced optimization challenges (Li et al., 2022), exponentially numerous local minima in shallow circuits (You & Wu, 2021; Anschuetz & Kiani, 2022), and barren plateaus in deep circuits (Mc- Clean et al., 2018; Ragone et al., 2024). In contrast, data re-uploading improves trainability through interleaved pa- rameterized and encoding gates, enhancing model expressiv- ity (Pérez-Salinal et al., 2020). This approach supports deep circuit training and enables single-qubit circuits to approx- imate multivariate functions given sufficient depth (Pérez- Salinas et al., 2024).
ple size can reduce generalization error but might increase training error. With these dynamic changes, it's difficult to determine the resulting prediction error (their sum), whereas our approach bypasses this limitation by directly analyzing the prediction error.
To establish this phenomenon, we analyze the model's pre- dictive performance by examining the expected output over the data distribution. Our analysis focuses on two key as- pects: the impact of the number of encoding layers and the number of repetitions. For the former, we employ tech- niques similar to Li's paper (Li et al., 2022), which origi- nally only permitted specific non -on-parameterized entangling gates (CNOT or CZ) between encoding layers. We extend these techniques by allowing arbitrary learnable parame- terized quantum gates between encoding layers. For the latter, Li's techniques cannot analyze repeated data upload- ing scenarios, we address this limitation by constructing approximating circuits to analyze cases with repeated data uploads. Importantly, repeated data uploading is crucial, as it significantly enhances the trainability of data re-uploading models (Pérez-Salinas et al., 2020; 2024; Yu et al., 2022; 2023). Furthermore, the data re-uploading paradigm with its good trainability addresses the trainability issues identified in Li's paper, and our work further extends this research by focusing on analyzing the predictive capabilities of these models.
The primary goal of machine learning is to achieve accurate predictions on previously unseen data. Researchers usually study trainability (training data performance) and general- ization (training vs prediction gap) separately (Mohri, 2018). But good trainability or generalization alone doesn't guar- antee good predictions. While mooding-variationn shows good generalization (Caro et al.. 2022), yielding small generalization error with few training samples. However, the predictive performance of data re-uploading models re- main less understood. In contrast to prior work, we directly analyze how these models perform on unseen data and estab- lish that, when the number of qubits is limited and the data dimension is high, their predictive performance degenerate to nearly random guessing.
QML
Our experiments confirm the theory, showing that differ- ences in encoding layers notably affect predictive perfor- mance, despite identical datasets, parameter counts, and training errors. Additionally, we explain why data re- uploading models perform well on MNIST (LeCun) even with deep encoding layers
1.1. Contributions
As shown in Fig. 1, we demonstrate that, regardless of the quality of training, the predictive performance of data re- uploading models on new data asymptotically approaches random guessing when the encoding circuit is deep. This typically occurs when the dimension of the data vectors far exceeds the number of qubits in the quantum circuits.
These findings offer critical insights into the architectural design of data re-uploading models, indicating that the ac- curate prediction of high-dimensional data is unlikely to be accomplished by few-qubit quantum circuits, notwithstand- ing their trainability and expressivity.
Our theoretical analysis shows that as encoding layers in- crease, predictions from data re-uploading circuits approach those from maximally mixed states, and repeated uploading cannot mitigate this problem.
1.2. Related Work
We introduce novel method to analyze the prediction error, bypassing the traditional decomposition into training and generalization errors. We directly analyze the expected output of the model over the data distribution. Our re- sults demonstrate that when using data rp-upddading models with deep encoding layers, the model's performance on the unseen new data approaches random guessing, regard- less of how good the training results are, and regardless of the choice of loss function, optimization method (gradient- based gradient-free), number of iterations, model pa- rameter count, or training sample size. After decomposing prediction error into training error and generalization error, increasing model complexity can reduce training error but might increase generalization error, while increasing sam-
Quantum Encoding The properties of quantum encod- ing strategies has investigated several fundamental aspects. (Schuld et al., 2021) examined how different encoding strate- gies affect the expressivity of models. Regarding robustness, (LaRose & Coyle, 2020) indicated that encoding impacts the robustness of quantum classifiers against quantum noise. Regarding generalization, previous research (Caro et al., 2021) derived generalization bounds related to different en- coding strategies for quantum models. Another study (Li et al., 2022) demonstrated that for angle encoding, the aver- age quantum state exponentially approaches the maximally mixed state as the encoding layers increases.
2

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Predictive Performance of Deep Quantum Data Re-uploading Models
Data Re-uploading The exploration of data re-uploading has predominantly focused its trainability. The con- cept was initially introduced in (Pérez-Salinas et al., 2020), where the advantageous trainability properties were exper- imentally demonstrated. Subsequent theoretical work by (Pérez-Salinas et al., 2024) established that increasing depth enables single data re-uploading models to approximate any multivariate function. Further theoretical foundations were provided by (Yu et al., 2022; 2023), who rigorously proved these models' advantages in function approximation com- pared to classical ReI.U neural networks. (Barthe & Pérez- Salinas, 2024) augmented these findings with a detailed analysis of gradient behavior and frequency profiles. These benefits have led to widespread use in quantum machine learning, typically with shallow encoding layers and multi- ple repetitions (Dutta et al., 2022; Wach et al., 2023; Cassé al., 2024), or on sparse
mixed state Pr A represents a state with no information.
Quantum states evolve through quantum circuits (mathemat- ically represented as unitary transformations U), transform- ing from to p' according to p' = U pl/.. Quantum circuits are composed of quantum gates, with typical single-qubit gates being rotation gates of the form Ro) = e -6///2 where P E {x, 3 are the Pauli matrices as follows:
-1 ) Z (1 0) , Y - (:
0
( 0
0 -)
x
In particular, any single-qubit gate can be represented as R(o1. 22 Φ3) = R_ R, (z Multi-qubit gates, also known entangling gates, entangle multiple qubits together. with the CNOT gate and CZ gate being typical examples. To extract information from quantum circuits for classical processing, measurements are performed using Hermitian matrices H, with the expectation value given by Tr pl. where H/ is called the observable.
et al., 2024; Rodriguez-Grasa datasets (Periyasamy et al., 2022; Jerbi et al., 2023). Regard- ing generalization capability, (Zhu et al.. 2025) established the relationship between the number of data re-uploading repetitions, training epochs, and the model's generalization performance.
2.2. Data Re-uploading
Data re-uploading was first proposed by (Pérez-Salinas et al., 2020), with its core idea being to enhance PQC-based quan- tum classifiers by alternately constructing data loading gates and data processing gates. In (Pérez-Salinas et al., 2020), both the single-qubit data encoding gates and single-qubit data processing gates take the form:
The investigation into the predictive performance of quantum machine learning models has been relatively scarce. (Caro et al., 2022) de- rived generalization error bounds for encoding-variatt v QML. (Wang et al., 2024) combined these generalization error bounds with the training error bounds of AdaBoost to establish prediction error bounds. In quantum kernel meth- ods, (Wang et al., 2021) established generalization error under conditions of quantum noise. For data re-uploading models, the predictive performance has not been well under- stood.
Predictive Performance of QML:
R(Ø)) = R(01, 92, 03) = (d (02) R2( = e -o- 12 -10.212 2/2
(I)
where Z and Y are Pauli matrices. The gate R(Φ) encodes data when Φ and processes data when s 0. In multi-qubit circuits, entangling gates (e.g., CNOT, CZ) link individual qubits to generate quantum entanglement.
2. Preliminaries
2.1. Quantum Computing
In this paper, we consider a more general architecture of data re-uploading circuit. The circuit loads data exclusively through single-qubit encoding gates as defined in Eq. (1), while allowing arbitrary parameterized unitary gates. Given that the limited-scale quantum circuit has N qubits, the entire data whose dimension is larger than 3N has to be uploaded via multiple encoding layers. A simple way is to divide the data into L chunks and upload chunks via L. encoding layers, where L > | D This data is then re- peatedly uploaded into the circuit P times. The complete architecture of the quantum data re-uploading circuit is il- lustrated in Fig. 2.
We first introduce elementary concepts in quantum com- puting. The state space of an N-qubit quantum system a 2 "dimensional complex Hilbert space H _ C2 The computational basis for this Hilbert space consists of {10), [1) ;...., 12N -))). where each basis state (i) repre- sents unique binary string of length 2 corresponding to the binary representation of integer i.
Information in quantum systems is stored in quantum states, which can be represented by a positive semi-definite matrix E C² x with property Trip] = 1. If Tr[p²) = 1, the quantum state is called a pure state; otherwise, it is a mixed state. For N-qubit pure states, they can be represented by unit state vector kp)) E C2" where p = ko) (1p) and (pl = l.) Mixed states are convex combinations of pure states. In particular, for an N-qubit system, the maximally
Without loss of generality, we assume that the dimension of data æ is D= = 3.N L. the entire data is divided into L chunks 211]]] each chunk Z[j] E R contains n three-dimensional data Di,, = [It,,,1, Ti,,, 31, and every Ii,n is encoded by the single-qubit encoding gate
x = [i|]]] [
3

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Predictive Performance of Deep Quantum Data Re-uploading Models
(a)
ab E RANZ
(b)
L chunks
Fi, E R
chunking
lil E R
(c)
1 chunk
10)
R((a
R R(2221)
R(
10)
R(ar.))
R(
R(
(1()p.)
U2(0p22)
Uz(0p1)
10)
R(ZLN)
(
R(
encoding layers
Figure 2. Data re-uploading encoding process. (a) The original data. (b) Divide original data into chunks. (c) Each data chunk encoded by an encoding layer. The entire data re-uploaded the circuit times, where parameterized gates each repetition can be arbitrary and can differ between repetitions.
R(a...)) in the I-th layer and n-th qubit. 1-th layer's encod- ing gate is denoted as (x))) Following (Li et al., 2022), we assume data elements follow independent Gaussian dis- tributions Ti,n,i N/ (Mt,n,i, ".,,,)-
its corresponding label ((m), independently drawn from same distribution. Here, X' denotes the feature space and Y represents the label space. This paper focuses on the deterministic scenario where each feature x sampled from Dx has unique label y(x). The machine learning model learns a hypothesis hş from dataset aiming to correctly predict labels for new features drawn from Dx
The parameterized gates after I-th encoding layer p-th repetition is denoted as (...). Note that each repetition uses the same data vector but with different parameter vec- tors. Let 0 (0) 1P]] represent the collection of all parameters, then the quantum gate of P-repeated data re-uploading circuit can be expressed as:
In this paper, consider both classification and regression tasks. For classification tasks, we focus on binary classifica- tion where the labels corresponding to are y(x) E {0, 1}. We employ Hy(x)) as the observable for feature æ, where Hy = [0) (0) and H1 = [1} {1) are single-qubit observables acting on the first qubit. Since the measurement results for both Ho and Hi are greater than and sum 1, these results be interpreted probabilities of belonging to each class. The hypothesis function learned by P-repeated data re-uploading models trained on dataset S is given by hg( = ( 0") with observable H Hy(w). where 0* is the parameters chosen during training. hg(x) indicates the probability that feature x belongs to the correct class.
V(z, 0)
Ui (((0p.)R((zm))) p=11-
I
where IT- A₁ A A1
Data re-uploading circuits starts from initial state Pu [0) (0). the quantum state evolves through P-repeated data re-uploading circuit V(x,, 0). The expectation value of mea- surement the evolved state with respect an observable
H is:
For regression tasks, we consider target functions f(x) N
[-1, 1] with observable H2, X n Zn. The hypothesis hs(x) represents the predicted value of f(x).
h Tr [HV O)po Vx,,
Different tasks require different observable H. In this paper, the cigenvalues observable lies in [-1,
To evaluate the predictive performance of hypothesis hgg define its prediction error distribution Dx for classifi- cation tasks as:
2.3. Supervised Quantum Machine Learning
R (Phs)
[1 (1-1-s())) hs x~
In supervised machine learning, we consider a dataset = [(x(m), ) containing M samples. Each sample (m)) (mm)). consists of feature vector a and
This prediction error evaluates the gap between the proba- bility that hypothesis hş predicts the feature the correct

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Predictive Performance of Deep Quantum Data Re-uploading Models
class and 1.
which has important applications in both quantum ma- chine learning (Liu et al., 2022) and quantum communi- cation (Fang & Fawzi, 2021). The D2 divergence can effec- tively analyze quantum state distinguishability in the Pauli basis, bounding observable measurement differences. (see App. B). For any N -uqubi quantum states p and maximally mixed state pr. D2 (pllpr)) E [o, N] (see App. F), reaching its minimum value of 0 if and only if p = pr. In this paper, when referring the quantum divergence, mean the D2 divergence defined in Eq. (3).
For new features, class prediction is performed by measur- ing observables Ho and H, then assigning the class with the higher measurement result. Crucially, In this paper, we assume that each feature æ inherently possesses a unique true label (x) This allows performance evaluation by com- paring the model's probability of correct prediction against (representing perfect certainty). If the prediction probability approaches random guessing levels(e.g. near 0.5 in binary classification), this statistically indicates model failure to learn meaningful patterns, even without label knowledge of new features.
3. Main Results
For regression tasks, we define the prediction error to evalu- ate the gap between the output of fy and f(x as:
This section presents our main theoretical findings on how encoding layers and repetitions affect the predictive perfor- mance of data re-uploading models. Subsec. 3.1 demon- strates exponential approaching of expected encoded states to maximally mixed states with encoding layers. Subsec. 3.2 shows the ineffectiveness of repeated uploading. Subsec. 3.3 derives theoretical bounds for prediction error.
(hs)
E x~>x [=) hs(x)]]
Meanwhile, we also care about the hypothesis's perfor- mance on the training set S ((x(1m) ) defining the training error for classification as:
3.1. Dependence on the Number of Encoding Layers
M M
(hs)
Σ 1 - (m)))
We first analyze how encoding layers affect the expected encoded state generated by single-upload (P = 1) data re-uploading circuits. The encoding transformation of in- put data x - (((=) introduces nonlinearity that complicates direct analysis of expected quantum states. To overcome this, adopt Pauli basis decomposition ap- proach, examining the evolution of Pauli coefficients of expected quantum states within this framework.
Similarly, for regression tasks, we define the training error as:
1 M
(bs))) = M
Σ m=1
(m) - -hg - (m))
Since the prediction error cannot be directly computed, it is typically analyzed by decomposing it into two components: the directly computable training error and the generalization error gen(hs). ). For classification tasks, this decomposition is expressed as gen(hs) problems follow the same principle). In practical applica- tions, we estimate the prediction error by evaluating the model's s performance on a test set.
For an N-qubit quantum state p. its representation in the Pauli basis can be expressed as:
Rº (hs) RG (hs) (regression
p 2N
Σ
aiP:
(4)
P.ELLXXYY2IS.
where a, TT [pP.] is the coefficient for Pauli basis cle- ment P. Since each P: is Hermitian, all coefficients a, are real numbers, with 4.V terms in total. Under unitary trans- formation U, the state UpT keeps the same Pauli basis structure, only changing the coefficients ag Our analysis will focus on these coefficient changes.
2.4. Quantum Divergence
Similar to the Kullback- divergence in classical in- formation theory, the quantum Pénz-Rényi divergence (Petz, 1986) measures the distinguishability between two quantum states:
When the encoding gate R(xr) is applied single-qubit state p. where æ [21, 22, sian distributions Ij (iu,, of) with of > a², E [3]. we can analyze the state by decomposing it in the Pauli basis {I, Z, X. with corresponding coefficients [ar, a' axx ay.. Considering the expected state after en- coding E[p E,, [R(z) po R(x)*]] [z) let [8,, Bz, 3x By] de- note the Pauli basis coefficients of E[p]. We establish that 83 + 88 + 1& & e (a + ax + a while 31 = aj remains unchanged. This demonstrates that the magnitude
22, 23] follows independent Gaus-
(Pilllpp)) = logz (Tr [pt -)1),
(2)
where E U (1, (oc). While not a formal distance metric, it enables critical quantum hypothesis testing anal-
yses (Audenaert, 2007: Nussbaum & Szkola, 2009). For analyzing quantum encoded states, we employ a = 2:
D2(Pil)o2) = logz (Tr [pip5
5

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Predictive Performance of Deep Quantum Data Re-uploading Models
of Pauli coefficients for non-identity terms decays expo- nentially with the variance o². As the encoding layers L increases, these non-identity coefficients rapidly approach zero, leaving only the identity term. Consequently, the quan- tum state converges to the maximally mixed state. The following theorem formalizes this phenomenon.
layers and P repetitions, we construct an approximating cir- cuit using N + (++NT((++)) qubits (Fig. 3). Let h 0) and hp 0) denote the outputs of the original and approximat- ing circuits respectively. The approximation error satisfies [ 0) - hr( 0)1 < á when q >> [log2 (3PLN/6)).
For both P = 1 and P > circuits, we construct ap- proximating circuits with outputs h1 (xx,)) 0) and hp 0) respectively. The P-repeated approximating circuit essen- tially applies multiple data- inteeeindeenndnnt unitary gates to the circuit with P 1. Since unitary operations cannot enhance the distinguishability between quantum states, P repetitions cannot improve the indistinguishabiliiy caused by deep encoding layers. That is, if [h (az 0) - -yyl < e, then [ 0) hr| < c. This leads to the following the- orem for data re-uploading circuits with P repetitions (see App. D).
Theorem 3.1. Consider an N-qubit data re-uploading cir- cuit with I, encoding layers and without repetition (P = 1), which encodes data æ E IR3NL into the circuit, where each data point follows an independent Gaussian distribution, ie., F1,n. ~ N (ut,,n. (ma)) and am 2 o². Let p(ær, 0) denote the N N-qubit encoded state. Then the quantum diver- gence between the expected state [[[p] = Ex [o( 0) and the maximally mixed state pI = z5r satisfies:
D2 (E[e)ler) & logg (1 + (2 - 1)e -0==
Theorem 3.2. Consider an N N-qubit data re- re-uploading cir- cuit with I, encoding layers and P repetitions, which en- codes data æ E RANT into the circuit, where each data point follows an independent Gaussian distribution, i.e., Fi,n, ~ ( o. and of.. 2 0². Let hp(æ 0) be its output with respect to an observable H whose eigenval- ues lie in [-1,1]. When L > 221(N + 2 + 2 In( }),, we have:
Intuitively, when quantum states become indistinguish- able, their measurement results for observables with eigen- values in [-1,1] 1] also become identical. Theorem 3.1 shows this occurs when L > I + 2) In + 2 (l) EE [x)] - hu] & e, where Ex (tsl/((x))) = Tr [p]] and h = Tr ps) This means that the expectation of hypoth- esis hss with respect to feature æ is nearly informationless (see App. C).
Elhr(ar 0)] h1 c:;
It is worth noting that Theorem 3.1 extends beyond the scope of Theorem 2 in (Li et al., 2022) by allowing arbitrary learn- able parameterized quantum gates between encoding layers, rather than being limited to specific non-parameterized en- tangling gates.
where hy = Tr pr) and pr = mally mixed state.
zx J is the N N-qubit maxi-
It is important to note that Theorem 3.2 extends to arbitrary repetition numbers, going beyond the non-repctition case addressed in Corollary 2.1 of (Li et al., 2022).
3.2. Dependence on the Number of Repetitions
3.3. Bounds on the Prediction Error
Due to repeated data uploading, even the Pauli coefficients of quantum states have nonlinear relationships with input data, making direct analysis methods inapplicable. To ad- dress this, we approximate the data re-uploading circuits with P using approximating circuit.
We establish that data re-uploading models with deep en- coding layers exhibit limitations in predictive performance:
Proposition 3.3 (Informal). When the conditions of The- orem 3.2 hold, the prediction error of the hypothesis hs generated by the data re re-uplodiing model satisfies:
For a data vector æ = [#1.".. [DD] E IRP, we focus on a single element Fa E [0, 2m] (periodicity follows from Eq. (I)). We expand z in binary form as -ا 6,2-
(1) For binary classification tasks:
|RC - 1/1 <G
where b, E 1) and leg/ & 2- Here. the first 3 bits encode the integer part and bits encode the fractional part. Let Za denote the approximation of xa with q + 3 bits.
(2) For regression tasks: RL(h)
>
E D>
[f(z)] -
For classification tasks, when RC (hs) = 1. it indicates that the hypothesis hy assigns equal probability to new features belonging either the correct or incorrect class. For regres- sion tasks, the lower bound implies that the prediction error depends solely on the target function and is independent of the learned hypothesis hg. In both cases, these results demonstrate that the model fails to extract useful informa- tion.
To approximate Rg (aa), we use working qubits (for final measurement) as targets and auxiliary qubits (for encoding data T.2) as controls. Controlled rotations C - R2 (2- are applied with auxiliary qubits (by) as controls and working qubits as targets, approximating Rg(xx) with Rg (ūa). This approach naturally extends to (x) gates.
For an N-qubit data re-uploading circuit with I. encoding
6

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Predictive Performance of Deep Quantum Data Re-uploading Models
xP
0))
Working Qubits
U1(Op.))
U2(Op.2)
R
Ur(0p.1)
3N(++3)
<
3N4+43)
Ancillary Qubits
Discard
Figure 3. Approximating circuit for data re-uploading circuits. The qubits in approximating circuit are divided into two parts: working qubits and auxiliary qubits. Firstly, 3.V L(q ancillary qubits encode the binary digits of feature E R into the circuit. Then, we use the data-independent control gate CR between the working qubit as target qubit and auxiliary qubits as control qubits to approximate the encoding gale in working qubits. Note that the approximating circuit is only used for theoretical analysis.
4. Experiments
The mean values H4z are class-dependent: for class one, Nd 22 (dmodd 8)] mod 2n,, while for class two, Paa 2x (8 md 8)) mod 2T. Each dimension d has vari- ance o 0.8 for both classes.
4.1. Divergence Experiments
(aj10
(b)10
We use data re-uploading model as shown in Fig. G.1. The training set contains 2000 samples, and the test set contains 1, 1,000,000 samples (see App. F), With normally distributed initial parameters, we trained models using cross-entropy loss with Adam optimizer (learn- ing rate 0.005) over 1000 epochs (batch size 200), se- lecting parameters with the lowest training emror for test. We computed divergence between maximally mixed state and average encoded states for both classes, then averaged these values as the final divergence result.
Experimental Setup:
10
10
Divrgeoss
10
10
10
10
upo
10
10
10)
(1110)
P-
preitritiss
10
10
Dieegooe
10-
10
10"
For fixed repetition P 1, we examine the depen- dence of quantum divergence on qubit number and encoding layers. As shown in Fig. 4(a). the divergence exhibits ponential decay with increasing encoding layers L. While training induces marginal increase in divergence as shown in Fig. 4(b), this effect is negligible compared to the expo- nential decay trend.
Results:
Figure Divergence versus encoding layer Panels (a, b): vary- ing qubit number N at fixed repetition P = 1; panels (c, d): fixed qubit number N Pre-training (before training, in panels (a, e)) and post-training (after training. in panels (b, d)) results are compared with theoretical upper bounds. Panels (a, h) share common legends, and panels (c, d) share com- mon legends: colors panels (a, c) represent different N or P, and line styles in panels (b, d) represent conditions (upper-bound, pre-traing pre-training, post- trainin)) Y-axis is logarithmic.
varying repetition
For fixed qubit number N 1, we examine the depen- dence of divergence encoding layers and repetition. As shown in Fig. 4(c), without training, more repetitions may lead to faster decay of divergence. However, as shown in Fig. 4(d), after training, the divergence with multiple rep- etitions approaches the upper bound of P = 1 given by Theorem 1. Notably, when the number encoding layer L 1, the divergence remains close to the theoretical up- per bound regardless of the number of repetitions, which explains why circuits with L 1 perform well in quantum machine learning tasks.
Data: To validate Theorem 3.1, measure the di- vergence in binary classification datasets. Each dataset sample (m), y((m) consists of D-dimensional feature vector (m) E RD and class label (m) The ele- ments of (m)) are independently drawn from Gaussian distributions, where the d-th element follows M(pa, a2).

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Predictive Performance of Deep Quantum Data Re-uploading Models
4.2. Classification Experiments
the model with different encoding layers and repetitions P. As L increases, the test error gradually increases 0.5 (equivalent to random guessing). As P increases, the train- ing error decreases, but the test error converges to 0.5 faster with increasing L. Fig. 5(c) and (d) display the difference between the model's s output with respect to the observable Ho and Tr [Hop]] for training and test data, correspond- ing to Fig. 6(a) and (b), As the test error approaches 0.5, the difference test data approaches zero, which verifies Theorem 3.2.
Data: To demonstrate the limitations of quantum data re- uploading caused by deep encoding layers, we consider simple linear separable classification problem. For D- dimensional data æ distributed in |- 2: 21 if 11114 Txx > 0.3D, the data belongs to class one, and if 1 x < -0.3D, it belongs to class two, where 1 is vector of ones.
(4) 0.5
(b)0.5
Erro
0.3
Differmue
0.3
4.3. Classification Experiments in Same Dataset
0.2
0.2
(4) 0.5
(b) 0.3
0.1
0.4
Dilleronce
0.2
Brror
(9) 0.5
(d)g.
Training
0.1
0.4
0,4
0.3
Test
Tiiferenne
Brrer
0.2
0.0
0.2
0.2
0.1
0.1
0.0
0.0
Figure 6. (a) Training and test error with same datasets, same rep- etition, but different encoding layers, (b) Difference between the output with respect to He on training and test data and Tr [H pr). Error bars represent the minimum and maximum values across 10 independent runs with different random seeds, with the central line showing the mean value.
3
L
Figure 5. (a) Training error, (c) Test error, (b), (d) Difference be- tween the model's 's output with respect to the observable Ho training data and test data compared to Tr pr]. respectively. Er- for bars represent the minimum and maximum values across 10 independent runs with different random seeds, with the central line showing the mean value.
Data:To eliminate the influence of data dimensionality on experimental results, we employed the same data as de- scribed in Subsec 4.2, with a feature dimension of D = 24.
Experimental Setup: We use a single-qubit data re- uploading model (Fig. G. for classification. A single qubit is chosen because it has received significant attention (Pérez- Salinas et al., 2020; Yu et al., 2022; Pérez- Srezz-Salinas et al., 2024) and serves as a typical example for examining the limitations of deep data re-uploading models. Besides, the single-qubit model can exclude the influence of trainability. To eliminate the impact of parameter count, we conduct comparative experiments using quantum circuits with fixed total encoding layers Lmax 8. For data with L chunks, we maintain the total circuit layers at Lmax by only encoding data in the first L layers and setting the data input to zero vectors for the remaining layers (using I. encoding layers).
We used data re-uploading circuits (Fig. G.1, excluding CNOT gates to prevent entanglement effects, as entanglement may degrade training and prediction performance (Ortiz Marrero et al., 2021; Leone et al., 2024)) 8),
Experimental Setup:
with the same number of parameters: (N = 1,
(N = 2, T, 4), (N 4, T, = 2), and (N = 8, T. = 1), all with P = 8 repetitions. Other experimental settings follow those in Subsec. 4.2.
Fig. 6 shows as encoding layers T, increases, the test error gradually increases 0.5, the corresponding dif- ference approaches zero. Despite similar training errors in same dataset between L 1 and L 4. the test perfor- mance differed significantly due to encoding layers.
Results:
In numerical experiments, the training set contains 600 sam- ples and the test contains 10000 samples. With normally distributed initial parameters, we trained models using cross- entropy loss with Adam optimizer (learning rate = 0.005) over 1000 epochs (batch size = 200), selecting parameters with the lowest training error for test. The experiments were repeated 10 times with randomly initialized parameters.
4.4. Real-world Dataset Experiments
Dataset: CIFAR-10-GAR (airplane/autoombiile. grayscale, 12x pixels), CIFAR- (airplane/automohiil, RGB, 12 x 12 pixels), MNIST (digit 0/1, 12 x 12 pixels).
We used circuits (Fig. G.1) with N = 8 and P 2, using encoding layers I, = 6 for CIFAR-
Experimental Setup:
Results: Fig. 5(a) and (b) show the training and test error of
8

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Predictive Performance of Deep Quantum Data Re-uploading Models
10-Gray and MNIST, and L 18 for CIFAR-IO-RGB. The training contain 600 samples and the test set contains 1000 samples per class. All other experimental settings follow Subsec. 4.2.
ond, as validated in Sec. 4, our conclusions remain valid even when the data elements not independent. Third, for certain tasks such regression, data elements can be ap- proximated as independent after feature engineering (such as principal component analysis).
Results: As shown in Fig. 7(a, b), MNIST outperformed CIFAR-10-Gray, which can be attributed to its near-zero variance (Fig. 7(d)). Direct encoding of RGB CIFAR-10 performed worse than grayscale CIFAR-I0 due additional encoding layers (Fig. 7(c)). As the number of repetitions in- creased, we observed that training accuracy increased while test accuracy decreased, leading to larger generalization error.
We adopt the Gaussian distribution due to its prevalence and convenience for proof, but our conclusions are not limited to Gaussian distributions. The key property we require is that the distribution of data satisfies E|cos(x))] 7 cos(p)) where |y| < 1. For example, with a uniform distribution over 11 a. a), we have [x)os(r)] = sin(a)/ a cos(p)) where |r| = |ii((a/aa < when + 0.
(b)
Finally, we note that our assumption implies each element provides independent information. When elements in high- dimensional data are strongly correlated, the limitations dis- cussed in this paper may not occur. However, such data es- sentially reduces to one-dimensional information, for more details, see App. H.
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Impact Statement
(c)
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Our paper aims to highlight the limitations of data re- uploading models with deep encoding layers in supervised quantum machine learning tasks. We provide theoretical guidance for the architectural design of data re-uploading models. Specifically, when using data re-uploading mod- cls to process high-dimensional data, employing a limited number of qubits with deep encoding layers is undesirable. Future work could focus on enhancing the predictive perfor- mance of large-scale data re-uploading models with shallow encoding layers.
CIRAR-IO
0.9
Tiksing 1P
Frequecy
Aganreyy
Dissesa 1P
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Thineng 1P
40
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Tea (P
20
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0
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2000
300 Epoch
400
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Figure 7. Results of real-world datasets: (a) Grayscale CIFAR- 10, (b) MNIST, (c) RGB CIFAR-10, (d) Standard deviations of Grayscale CIFAR-10 and MNIST. Solid and dashed lines represent the mean training accuracy and test accuracy, respectively, across 10 independent runs with different random seeds. Shaded areas indicate the minimum and maximum values over these runs. Colors represent different number of repetitions (P 2,4, 8)..
A Acknowledgments
This work supported by NSFC (Grant No. 62173201) and the Innovation Program for Quantum Science and Technol- ogy (Grants No. 2021ZD0300200..
Additionally, we conducted regression experiments, ana- lyzed the relationships among prediction error, training error, generalization error, and accuracy (App- G), and examined how increasing sample size and model complexity (repeti- tion number) impacts these three types of errors.
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Predictive Performance of Deep Quantum Data Re-uploading Models
A. Terminology about data re-uploading
this appendix, we clarify several key terminologies related to data re-uploading in this paper.
Data re-uploading encoding: The data re-uploading encoding discussed this paper refers an encoding paradigm that maps classical data into quantum circuits, characterized by two key properties:
Insertion of parameterized quantum gates between encoding gates.
Repeated uploading of data through identical encoding gates into the quantum circuit.
This encoding paradigm encompasses two special cases:
When no parameterized quantum gates exist between encoding gates (only identity or fixed quantum gates), with parameterized gates appearing only after all encoding gates, the data re-uploading encoding becomes equivalent to angle encoding (encoding data through rotation gates) in the encoding-variational paradigm;
When data is uploaded only once without repetition, which we consider as a special case of repeated uploading with repetition count equal to one.
When data re-uploading encoding satisfies both special cases simultaneously that is, data uploaded only and only identity or fixed quantum gates (such CNOT or CZ) exist between encoding gates the data re-uploading encoding discussed in this paper reduces to the scenario discussed in the paper (Li et al., 2022).
Data re-uploading circuits: The quantum circuits that satisfy the data re-uploading encoding paradigm. Data re-uploading circuit using parameters with respect to data æ can be represented as a unitary operator V (xx 0), which encodes data æ into a quantum state p(z, 0) Vax, Pn (x, 0y*, where Po is the initial state of the data re-uploading circuit.
Data re-uploading models: The machine learning models with data re-uploading circuits is data re-uploading models. For different machine learning tasks, the data re-uploading models will choose different observables to measure the encoded state p(æ, 0) generated by the data re-uploading circuits. The output of the data re-uploading model is the expectation value of the observable // with respect to the encoded state, i.e., (1x 0) 'Ir p(æ, 0)]. The relationship between the data re-up ce-uploading circuit and the data re-uploading model is shown Fig. A.1.
Data re-uploading model hp(x, 0)
H
10)
R(
R(
R(
7
(0)
R(
RI R(
R ((
(1((
U2(0p2)
Uz(0p1)
(0)
R(
M(
R(
Initial State Po
Data re-uploading circuit V(x. 0)
Encoded State p(x, 0)
Figure A.I. The data re-uploading circuit and data re-uploading model.
Encoding Layer: data re-uploading circuit, we define the number of encoding layers as the number of encoding gate layers used in repetition cycle. Notably, in data re-uploading circuit with P repetitions, only L distinct encoding layers are repeatedly used. The number of encoding layers in the circuit architecture corresponds to the number of data chunks, where each data chunk requires one encoding layer for uploading.
Repetition: In a data re-uploading circuit, the repetition count P refers to the number of times the data uploaded into the quantum circuit. While the same encoding layers are used across all P repetitions, the parameterized gates inserted between encoding layers may have different structures and parameters.
12

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< naturally follows Predictive Performance of Deep Quantum Data Re-uploading Models
Depth: In data re-uploading circuit, if the structure of parameterized gates between any encoding gates is identical (though parameters may differ), the circuit can be unfolded into PT. layers with identical structures but different parameters. The depth of such circuit refers to the number of these layer structures, which equals PL. The relationship between encoding layer, repetition, and depth is shown in Fig. A.2.
(a)
P Repetitions
(b)
L Encoding Layers
R 2
Ricssl
Rinaal
Vic. mni
Vix. x))
Vaar. pp)
30
U(0,a)
Rlaysl
010,al
Rirsavi
Riraal
(c)
Unfolding
R P
Ri
Rioezaa
07 (P.))))
Rlone 1o
Riaa
Depth is PL
Figure A.2. (a) Illustration of Repetition, (b) Illustration of Encoding Layer, (c) Illustration of Depth.
B. Measures of Quantum State Distinguishability
We introduce several commonly used measures of distinguishability for quantum states; trace distance, fidelity, affinity, and Petz- Rézyi divergence. These measures play crucial roles in quantifying the distinguishability between quantum states. In partieular, we establish the relationship between Petz-Rényi-2 divergence and trace distance, which serves as the foundation for our theoretical analysis in this paper.
Definition B.1 (Quantum Trace Distance). The quantum trace distance between quantum states Pi and is defined as:
T(p,, = 2 1 P2ll1.
where || ||| denotes the Schatten 1-norm. The quantum trace distance satisfies all properties mathematical distance: non-negativity, symmetry, and the triangle inequality. Furthermore, T'(i,, p2) E [0, 1] with equality to 0 if and only if P1 P2.
The trace distance provides an upper bound on the distinguishability of quantum states through measurements with respect any observable H with eigenvalues bounded [-1, 1]. Specifically, the difference in expectation values between states P1 and p2 is bounded by the trace distance. This fundamental relationship follows directly from Hölder's inequality (Watrous, 2018):
Tr [HPp - Tr p2] < |H] - P2l1i = (2),
where || Iloo represents the Schatten co-norm (spectral norm). The spectral norm constraint ||
from the eigenvalues of H being bounded in [11,
Definition B.2 (Quantum Fidelity). The quantum fidelity between quantum states P1 and Pa is defined as:
F(P1.P2) = Tr [VVVipe
The quantum fidelity satisfies F(p1,p2) E [0, 1], with equality to and only if P1 = P2-
The Fuchs-van de Graaf inequality (Watrous, 2018 establishes that the trace distance and fidelity between quantum states Pr and satisfy:
1 F(pr,p) T(PipP)) & VI- F(
(B.1)
13

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Predictive Performance of Deep Quantum Data Re-uploading Models
Definition B.3 (Quantum Affinity.. The quantum affinity between quantum states Pr and P2 defined
P2) = [F/Viv/2ll
The quantum affinity satisfies P2) E [0, 1]. with equality if and only if p1 = P2- Given that F(p1, = VV = Tr [V PT
quantum affinity and fidelity are related through:
F2),, = Th VPi Tr Ply P2),,
(B.2)
Definition B.4 (Quantum Pétz-Rényi-a Divergence). The quantum Petz-Rényi-a divergence between quantum states P1 and P22 is defined as:
1 loge (Tr (p](2~1))
Da(pille)) a
(B.3)
where (I, with the support condition supp(p1) supp(e2). The quantum Petz-Rényi-a divergence satisfies (p( [p22) > 0, with equality if and only if p1 /2.
Definition B,5 (Quantum Relative Entropy). The quantum relative entropy between quantum states Pi and P2 defined as:
S(Pllpe) = Tr(p1 (log2 P1 -loga P2)1,
where the support condition supp(p1) S supp(p2) must be satisfied. The quantum relative entropy satisfies S(pillp2) > with equality if and only if P1 = P2-
The quantum Petz-Rényi-a divergence has several important special cases that are worth highlighting.
For a 1, the quantum Petz-Rényi-a divergence reduces to the quantum relative entropy:
Di(Pill22) lim (a(pillp2) S(1llp2)).
a-1
For 2. it relates quantum affinity:
D3 (Pille))) = -2 logg (Tr (VPiv/2) = -2 A(P1,P2)
For a = 2, we obtain the divergence considered this work:
D2(pille2) loga (Tr [pipz']).
(B.4)
As (a(Pillp2) monotonically increasing on (0, +o0) (Mosonyi Hiai, 201 1), we have:
- -log2 A(pi, P2) D2 (pillP2) & D2(p1 llp2)-
(B.5)
Lemma B.6. For quantum states p1 and p2, their trace distance and Petz-Rényi-2 divergence satisfy:
T()) & V1 2D20e1lee)
Proof. Combining the trace distance-fidelity inequality in Eq. (B.1) with the fidelity- relation in Eq. (B.2), obtain:
T(p,, P2) < /1 F(11.22) & /1 A(p1.
The result follows from the affinity-divergence inequality in Eq. (B.5).
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Predictive Performance of Deep Quantum Data Re-uploading Models
C. Proof of Dependence on the Number of Encoding Layers
C.1. Pauli Basis Decomposition
Let Qi and P. denote the Pauli basis clements before and after applying unitary gate U, respectively. The transformation of an --qubit quantum state p under U can be expressed as:
UpU
E
U(a:Qi)tt
2N
QiE[1XXYY,2joN
Σ
BeP;
2N
PE(
Throughout the analysis, we adopt the Pauli basis ordering: (1, 2, X, Y for single-qubit systems and its N-fold tensor product {I, X, y for -uqubit systems.
Denoting a and ]3 the vectors of Pauli basis coefficients before and after the unitary transformation respectively, there exists Pauli basis coefficient transfer matrix H such that H.. The transfer matrix H be expressed collection of column vectors H hh
haw), ]. where each h, = Abij representing the transformation of the j-th original Pauli basis element. The transformed Pauli coefficients / are obtained through a linear combination of these column vectors: B E}=a,hh
haNr] is 4th -d-mmensional column vector
Each element huj of the column vector hj quantifies how the original Pauli basis element Qj contributes to the coefficient of the transformed basis element P. under the unitary operation U. This relationship is formally expressed as:
U(a,Q;)0t = Σ
0ryhigh Ps.
(C.1)
Next, we will first study the form of the transfer matrix for general quantum gates in Subsec. C.2. Then, in Subsec. C.3 and Subsec. C.4, we will analyze the properties of the transfer matrix for encoding gates under data expectation. Finally, we prove that the quantum divergence between the expected state of data re- re-uploading circuit and the maximally mixed state decays exponentially with the number of encoding layers L in Subsec. C.5.
C.2. General Properties of Quantum Gates in Pauli Basis
Firstly, we consider the general property of the Pauli basis coefficient transfer matrix H determined by arbitrary quantum gates.
Theorem C.1. For an N-qubit quantum gate U, the corresponding Puuli basis transfer matrix H exhibits the following
properties:
(1) The transfer matrix H orthogonal.
(2) The transfer matrix H can be written as a block matrix in the following form:
H
where the top-left element is 1, and matrix H. is also orthogonal.
Proof. (1) According to Eq. (C.1), we have:
N
UQ = Σ Pi
Taking the trace of the square of both sides and utilizing the properties of Pauli matrices Tr [P?] 2 and Tr [P,P]] =
15

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Predictive Performance of Deep Quantum Data Re-uploading Models
for * Pt, we obtain Tr [UQQ
2 for the left side, while the right side yields:
2
4
2 = TF
Σ
hisP: P.
=
Σ K Tr [P:)
Σ
hight Tr (PPP)
4N
2
Σ
hij = 22 12 h.
h-j
f=1
Therefore, h, h; 1. Next, considering the action of quantum gate U on another Pauli basis element Q3 of the original qubits:
N
UQLUI=
Σ
hm Pt.
1=1
Then.
4N
(UQ,U)) = Σ highan (P:)² +
Σ
hight P.Pt.
P.RE(LXXXY.2JO PPP
Taking the trace of both sides and using the properties of Pauli matrices Tr [P?] = 2 and Tr [P:P] for P. * P. we obtain:
4ad hig be TI [(P:)]]
0 [U((U,,)))
=
Σ
E
high TI (P.P]]
R.NEELXXYYJJM, PPR
2 N
highak = 22" h2 h
This shows h2 hs = h h2 = 0 for j * k. Thus,
- =1,
proving H is orthogonal.
(2) Since the first element of the Pauli basis /@N the vector h the first column of matrix H represents the influence of the original identity operator J&N on other Pauli basis elements after the action of gate U. From the unitary property UUt obtain:
h = [..0..... ,0,
Now consider the first row elements hıf of matrix H. For the j-th Pauli basis element Qj (where Q, + I0SN) of the original quantum state, after the action gate U:
4
UQ,Ut= 11
Σ
PigPt
Taking the trace of both sides and using the property Pauli matrices Tr [P.P]] = Tr [Q.] = 0, JON ,Q, 7 JON we obtain hı = 0.
Therefore, matrix H can be written as:
H =
H
and since H is orthogonal, H is also orthogonal
16

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Predictive Performance of Deep Quantum Data Re-uploading Models
For the first property, the orthogonality of transfer matrix H reflects an important physical principle: unitary operations preserve the purity of quantum states. When express a pure state p in the Pauli basis with coefficients a, its purity condition gives:
4 Tr
Trip-] =
Σ
a
[P:]
+
Σ
a,y Tr [PPP]]
4N
R.
1
Σ
2
a =1
2
2N
where we have used the Pauli basis properties that Tr[P,] = 2d and Tr[P.P,] = for P. * Pj. Under unitary evolution, the transformed coefficients 3 = Hox must maintain this unit norm, i.e., | = a Ha = ||ax 12. which is guaranteed by H being orthogonal.
The second property demonstrates that the transfer matrix preserves the fundamental quantum property of trace preservation. In the Pauli basis expansion, only the identity matrix 1&N contributes to the trace quantum state. Consequently, the coefficient associated with the identity term must remain invariant under unitary evolution. This fundamental constraint manifests in the block-diagonal structure of the transfer matrix H, where the identity component is decoupled from the other Pauli basis elements.
C.3. Properties of Encoding Gate in Pauli Basis
We now analyze the transfer matrix of the encoding gate R(x) defined in Eq. (I), where R(x) (ra Since single-qubit states can be decomposed in the Pauli basis {I, Y. }}. the transfer matrices of (x) and Ry (z) are determined by their action on this basis. We begin by establishing the fundamental commutation relations of Pauli matrices:
Let X, Y, Z be Pauli matrices, then: Lemma C.2.
[ 21Y, [Z, Y] = -2iX;
YY, Z 21X, [Y -21Z,
where [s.]] denotes the Lie bracket.
The following lemma establishes the fundamental transformation rule of Ry(x) gate acting on Pauli basis states. Lemma C.3. For single-qubit Pauli rotation gate Rr(œ) acting single-qubit Pauli basis Q, where P (X,Y,
Q E X,Y, we have:
1 - sin cos(z))
Q == or P Q # and P + Q
Rr(x)QRR(z)*
where H ii/ Q}, and [1y]] denotes the Lie bracket.
Proof. Since
I'
I 2
Rp(x) = co8
1 - isin
P.
2
we have
Rp(z)QR((2) coss (2) I i isin
P
Q
I + +isiin
41
P
2
2
coss ( sin? (2) PQP isin
Q+
2 cos
2
PQ icos
2
2
QP.
By the properties of Pauli matrices, when Q or P = Q, we have
PQP Q and
PQ = QP, therefore
Rp())QRp(z) = cos²
(5)
27
Q + sin?
I
Q = Q
/
When P * Q and Q + I, we have PQP = -Q, thus:
Rp(z)QRp(z){ = |cos² (5) sin 2 I { Q
(
isin
cos
I (5)
(PQ - QP)
2
= cos(x)Q sin(x) H,
17

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Predictive Performance of Deep Quantum Data Re-uploading Models
where H -12((Q-QQQ))
Based on the fundamental commutation relations of Pauli matrices in Lemma C.2 and the transformation rules for single- qubit Pauli rotation gates in Lemma C.3, we can derive the transformation rules for R2 and (z(z) gates acting on Pauli basis states.
By Lemma C.3, for Pauli basis Q or Z, we have R2 (z) = Q. When Q - X or Q = Y, let X you denote the Pauli basis of the original quantum state, and Xnew Ynew denote the Pauli basis after the quantum gate Rg (2) is applied. Combined with Lemma C.2, we have:
(x (2(x) = cos(2) (x) + sin(2) (2) new,
R₂ (x) (z)x) sin(x) + cos y
Therefore, the transfer matrix of quantum gate Rg (x) is:
0
0
0
0
0
(x) =
(C.2)
cos(2)
sin (a)) cos(Ir)
sin (x)
Similarly. by Lemma C.3, for Pauli basis Q = I or we have Rg (x) (x) Q. When Q= Z Q X, let Z X denote the Pauli basis of the original quantum state, and Znew X new denote the Pauli basis after the quantum gate R2 (x) applied. Combined with Lemma C.2, we have:
R2 Ry(()) cos(s)) (2))z)))e) Zeew + sin(()) new Xx) R (ar)) - sin(a)22me + cos((r) X new
Therefore, the transfer matrix of quantum gate Ry (x) is:
0 cos(x) sin (x)
0
(x(x))
(C.3)
0 sin x))
cos (I)
0
1
C.4. Properties of Expectation of Encoding Gate in Pauli Basis
The transfer matrix of R2 (z) and Rg (z) also satisfies the properties stated in Lemma C. However, under the expectation over data distribution, the transfer matrix of R. (x) and Ry (z) are not orthogonal and cannot preserve the purity of quantum states.
Now, we introduce the following lemma to calculate the expectation of cosine and sine functions over Gaussian distribution.
Lemma C.4. Let random variable x N(,,o then
B[cos(x)) = = cos(p); E[sin(x)) = - sin(n).
Based on Lemma C.4 and transfer matrices of R2 (x) and (x) in Eq. (C.2) and Eq. (C.3), we can get the expected transfer matrices R2 (x) and (x) under the data distribution.
Lemma C.5. Assume the data x Np, o)), then the expected transfer matrices of Rz(x)) and Ry under the data
18

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Predictive Performance of Deep Quantum Data Re-uploading Models
distribution are:
0
0
0
1
0
[[T,()))
0
cos(11)
- 4 sin(ls)
0
0
sin(n)
o cos(11)
1
0 - sin(p)
0 e
cos(pL
[T)
0
sin(w)
e $ cos(11)
0
Therefore, get the expected transfer matrix of R(x) under the data distribution.
Lemma C.6. (From the work (Li al., 2022)) Assume the data æ [x1, 22, Isl follows independent Gaussian distributions, ie., 2; ~ NN of), E [3]. The expected transfer matrix of encoding gate R(xx) under the data distribution is:
[[(()]]
tea
tzs
ty
tzy
tzy
tyy
where
t2 = A2 cos (H2).
taz A2 sin (12) Ag cos (1²))
Lzy A2sin (u2) Aa sin (ua)
taz A2 sin (W2) A
(M1),
taw Ag cos (u2) Ai cos (u1) A3 cos (ua) Ai sin (pL) Ag sin (wa),
try = A2 cos (12) A cos (H1) Ag sin (ML) + A1 sin (11) Ag cos (ps),
lyz = sinin (μ2) A sin (u1),
tye = A2 cos (12) Aj sin (H1) Ag cos (μa) - Ai cos (μτ) A3 sin (H3)
tyy Ag cos (W2) A sin (F1) Ag sin (Ha) A cos (p)) A3 cos (J1z)
where A, = e
Proof. Since
E [T))
T. (x))
T3 (xx)) (T,ss)
E B[,,
According to Lemma C.5, we have:
1
0
0 coe (pz) Ag sin(w2) 0
o
07
3((())
0
Ag cos(as)
Ag sin( Ag cas(ps)
Ag sin As co2(22)
0
con(ar)) An sin(es)
sin(m)) An cos(a1)
0 As sin(ss)
0
According to the Lemma C.I, the expected transfer matrix E [x)) in Lemma C.6 can be written as a block matrix where
T
Ezg
tzz
tys
T
tzx
tyz
(C.4)
Lzy
Lzy
Lyy
The properties of matrix mainly depend on the properties of matrix T, so we will focus on analyzing matrix next. Lemma C.7. (From the work (Li et al., 2022), Lemma S3) Given a Hermitian matrix H E C with all eigenvalues not exceeding X, for any matrix Q E Caxn with maximum singular value not exceeding s. the maximum eigenvalue of matrix Qt HH does not exceed $2) X
19

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Predictive Performance of Deep Quantum Data Re-uploading Models
Lemma C.8. Assume the encoded data a: [x1, #2, #3) follows independent Gaussian distributions, i.e., ai N (pp o
with of 2 o2, [3]. The expected transfer matrix of encoding gate R(x)) R2(x3))B) (22 R2 (2()) can be written as
1
B[T(x)]=
where the maximum eigenvalue of matrix TTT T does not exceed e e and matrix TTT is Hermitian.
T
Proof. According to the proof of Lemma C.6, matrix T can be decomposed as:
Ag cos(uce)
- Ag sin( costra))
(o)(
Ag sining) Ag cos(as)
Az sin(p?)
A con(a))
A, aintmr) cos(us)
sin(My)
sin(Me)
A2
cos(ir)) sin( 0
sin(ue) coo(ar))
(oo( sin(ss)
ein(ug) coc(
As
confos))
As sin(m1)
As
An sin(Me)
Al cos(p1)
1
cos(
sin(
Let Q
A1 cos(p1)
A1 sin(ur)
=
sin(
cos(
we have:
A1 sin(
Ai cos(12)
0
A2
TT=
(A2Ag) 2
RQ.
A
Since (TTT)- TT matrix TT is Hermitian. As the maximum singular values of matrices Q and R do not exceed 1,
A3
and the maximum eigenvalue of matrix
0
AA2
is max (axx.AA,43) by assumption, A2, A3 are both greater
A3
than e 3. According to Lemma C.7, the maximum eigenvalue of matrix TTT does not exceed -=
C.5. Effect of Encoding Layers
The following theorem establishes that the number of encoding layers L increases, the expected encoded state approaches the maximally mixed state exponentially.
Lemma C.9 (From the work (Li et al., 2022), Lemma S2) Given Hermitian matrix H C with all eigenvalues greater than X, and an n-dimensional vector x, we have
x Hx |
where || 1l2 denotes the 12-norm.
Theorem C.10 (Theorem 3.1 in the main text). Consider an N N-qubit data re-uploading circuit with L encoding layers and without repetition (P= = 1), which encodes data æ E L into the circuit, where each data point follows an independent Gaussian distribution, i.e., Di,a N (pi,n, T.m..)) and oi... 7 a², Let plæ, 0) denote the -qubit encoded state. Then the quantum divergence between the expected state [ppp] = En (p(x,, 0)] and the maximally mixed state 22 satisfies:
D2 (BTp)
< logz + (2r - -1)e -1ee-Le -L00
Proof. Let be the Pauli basis vector of the initial state po = (10) (N of the N-qubit data re-uploading eircuit. Since (0) (0) 1 (I + Z), we have
α 1 0
n=1
Let B3 be the Pauli basis vector of the expected encoded state E.p [p(x,, , 0)] with respect to data æ, i.e.,
Ejpxx 0)]
1
Σ
B.P.
2N
DE([IXXYY2255x
20

---

Predictive Performance of Deep Quantum Data Re-uploading Models
therefore,
(E
4"h
(Bip))²
P.
E
B,B,
4N
PcP.ee(.X.. 8x P.
Since Tr
[P]] = 2 and for different Pauli matrices P; + P.. Tr [PPP]] = 0, we have:
Tr [[]A))
8
2N 3 B.
2N
Let Hi be the transfer matrix of the 1-th layer parameterized gate, and E2y [T(1)) be the expected transfer matrix of the 1-th layer encoding gate with respect to data:
E
[T
E
[T(at,))
then we can obtain
β Hy [T(T)))) ... H2 E (T(M)) E [T(p))) mic.
1 a² By
Since the first element of a corresponds to the coefficient of identity matrix Ja which is 1, we have a
Lemma C.I, Hi, Buy [T(n)) E [L] can be written as block matrices:
['
H
E [T()))))]
Hi
21
Ti
where Hj orthogonal, the maximum eigenvalue of T, Ti does not exceed e -o² and T, Ti is Hermitian, therefore
B [1 (a:) 1 =1+ TT T, T Tr. Hi T
HTT H1T
a
By repeatedly using Lemma C.7, the maximum eigenvalue of matrix J.T... ... ...TT Tz ... T.T does not exceed e T.o? and a has 2 elements equal 1, so ||a
2 1. According to Lemma C.9, we have:
T (2 -o Lo²
Therefore,
1 logg (2 Tr [B(p]])
(2(E(pller) log,
Tr E[p]².
2N
loga ( loga 1 (2 1)e -)c-Loo))
Corollary C.11. Consider an N-qubit data re-uploading circuit with L encoding layers and without repetition (P 1). where the encoded data follows the independent Gaussian distribution defined in Theorem 3.1. Let (x, 0) be its output with respect to an observable H whose eigervalues lie in [-1, When L > 2 + 2) In + 2 In (n))), we have
h (xx 0)) li S
J is the maximally mixed state of N gubits. where h, = Tr and Pr 2N
21

---

Predictive Performance of Deep Quantum Data Re-uploading Models
Proof. Let [[pp B 0)] By Theorem C.10, when > + [NN and the maximally mixed state Pr satisfies:
+ 1n + +2 (n) the divergence between E(p]
In
(2 1)c²
(2(El|ller) & +1+ (2
10--->o Lod
< 11
2N+2
2N E² 4 4+
2
By the relationship between distance and divergence given in Lemma B.6, have
P(E(p).P)) <
4+ V4+6 &
Therefore, by Hölder's inequality (Watrous, 2018), we have:
(Tr [HE]p]] Tr [Hr]] < || prlli < (T(())),pp) E
where | [ Dc is the Schatten-ro: norm (spectral norm) of H.
D. Proof of Dependence on the Number of Repetitions
D.1. Periodicity of Encoding Gates
Consider the encoding gate defined in Eq. (1) which consists of rotation gates (2(x)) and (x).. For any Pauli operator Rp(x), P {Z,Y) Y acting quantum state can derive:
Rp(x)pRR(xx)
cos?
p+ + sin?
1 2
PpP isin
Pp icos
sin
2 pP
+ cos(rr)
1 cos(2) xx)
P 2
PpP-
iisn sin(x)pP.
From this expression, it evident that the mapping x + Rp(z)pRp(z)1 P {2,Y} exhibits periodicity with period 2n.
D.2. Approximating Circuit
For an approximate data ãa = Σ - 6,2 where bj E {0, 1}. we can construct a controlled rotation gate C - Rs (2- with auxiliary qubits (by) as controls and working qubits as targets. This controlled rotation gate approximates the rotation gate in working qubits as:
R₂
Σ
6,2-d
R2(xs- Eg) R,,(),,
Similarly, for the Ry gate, we can approximate it using controlled rotation gates C - (2-5) with auxiliary qubits (b,) as controls and working qubits as targets. This controlled rotation gate approximates the rotation gate in working qubits as:
9
Ry
Σ 3
6,2-
RyyxIs kq) (āa(
where cg = Fdd Ta-
In the original data re-uploading circuits shown in Fig. 2, the chunk data in the I-th encoding layer is 2[j [1,,, ... xI,] Here, 인n represents the (q + bit binary approximation of each data component in æmn- The quantum state corresponding to binary string कn in the approximating circuit ( (L and the initial state of approximating circuit is:
po (x) Pw.o &
m (zml
22

---

Predictive Performance of Deep Quantum Data Re-uploading Models
where Pw.0 the initial state working qubits, typically [0) (0). Since each encoding layer in the data re-uploading circuit uses quantum gates defined in Eq. (1) for data encoding, the data-independent controlled gates used for any #ig identical. Denoting the controlled gate as CR and the quantum gate applied in the approximating circuit during the p-th repetition as Vp we have:
V
bi)
II
Ui (0 CR.
1=1
where 0pp = p00p11 .. 0p.L]. After P repetitions, the quantum state of approximating circuit is:
òp(x, 0) = Vp (0(p) ... V1 (0(1) Ao (a) Vi (0m)) ........ .. (0(P))',
(Qp))'...1
and the corresponding quantum state on working qubits is:
( 0) = Tr Vp (0(p)) ... Vi (0(1) م (z: Vi (0m)) "..
... Vr " (01P))
where Tr [:] denotes the partial trace over the auxiliary qubits that contained the information of data a. Pu, (a, 0) equivalent the quantum state:
L Pr( 0) IIII Us (0p.) Ri (ā)1)
P L R R.
Ui (E))
(0
Po
p=1
where Po = =P = =10) (0). Therefore, for any observable H acting only on working qubits, we have:
Tr [Hăr(a, 0)] Tr (H 0)] Tr (H 0)]
(D.1)
D.3. Approximation Error Analysis
To investigate the approximation error between the approximating circuit in Fig. 3 and data re-uploading circuit in Fig. 2, we give the following definitions and lemmas:
Definition D.1 (Diamond Distance). For quantum channels NA-1 and MA-8, their diamond distance is defined as:
W MA-R-Mll =
[[ & (IR @ M M (PRA))l PRA PRAAD(
sup
where || 11 is the Schatten-1 norm.
Lemma D.2 (From the work (Watrous, 2018), Proposition 3.48). For any completely positive and trace ance-preserving maps A, C, where B and 'D map from n-qubit systems to m-qubit systems, and A and C map from m-qubit systems to k-qubit systems, the following inequality holds:
|B CDI|o < ||A Cllo -DIlo-
Lemma D.3 (From the work(Caro et al., 2022), Lemma B.5). Let (p) = Uptt, v(p) VpVt be unitary channels, then
2 |
Vll & |IU - Vl
where || I1.o is the Schatten-co norm (spectral norm).
Lemma D.4. Given any Pauli matrix P E {X, Y, Y. Z} and two parameters Φ1, &2 E [0, 2m,) construct rotation operators
R( = - R(g2) = - then the Schatten-co norm between these two rotation operators satisfies:
[((0)) R( < - 101 Q2l.
23

---

Predictive Performance of Deep Quantum Data Re-uploading Models
Proof. By the definition of rotation operators, have:
Φ1 -
)
2 1-
sin 11
(
@2 2 P.
]
R(bi)= - R(db) cos
0Os
sin
2
The singular values of this matrix are sin @ 22) } with multiplicity 2". Using the inequality [sin(>)] s |x| for all real x, we obtain:
Φ1
02
[[ R( 2 sin
< 21 -22--
2
This completes the proof.
The following theorem illustrates the relationship between the approximation error of the approximating circuit and the number of binary approximation bits used for the data components.
Theorem D.5. Consider an N N-qubit data re-uploading circuit with L encoding layers and P repetitions. Let hp(æ, 0) be its output with respect to an observable H whose eigenvalues lie in [-1,1]. There exists an approximating circuit as shown in Fig. 3 with output hp 0) with respect to H on working qubits. When the number of approximation qubits used for each data satisfies E [log2 (3PLN/8)], we have:
hr(a,, 0) - hp 0) &
Proof. Define the quantum channel of unitary gates as Up,(pp = (i((0p)))pi000p)) For an N-qubit eircuit with encoding layers L, define the channel of single data encoding rotation gate (p( Rp ((( where
P= P=Y,7. Z. The encoding channel for the n-th qubit at layer I is Ri, (x = R2 (a ( R= (1, (1, n, 1), and the encoding channel for layer is R1(2)1) -I R, (xI.n))
-T p= IT (x and the channel of the approximating
Let the channel of the data re-uploading circuit be E
circuit be E = - TT
L = Rp Then hr( 0) Tr (HTE())) hr( 0) Tr [TE(p))) therefore
h2(2 0) - hp( 0) [HH)) - IIr [HE(P))]
| || EE
(D.2)
I.
11 Up P P [(((())))) - (1
(D.3)
|| (II,.) Re, (01,
(D.4)
N 3
(L,,)))) - Rim,
(D.5)
N
3
2P
(21,))) - Rim.l
(D.6)
N
Mm, - 3lmmll
(D.7)
3/
(D.8)
where Eq. (D.2) follows from Hölder's inequality, Egs. (D.3), (D.4), (D.5) follow from Lemma D.2, Eq. (D.6) follows from
Lemma D.3, and Eq. (D.7) follows from Lemma D.4. Therefore, when lcyl < 3PI.N. i.e., q 3 [log2( (3PLN) )} we have:
hp(æ, 0) hp(z, 0) &
24

---

Predictive Performance of Deep Quantum Data Re-uploading Models
D.4. Effect of Repeated Data Uploading
Next, we will prove that repeated data uploading cannot mitigate the limitations in predictive performance imposed by encoding layers.
Theorem D.6 (Theorem 3.2 in the main text). Consider an N-qubit data re-uploading circuit with L encoding layers and P repetitions, which encodes data E R3NL into the circuit, where each data point follows independent Gaussian distribution, i.e., FL,, N (M am) and of o², Let (x, 0) be its output with respect to observable H whose eigenvalues lie in [-1,1]. When 3 ze IIN + 22)
In2 + 2 In( 1 }1, we have
Bl 0)1 - h1 E
where h1 Tr pr] and pı 22 is the N --qubit maximally mixed state.
Proof. Let p) denote the parameters used in the p-th repetition of the data re-uploading circuit, with (2 being the corresponding quantum gate in the approximating circuit used to approximate the quantum gate in the p-th repetition of the data re-uploading circuit.
We define 0 [1:p] as the complete set parameters across all P repetitions, and ( (1:r)) as the final quantum state of the approximating circuit after repetitions of data re-uploading. Let P1 (ax 011) be the quantum state of approximating circuit corresponding to data re-uploading without repetition (P 1). Their measurement results with respect to an observable H on working qubits are:
11 0) Tr [HV((())))
Tr [Hii(z, 011))
hp(x, 0) Tr [HV)(0(p)) ... V1 ( (() ... (p(0))))
Tr 01.
Therefore,
hr(x,0) 0) = Tr 1
=Tr [V2(22))))) (l HVp(0m (0) 01))
=Tr (0 (2:00)))))) 01)]
where
(0 = (2( (6)) (V)) (v(00)))
Since (0))) are all unitary matrices, H' (0:r))) remains an observable with eigenvalues in [-1,
Let the outputs of the original data re-uploading circuit without repetition and its approximating circuit with respect the new observable H' = H' (0)::)) be:
(1 0) Tr [H' P1 p1 1n)]]
Ri (x, 0) = Tr [H' pz (zx 0n)]]
The output of the new observable H' with respect to the N-qubit maximally mixed state = 2v is:
' Tr Tr [V)(0 " (V+(()))))))))))))))) [(00)))))
Tr [HV (r)) ... V2(0 ... v=(0
=
Tr [H h1.
Therefore, combining Theorem D.5, when > Toga 32N 1. [h (z. 0) - M (a. 0)1 < 6, and according to Corollary C.11,
when L > 2 [N + 2) In + 2 1m (1)], we have Eg [hi (z. 0) - hrl < 6, so
E Elha (z. 0) hy < B (z. 0)]
ETh (c. 8) + (z, 0) hul
z
< c 5,
25

---

Predictive Performance of Deep Quantum Data Re-uploading Models
and
E|K (x, 0) [[Tr 0)]} E [r HV [Vp((pp) (000)p)))...v11))))0))))))))) (=(0))(())(((00))) ....ii........... (p(0 = [lhp(æ,0)) 0)]
Therefore, [E [E,,,, 0)] hr| < e + 6, and according to Theorem D.5, [xp( 0) - hp(a, 0)1 < so
E E 0) h1 & c + 26.
Since ó can be arbitrarily small as + we have Bm[hp( 0)] hrl E.
E. Proof of Bounds on the Prediction Error
E.1. Classification Problems
For binary classification problems, consider a training set S consisting samples (æ, y) where x Dx and class labels {0, 1}. The hypothesis hy generated by the data re-uploading model trained on dataset S defined as:
h5( = Tr (()) (a, 0*) Po (2, 0-)"],
where 0 are the parameters chosen during training, for y(x) E {0, 1,, Ho 10) (0), H1 = |1> (4) hs(#) E [0, represents the probability that the model predicts feature x belongs to the correct class. The prediction error is defined as:
RC (hs)
IR [1 hs(+)]]. x=>xx
(E.1)
Proposition E.1 (Proposition 3.3 in the main text). Consider binary classification tasks where the features æ are drawn from distribution x and the class labels belong to y {0. 1}. Using the data re-uploading model with corresponding observables Ho = [0) (0) and Hi [1} (1), the hypothesis generated by the data re-uploading model trained on dataset S is given by hx) = Tr [yyte) (ax 0" Po V y.) Here, 0" represents the parameters chosen during training, and (()) denotes the label associated fiature a. The binary classification prediction error is defined in Eq. (E.1). When the expectation of hz())) over Dx satisfies |EaND [hs(x))
hrl & e, the prediction error of this hypothesis under distribution
Dx is bounded by:
R (bs)))
1 2 < c.
Proof. Since for Ho = 10) (0), H1 11) (1). we have h2 Tr [H(())P]] 1 therefore
(hs) 2
=
E B>Dxx
[1 - hs
2
E [/s (z))
2
h1
E
[ms(x)]
<
D>
E.2. Regression Problems
For multidimensional functions f(x) where æ ~ Dxx and function values are bounded in [-1,1], we consider the N-qubit data re-uploading model with the observable Hr. X v= Zn. The corresponding hypothesis function is:
hs(x)) Tr V 0* Vt 0²)]
26

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Predictive Performance of Deep Quantum Data Re-uploading Models
where 0* are parameters chosen during training. We consider the prediction error:
R (hs)
E [=) (=)(=))]-
(E.2)
a--Dx
This prediction error measures the difference between true labels fx) and model outputs hg(2:).
Proposition E.2 (Proposition 3.3 in the main text). Consider regression tasks where the input features a are drawn drawn.rrom distribution Dx and the function values are bounded in [-1,1] Using the data re-uploading model with the observable Hr - - Zn having eigenvalues in [-1,1], the hypothesis generated by the data re-uploading model trained on dataset where 0* are parameters chosen during training. The regression prediction
( Tr [L (xx, 0*) P) V (x, 0²)]. error is defined Eq. (F.2). When the expectation of his (x)) satisfies [Bee- [x)) ] h1l E 6, the prediction error of this hypothesis under distribution Dx satisfies:
Rh(hs)
>
E
[f(x)]
c.
m~Dx
Proof. Since for the observable H = hm Zn. we have h1 = Tr [HLPi] = 0, therefore
RC (hshs
E z~ix
[1s((r)) f(=))]
E [hs(x)]]
E
[f(x)]
Dx
x~xx
E [f(x)]
cc
Dx
F. Proof of Difference Between Average and Expected Performance In this appendix, we will explain the reason why we can use average performance as surrogate of the expected performance.
F.. Useful Lemma
Lemma F.1. For any N-qubit quantum state p. the following inequality holds:
< Tr [ 1
2
Proof. Firstly, According the definition of purity, Tr [p3] < 1 is obvious.
Then, let A1, A2N be the 2 cigenvalues (with multiplicity) of the N-qubit quantum state. All quantum states satisfy the constraint Σ >₂ = According to the AM-GM inequality:
Tr |0' = 2
2N VA1 -22N.
The equality holds and only A1 A , (2>), and Tr [p²] = 2 (1/2N)2 = 1122.
AzN, which corresponds to the N-qubit maximally mixed state where
As the definition of Petz-Rényi-2 divergence in Eq. (B.4), for any N -uqubi quantum state p. we have:
Dz(pller) log, (rr pp²PT log, (2 Tr (p²]) N.
Thus, for N -ubub quantum states, the Petz-Rényi-2 divergence between any quantum state and the maximally mixed state upper bounded by N.
27

---

Predictive Performance of Deep Quantum Data Re-uploading Models
F.2. Difference Between Average and Expected Petz-Rényi-2 Divergence
The following lemma tells that in order to make the Petz-Rényi-2 divergence between the average quantum state and maximally mixed state close to the Petz-Rényi-2 divergence between the expected quantum state and maximally mixed state, need to make the number of features M large enough.
Lemma F.2. Consider a dataset S {(x(") yym))) } where the features (m) are independently sampled from distribution Dx- Let p(z: 0) represent the pure quantum state encoded by data re re-uploading circuit with parameters O. where the feature x is encoded into the quantum state, and O are independent of the features in S. The average state is given by Pas = M I 0(x (), while the expected state is E[pp = RE-Dx 1plxx, 0)} For any E (0, 1). as M + cc, we have:
DD(PMMllrr) - D2 D2(Ellllpr) E
almost surely.
Proof. The Petz-Rényi-2 divergence between quantum state and the maximally mixed state P1 2 is:
D2 (pller) logz (2 Tr [p²)) N N logz (Tr [o²1).
(F.I)
Therefore,
D2(PMsllrr) D2(B(p)(lrr) = |log2 (Tr [pM]] loga (Tr [E(]]]])]
Tr [PM
(F.2)
log2
Tr [E[p]]]]
For notational simplicity, we omit the parameter in the quantum states, i.e., p(x)) p(xx, Let [(x(1))... ... ,x(x)) (a))) = Tr [p]]] = Me. Emn= Tr [p(x((m) p(a(()))] (((a(m)
For any n E [M], when replacing (n))
with (m) and its corresponding quantum state p' (((x(1)) p((x(1)))) have:
s(x(11, ... (n)
x f(x(1), ... (.00))) x (10))
M
Ir (( (m)
((2) p(
Tr (ole (' (x)
M
m==1
Tr (m)) {p(=(")) 0 (z((\))))
(F.3)
M
M
Σ
p((o(m))
/(a /² (((o("))
M
2
M
where -I- oo and || |l are the Schatten oc-norm and Schatten 1-norm, respectively, and Eq. (F.3) due the Hölder's inequality (Watrous, 2018).
Let X = (1) ,,)),,,),,.........).,,.)...
... ,0,,,,,,,,,))) (A))) by McDiarmid's inequality (Mohri, 2018), we have:
[ (m))) ((0)
-M12
IP X-DM
EE X
[f(
x (x)
(M))
St t < 2exp
2
Therefore,
Tr [+
Tr [phr]]
Ec
(F.4
X-D
holds with probability at least Me /2
28

---

Predictive Performance of Deep Quantum Data Re-uploading Models
We expand the trace of the squared average state follows:
M
Tr
PM
Tr
M
MyR=
1
Tr ((lo(tn) )2)
Σ Tr (llelm)))((al))) mgn
M
Σ (p(aaa
M
M
where have used the fact that each (mx( is pure state, implying p(x(m))2 p(x(m)) and Tr(p(x(m)) 1 for each m..
Given the independence of (m) and (()), we can derive the expectation of the trace of the squared average state as:
E
Tr
[M]]
1
Σ
Tr
E XWDD
ja
XwD AM
M
M
1
Σ
Tr
(m(
[o(Z(()))
M
M
Dx
1
Σ
Tr [p
M
M
ma
1
(M²
MM Tr [BP
M
M
1
1
Tr [|
M
M
This leads the following expression for the difference between the average state and expected state:
Tr
P
PM
}
Tr P
[ (1- M
Blp
X Dmm
=
M
Since Tr [[]]]]] is positive, According to Eq. (F.4,, with probability at least 1 - 2e -Me22212 we have:
Tr P
1
11 c
Tr p
M
Tr E|p
Then, applying Lemma F.I, with the same probability bound, we have:
Tr P
1 M
2N -1
+
1
< &
Tr E
Tr
Ep
M
We analyze the difference in Petz-Rényi-2 divergences in two cases based on the ratio Tr [pMu] / Tr [E[p]
Case 1: When the ratio is greater than equal to 1, Eq. (F.2) implies that with probability at least 1 2e -e 122
Tr [RM]
loga (1+c+
2N -11
D2(Pmller) D2 -D2((()le))) loga
(F.5)
Tr [[
M
2N M
Case 2: When the ratio is between 0 and and
O, we have with the same probability:
Tr PP
(1- 2
(D((Muller) D2(E(p]]ler) -logg
< - <
(F.6)
Tr [E]
M
29

---

Predictive Performance of Deep Quantum Data Re-uploading Models
Asymptotically, M- oo with fixed N, based Eq. (F.5) and Eq. (F66, the difference in Petz-Rényi-2 divergences converges as:
Dz(Pmller) - max + e), log(1 +)} E olE).
Note that when M + 0o, 1 1 2 > O is always satisfied.
E3. Difference Between Average and Expected Model Output
The following lemma shows that can approximate the output of an observable H on the expected quantum state by using the output of H on the average quantum state.
Lemma F.3. Consider dataset S = { m- } where features (m)) are independently sampled from distribu- tion Dx- Let p(æ, 0) represent the pure quantum state encoded by data re-uploading circuit with parameters 0, where the feature x is encoded into the quantum state, and are independent of the features in We define the average state as PM = M ΣM 2m=1 (m) 0) and the expected state as [Epp = Dx (p(ax, 0)). For any observable with eigenvalues
bounded 1-11] 1] and any E (0, 1), the following inequality holds:
Tr [HPM]] Tr [HE[[p]]
E
holds with probability at least 2e- 12
Proof. For notational simplicity, we write p(a) p(æ, 0), omitting the explicit dependence parameters 0. Let X (2) ,.... (d))) where each feature vector is independently sampled from Dx. Since the parameters O are independent of the features, the sequence 1 Tr|H p(z (m) } forms a set of independent and identically distributed random variables. Furthermore, as the eigenvalues of H lie [-1, 1], each term satisfies A < (r(Hp(x(m)) < M for all E MM.. Applying Hoeffding's inequality (Mohri, 2018), obtain:
M Tr
Σ
Hp
TT [Tpptt>)]
<C)
31 2e 2
X-DA
2~Dx
Therefore, by the definition of average encoded state and linearity of trace operator:
Tr(H(Px)] Trr(HE(]l]] Se
holds with probability at least - 2e Mr²
G. More Experiments
In experiments, we employ the data re-uploading circuit as shown in the Fig. Figure G.I, where the parameterized gates consist of single-qubit rotation gates defined Eq. (1) and two-qubit CNOT gates arranged in a ring connectivity pattern.
xP
10)
R(
R(Op....)
R(EL,1
R(0 p.L11)
10)
R(x122
p((2p.1))
R(XL2)
R(O p(..).)))
10)
R(ELN)
P(IN)
R(
R(O p...)
Figure G.I. Data re-uploading circuit used in experiments. The single qubit parameterized quantum gates are same with original data re-uploading circuit (Pérez-Salinas al., 2020), and the entangling gates CNOT gate with ring connectivity. Each parameter vector pp... is indexed by the repetition number p. encoding layer and qubit number n.
30

---

Predictive Performance of Deep Quantum Data Re-uploading Models
G.1. Regression Experiments
Data: For regression tasks, we consider the multivariate function defined in [-1, 1jp:
1 ( 1+ tanh Σ
D
f(x) -
2
(G.1)
where D is the dimension of the data æ and (ax E [0, 1]. For each dimension &i, draw samples uniformly from [-1, Experimental Setup: In experiments, setup as in Subsec. 4.2, to eliminate the impact of parameter count, we conduct comparative experiments using quantum circuits with fixed total layers Lmax = 10. For encoding layers < Lmax. we maintain the total circuit layers at Lmax by only encoding data the first L layers and setting the data input to zero vectors for the remaining layers (Using L encoding layers). The observable used experiments is H = Zi 8 Z2, where Z; is the Pauli-Z operator on the i-th qubit.
two-qubit data re-uploading circuits shown in Fig. G.I. Following the same
In experiments, the training contains 600 samples, and the test contains 10000 samples. Following the experimental setting in Subsec. 4.2, initialize the circuit parameters f from normal distribution, A((0, 1) and optimize using Adam with learning rate 0.005 and mini-batch size 200. The loss function is Mean Squared Error (MSE). Each experiment is trained for 1000 epochs and repeated 10 times with different random seeds. The results are shown in Fig. G.2.
The experimental results are similar to those in Subsec 4.2. As shown in Fig. G.2 (d), as the number of encoding layers L increases, the output with respect to observable Hr. progressively approaches the output of maximally mixed state, leading to corresponding increase test error as shown in Fig. G.2 (c). According to Theorem E.2, the prediction error will be greater than D [f [=)]] For the function fx) defined in Eq. (G.1), its expected value Ex>Dx [f()] = 0.5. Therefore, the prediction error will be greater than 0.5
Results:
(a)
(b).55
(c)
(d)
0.5
0.5
0.6
P-2
0.6
0.4
0.4
Difference
Difference
Brror
0.4
0.3
Error
0.4
0.3
0.2
0.2
0.2
0.2
0.1
0.1
0.0
0.0
0.0
0.0
2
6
9
2
2
3
4
2
3
6
Figure G.2. (a) Training error; (c) Test error; (b) Difference between the output with respect to H, Zι Z2 on training data and Tr [H))). (d) Difference between the output with respect Hy on test data and Tr pH) Error bars represent the minimum and maximum values across independent runs with different random seeds, with central line showing the mean value.
G.2. Impact of Dataset Size and Model Size
In traditional machine learning theory, prediction error is typically decomposed into two components: training error and generalization error (the gap between prediction error and training error). It is commonly believed that increasing model size can reduce training етrоr but may increase generalization eπror, while increasing the size of the training dataset can reduce generalization error. The goal is to find a balance that achieves both low training error and low generalization error, thereby obtaining a small overall prediction error.
difficult to determine the prediction error solely by determining the training error and providing an upper bound on the generalization error. However, when the prediction error determined, we can make the following inferences: under the same model complexity, increasing the training dataset size can reduce the generalization error, thereby increasing the training emor; on the other hand, with the same training dataset size, increasing the model complexity can reduce the training eиror, but simultaneously increases the generalization error.
Data: We employed the same data as described in Subsec 4.3, with a feature dimension D 24.
Experimental Setup: To experimentally validate our inferences, we used single-qubit data re-uploading circuit (Fig. G.1)

---

Predictive Performance of Deep Quantum Data Re-uploading Models
with encoding layer L 8 (corresponding to the data dimension of 24). We investigated the effects of training size and model complexity by systematically varying both the number of training samples and circuit repetitions. Other experimental settings followed those in Subsec. 4.2
a
(b)
(c)
(d)
Training
0.5
0.3
0.9
0.9
Test
Error
0.4
Error
0.4
Accuracy
Accuracy
0.7
0.7
0.3
0.3
0.2
0.2
0.5
0.5
600
1200
2000
5000
8
16
32
64
600
1200
2000
5000
16
32
64
N
N
P
Figure G.3. (a) Training and test error under same model complexity but different training set size; (b) Training and test error under training size but different model complexity. (c) Training and test accuracy corresponding (a); (d) Training and test accuracy corresponding to (b). Error bars represent the minimum and maximum values across 10 independent runs with different random seeds, with the central line showing the mean value.
Results: First, with fixed model complexity (repetition number P 8), varied the training size [600, 1200, 2000, 5000]. The training and test errors and accuracy are shown in Fig. G.3. We observe that the training set size increases, the generalization error (gap between training error and test error) decreases while the test error remains constant, resulting in increasing training eπror.
Then, with fixed training set size 5000, we increased model complexity by varying P as [8, 16, 32, 64]. The training and test errors and accuracy are shown in Fig. G.3(b) and (d). observe that the training error gradually decreases while the test eтrоr remains unchanged, and the generalization error increases.
G.3. Accuracy Results
(a)
(b)
(c)
(d)
P-1
0.4
0.4
0.9
Accuracy
Accuray
Beror
0.3
0.8
Error
0.3
0.8
16
0.2
0.7
0.2
0.7
0.1
0.6
P
P
0.0
0.5
P
P
0.0
0.5
2
3
5
6
1
L
Figure G.4. (a) Training error from Fig. 5.(a); (b) Corresponding training accuracy for panel (a); (c) Test error from Fig. 6.(c); (d) Corresponding test accuracy for panel (c). Error hars represent the minimum and maximum values across 10 independent runs with different random seeds, with the central line showing the mean value.
In classification problems, accuracy is another commonly used evaluation metric alongside error. We present the training and test accuracy corresponding to the error results shown in the main text. Figure G.4 shows accuracy corresponding to Figure while Figure G.5 shows the accuracy for Figure 6. When the test error is around 0.5, the corresponding accuracy also near 0.5, indicating random-guest level performance.
H. Counter Example
In this appendix, we present a counter example that does not exhibit the limitations of data re-uploading with deep encoding layers as discussed in this paper. We demonstrate that this is due to the fact that the dataset essentially contains only few informative dimensions.
32

---

Predictive Performance of Deep Quantum Data Re-uploading Models
(6) 0.5
(b) 1.0
Training
Tist.
0.4
Accuracy
Eroor
0.3
0.6
0.2
L
L
Figure G.5. (a) Training error from Fig. 6.(a): (b) Corresponding training accuracy for panel (a). Error bars represent the minimum and maximum values across independent runs with different random seeds, with central line showing the mean value.
Data: Consider a binary classification dataset where each sample (x("), (m)) consists of a D-dimensional feature vector (m) E RD drawn from a Gaussian distribution [u, Σ). The mean vectors μ are class-dependent: for class one, the mean of the d-th dimension is given by pa
2x
mod 8)] mod 2T,, while for class two, the mean the d-th dimension 16
[2 16 (8 + (d mod 8))] mod 2m.
Ha =
Both classes share the same covariance matrix Σ A + N, where a diagonal matrix with each diagonal element equal 0.8, and N is matrix with zero diagonal elements and all off-diagonal elements equal to 0.792(0.8 0.99).
Experimental Setup: We employed the experimental setup as in the Subsec. 4.2.
(a)
(b)
0.4
0.4
Aceurocy
Difference
Error
0.8
0.2
0.6
0.0
3
(d)
(0)
1.0
(T)
Accurooy
Difference
Error
0.8
0.6
3
5
3
5
2
3
4 L
6
L
L
Figure H.I. (a) Training error; (b) Corresponding training accuracy; (c) Difference between the output with respect to Ho 10) (0) on training data and Tr pr]: (d) error; (e) Corresponding test accuracy. Difference between the output with respect Ho on test data and Tr [H pr]. Error bars represent the minimum and maximum values across 10 independent runs with different random seeds, with the central line showing the mean value.
The experimental results presented in Fig. H.1, showing that as the number of encoding layers L increases, the test error does not approach 0.5. Both the test accuracy and training accuracy remain around 0.9, and the difference between the model's output on the test set and Tr [H pr] stays around 0.4, without exhibiting the limitations of data re-uploading with deep encoding layers discussed in this paper.
Results:
In fact, by diagonalizing the covariance matrix Σ QDQ and applying a linear transformation = QQ (X -+) to the multivariate Gaussian random variable, we obtain a transformed variable y that follows a distribution A/ D). Here, D
33

---

Predictive Performance of Deep Quantum Data Re-uploading Models
diagonal matrix where only one element has significant magnitude while all other elements negligible (0.008). To illustrate this, consider the case when D 24: the variance of the dominant dimension is 19.16, whereas the variances of the remaining 23 dimensions are all 0.008. This indicates that the multivariate Gaussian data is essentially determined by single dominant dimension (corresponding to the variance 19.16), while the other dimensions remain nearly constant.
The key insight is that while the data appears to be high-dimensional, it actually contains only one dominant dimension that carries meaningful information. This explains why the test error does not approach 0.5 even with increasing encoding layers, the effective dimensionality of the data is much lower than its apparent dimensionality.
34


Reviews:
[{"review_id": "uYCXF8ybvG", "reviewer": "Reviewer_1fvX", "summary": "The authors examine the predictive performance of data re-uploading models, a class of variational quantum circuits which has attracted significant attention in recent years. They prove that under certain theoretical assumptions about the data generating distribution, these models' have the property that as one increases the numebr of data encoding layers, the average output becomes indistinguishable from a maximally mixed state.  This property can be used to give lower bounds on the predictive accuracy of such models. The authors also validate their findings with experiments on both synthetic and real world datasets. They argue that their results imply that data reuploading models should be designed with large width rather than large depth.\n## \nUpdate after rebuttal: I would like to thank the authors for their responses. I have decided to maintain my score.", "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 you for the insightful questions and valuable feedback. Our responses to your questions are as below.\n\n## 1. The results suggest that any data re-uploading model (in the regime studied by the authors, and with gaussian inputs etc) will be roughly speaking \"mean zero\". Could one not add a classical bias term which will be trained alongside the data reuploading model to mitigate this? \n\nWe thank you for this constructive suggestion. We thought of this possibility, and found that adding a classical bias term would improve the model's overall predictive performance to some extent, but the data re-uploading model has no contribution to the improvement. In other words, when the data re-uploading model is in a \"mean zero\" state, this model becomes informationless, and the model's prediction is solely dependent on the information provided by the classical bias term for classification and regression.\n\n## 2. The authors study a regime where the dimensionality of the data grows linearly with L. Presumably the results would not hold if the dimensionality of the data were fixed? \n\nThis is an important question to be clarified. Our answer is yes. Actually, in Section 4.3 (Classification Experiments in Same Dataset) of the main paper, we conduct experiments using the same dataset with fixed data dimensionality ($D = 24$). Encoding data of this dimensionality into quantum circuits requires 8 encoding gates (since each encoding gate can encode three classical data elements). We adjust the number of encoding layers by varying the width of quantum circuits (number of qubits).\n\nWe investigate four different configurations: 8 qubits with 1 encoding layer, 4 qubits with 2 encoding layers, 2 qubits with 4 encoding layers, and 1 qubit with 8 encoding layers. Figure 6 shows that as the number of encoding layers increases, the model's predictive performance gradually degenerates to the level of random guessing (with prediction error approaching 0.5). Additionally, the model's average output on the test set converges towards a maximally mixed state, indicating that the model becomes informationless.\n\n## 3. In classical ML, one can also show that taking the number of layers of a NN to infinity, at least if one does this naively, results in pathological behavior-how does the phenomena described in this work compare to that?\n\nWe thank you for raising this interesting comparison with classical neural networks. It should be pointed out that the number of layers in classical neural networks and data re-uploading models generally refer to different concepts. In classical neural networks, the number of layers typically refers to parametric layers, while in data re-uploading models considered in this paper, the number of layers refers to encoding layers that are used to encode data into the quantum circuit. Classical neural networks input data all at once and continuously learn from it through parametric layers, whereas data re-uploading models partition the data and encode it through encoding layers, which allows encoding high-dimensional data using a limited number of qubits.\n\nA comparable quantum machine learning paradigm to classical machine learning is the encoding-variational paradigm, where data is first encoded into the quantum circuit all at once and then learned through parametric layers. Similar to classical machine learning, pathological behavior emerges as the number of parametric layers approaches infinity. In the quantum machine learning community, this pathological behavior is known as the Barren Plateau phenomenon, where the gradient of the loss function with respect to the model parameters vanishes as the number of parametric layers becomes too large, causing severe trainability issues, as discussed in reference [1]. The data re-uploading paradigm addresses this by placing parametric layers between encoding layers, potentially alleviating the trainability problem. However, the predictive performance of data re-uploading model remains unexplored, which is the central question we aim to answer in this paper.\n\n\n\n[1] Cerezo, M., Sone, A., Volkoff, T., Cincio, L. and Coles, P.J., 2021. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nature communications, 12(1), p.1791.", "claims_and_evidence": "The authors' claims regarding the limitations of data re-uploading models are thought provoking, and their presentaion is clear, but I am not fully convinced that they imply that deep data re-uploading models cannot be made to work in practice, perhaps with some tweaks analogous to those used in the classical ML literature for deep models (residual connections, batch normalization etc.).", "methods_and_evaluation_criteria": "The methods use seem appropriate for the problem at hand.", "theoretical_claims": "I did not check the correctness of the proofs.", "experimental_designs_or_analyses": "I did not check the soundness of the experimental results.", "relation_to_broader_scientific_literature": "The authros provide a solid contribution to the theoretical undertanding of data re-uploading models, a class of variational quantum circuits which has received widespread attention.", "essential_references_not_discussed": "I am not aware of any such references.", "other_strengths_and_weaknesses": "I believe the paper represents a solid contribution to the theory of data reuploading models- its results are certainly thought provoking, and should spark discussion about their implications for practice.", "other_comments_or_suggestions": "I have no other comments.", "questions_for_authors": "1.The results suggest that any data re-uploading model  (in the regime studied by the authors, and with gaussian inputs etc) will be roughly speaking \"mean zero\". Could one not add a clasical bias term which will be trained alongside the data reuploading model to mitigate this? \n2. The authors study a regime where the dimensionality of the data grows linearly with L. Presumably the results would not hold if the dimensionality of the data were fixed?\n3. In classical ML, one can also show that taking the number of layers of a NN to infinity, at least if one does this naively, results in pathological behavior- how does the phenomena decribed in this work compare to that?", "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": "fk1H4g7Hot", "reviewer": "Reviewer_QK5P", "summary": "The paper investigates the effectiveness of quantum machine learning models that use data re-uploading circuits.\nThese models have gained attention for their expressivity and trainability, but their ability to make accurate predictions on unseen data remains under-explored.\nThe study highlights a limitation in deep quantum data re-uploading models and provides guidance for designing better quantum machine learning architectures.\nIt suggests that increasing the depth does not necessarily improve model performance, especially when dealing with high-dimensional classical data.\nInstead, wider quantum circuits may be the key to more effective quantum learning models.", "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 you for the thorough and insightful comments, which helped us improve the clarity and depth of our manuscript. Below we elaborate on the differences between our work and (Li et al. 2022) in terms of overall framework and proof techniques.\n\n## Framework for analyzing prediction error\n\nRegarding the overall framework, we introduce a novel method to analyze the prediction error, bypassing the traditional decomposition into training and generalization errors. We directly analyze the expected output of the model over the data distribution. Our results demonstrate that when using data re-uploading models with deep encoding layers, the model's performance on unseen new data approaches random guessing, regardless of training quality, loss function choice, optimization method (gradient-based or gradient-free), iteration count, number of parameters, or training sample size.  In contrast, the results in (Li et al. 2022) are limited to showing that the gradient is small for quantum machine learning models when optimizing with cross entropy loss function.\n\nIndeed, directly analyzing prediction error, rather than decomposing it into training and generalization errors, is important. As shown in Appendix G.2, traditional machine learning theory suggests that increasing model complexity reduces training error while increasing generalization error, and increasing training sample size increases training error while reducing generalization error. However, since both components change dynamically, it is challenging to theoretically determine their sum (the prediction error). Our analysis method confirms that regardless of how training and generalization errors vary, the prediction error consistently approaches random guessing.\n\n\n## Proof techniques\n\nRegarding proof techniques, when analyzing the expected output of the model over the data distribution, our analysis focuses on two key aspects: the impact of the number of encoding layers and the number of repetitions. For analyzing the number of encoding layers, we employ techniques similar to (Li et al. 2022). However, Li's results (Theorem 2) only allow specific non-parameterized entangling gates (CNOT or CZ) between encoding layers. Our results (Theorem 3.1 in the main paper) allow arbitrary learnable parameterized quantum gates between encoding layers (Proof in Appendix C).\n\nFurthermore, Li's techniques cannot analyze scenarios involving repeated data uploading. Corollary 2.1 in Li's paper, directly derived from their Theorem 2, only examines the relationship between the model's expected output and maximally mixed states **without** repeated data uploading. In contrast, our result (Theorem 3.2 in the main paper) cannot be directly derived from Theorem 3.1; instead, it establishes the relationship between the model's expected output and maximally mixed states **with** repeated data uploading through a non-trivial extension. We address this limitation by constructing approximating circuits to analyze such cases (proof provided in Appendix D), a contribution absent in Li's paper. Importantly, repeated data uploading is crucial because it significantly enhances the trainability of these models[1].\n\n\n## Overall\n\nWe thank the reviewer for prompting us to clarify these important distinctions. In fact, the final result in (Li et al. 2022) (Proposition 4) highlights the trainability issue in quantum machine learning models lacking repeated data uploading and learnable parameters between encoding layers. The data re-uploading paradigm addresses these limitations by incorporating two key elements: learnable parameter gates between encoding layers and repeated data uploading[1,2,3]. The improved trainability of data re-uploading models is a key factor driving their widespread adoption in the field.\n\n\nOur theory presents a novel perspective: even though data re-uploading models with deep encoding layers may exhibit good trainability, their predictive performance approaches random guessing. This phenomenon has not been previously discovered, as earlier works [1,2,3] and (Li et al. 2022) primarily focused on model trainability.\n\nWe appreciate the opportunity to improve our manuscript and will revise the Contributions section in the main paper to clearly highlight our contributions and the key differences between our proof techniques and those in (Li et al. 2022).\n\n\n\n\n\n\n\n[1] Pérez-Salinas, A., Rad, M.Y., Barthe, A. and Dunjko, V., 2024. Universal approximation of continuous functions with minimal quantum circuits. arXiv preprint arXiv:2411.19152.\n\n[2] Pérez-Salinas A, Cervera-Lierta A, Gil-Fuster E, Latorre JI. Data re-uploading for a universal quantum classifier. Quantum. 2020 Feb 6;4:226.\n\n[3] Yu, Z., Chen, Q., Jiao, Y., Li, Y., Lu, X., Wang, X. and Yang, J.Z., 2023. Provable advantage of parameterized quantum circuit in function approximation (No. arXiv: 2310.07528).", "claims_and_evidence": "- The paper provides theoretical proof showing that as encoding layers increase, the encoded quantum states converge toward a maximally mixed state, leading to predictions that approach random guessing.\n\n- Theoretical analysis shows that increasing the number of repetitions cannot reduce the loss of distinguishability caused by deep encoding layers. Experimental results confirm that increasing repetitions does not improve the performance.\n\n- The study presents comparative experiments using circuits with different numbers of encoding layers but the same total parameter count. The experimental results show that circuits with fewer encoding layers generalize better.", "methods_and_evaluation_criteria": "- The paper provides mathematical analysis between the quantum states and a maximally mixed state, which is intuitive to show the performance of the models when the depth of the encoding is increased.\n\n- The paper uses both synthetic and real-world datasets to show the effect of encoding depth and theoretical limitations in practical scenarios.", "theoretical_claims": "The proofs of the theoretical claims are checked. The proofs are based on the assumption that the input features are independent Gaussian-distributed, which is acceptable.", "experimental_designs_or_analyses": "As mentioned in \"Methods And Evaluation Criteria\", the paper uses synthetic datasets to evaluate the theoretical claims.\nThe paper also uses real datasets to evaluate the limitations in practical scenarios.", "relation_to_broader_scientific_literature": "The contributions of the paper can help to understand the expressivity in data encoding designs.", "essential_references_not_discussed": "There are no additional related works that are essential to understanding the key contributions of the paper.", "other_strengths_and_weaknesses": "What I am concerned about is the originality and novelty of the theorems 3.1 and 3.2.\nThis is because these theorems are mentioned and proved in (Li et al. 2022) (see Eqn (5) in Theorem 2 and Eqn (7) in Corollary 2.1).\nAlthough the paper notes that the contributions are expanded from (Li et al. 2022), the novelty should be emphasized in the main paper.", "other_comments_or_suggestions": "No other comments or suggestions.", "questions_for_authors": "I think in the main paper, the authors should highlight the differences between the proof from (Li et al. 2022) and the authors' paper.\nThis could make the reader less confused since the paper claims that the limitation can be proved in a broader case.", "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": "3MRxzsYNt1", "reviewer": "Reviewer_aAWZ", "summary": "This paper provides further theoretical insights into the reuploading approach of parameterized quantum circuits. They authors claim the divergence (and predictive error) are worse with increasing numbers of layers. The has implications for near term devices, since the number of qubits is much less than the dimensionality of the data often. Additional empirical results are shown to further highlight these claims.", "questions": "", "limitations": "", "rating": 4, "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 greatly appreciate the reviewer's thorough and constructive feedback, which has helped us improve the clarity and quality of our manuscript. We address each point below.\n\n## figure 4 it looks like the pre-training is missing/overlapping and is hard to identify.\n\nWe thank you for pointing out this issue that was not clarified in the figure caption. Figure 4 is organized into left panels (a,c), showing pre-training (before training) divergence results marked by circles, and right panels (b,d), showing post-training (after training) divergence results marked by squares. We will update the caption of Figure 4 in the next version.\n\n\n## for Figure 5, it would be worth having the error bars to give an estimate of if the variance is important.\n\nWe appreciate this valuable suggestion. We have added error bars to the figure. The maximum, minimum, and mean prediction errors in 10 independent runs consistently demonstrate that as the number of encoding layers increases, the predictive performance approaches random guessing. Additionally, as the repetition number $P$ increases, the variance gradually decreases. We will update the figure with error bars in the next version.\n\n\n## one that would be worth adding is https://arxiv.org/abs/2501.16228.\n\nWe thank you for bringing this relevant reference to our attention. We will cite and add it to the related work section in the next version.\n\n## One point that could be clearer is the impact of the assumptions about the data. Specifically, in 3.1 data is drawn from a Gaussian for each point. Although this allows the Pauli decomposition, it would be beneficial to further discuss the implications/strength of this assumption.\n\nWe adopt the assumption of Gaussian distribution because of its prevalence and convenience for proof. In fact, our conclusions are not limited to Gaussian distributions. The role of the Gaussian distribution in our proof is mainly reflected in Lemma C.4 of Appendix C. As long as the distribution of data $x$ satisfies $\\mathbb{E}[\\cos(x)] = \\gamma \\cos (\\mu)$, where $|\\gamma| < 1$, our main point still holds. For example, when $x$ follows a uniform distribution over $[\\mu-a,\\mu+a]$, we have $\\mathbb{E}[\\cos(x)] = \\sin(a)/ a \\cdot \\cos(\\mu)$, where $|\\gamma| = |\\sin(a) / a| < 1$ when $a \\neq 0$.\n\n\nWe will elaborate on this point in more detail in the Discussion section of the next version.\n\n\n## Fig G1 gets reference but the link when clicked on goes to Figure 1 not Figure 1 of appendix G\n\nThank you for your careful attention to detail. We have fixed this issue.\n\n\n## in Figure 7, I think it would benefit from more color scaled lines showing different repetitions\n\nWe appreciate this constructive suggestion. Building upon our original experiments (data re-uploading model with repetition $P=2$), we have added more experiments with $P=4$ and $P=8$, and updated Figure 7 using color scaled lines. The results continue to show that MNIST has the highest test accuracy (fewer encoding layers, smaller data variance), followed by grayscale CIFAR10 (fewer encoding layers), while RGB CIFAR10 has the worst test accuracy, consistently around 0.5 (deeper encoding layers). Furthermore, as the number of repetitions increases, the training accuracy improves while the test accuracy decreases, indicating larger generalization error. This aligns with the findings reported in https://arxiv.org/abs/2501.16228. We will include this updated Figure 7 with color scaled lines in the next version of our manuscript.\n\n\n## Figure 6 would benefit from more datapoints (adding more points to the line).\n\nThank you for this thoughtful suggestion. For Figure 6, our experiments used fixed data dimensionality ($D = 24$) requiring 8 encoding gates (since each encoding gate can encode three classical data elements). We have already explored all possible configurations under the same parameter settings: 8 qubits with 1 encoding layer, 4 qubits with 2 encoding layers, 2 qubits with 4 encoding layers, and 1 qubit with 8 encoding layers. While adding more data points isn't feasible given these experimental constraints, we sincerely appreciate your careful review and valuable suggestion for improving our work.\n\n\n## In 4.3, what is the need to keep entanglement effects out? Does this meaningfully change the trend/statistics?\n\nWe thank you for this thoughtful question regarding our experimental methodology. We chose to exclude entanglement effects from our experimental design because prior studies [1,2] has shown that entanglement can significantly degrade both training and prediction performance. By eliminating this confounding factor, we were able to isolate and study specifically how the number of encoding layers impacts model behavior.\n\n\n[1] Ortiz Marrero, C., Kieferová, M. and Wiebe, N., 2021. Entanglement-induced barren plateaus. PRX quantum, 2(4), p.040316.\n\n[2] Leone, L., Oliviero, S.F., Cincio, L. and Cerezo, M., 2024. On the practical usefulness of the hardware efficient ansatz. Quantum, 8, p.1395.", "claims_and_evidence": "In general, the claims are supported by evidence. The core claims of this paper are theoretical, and the experimental aspects demonstrate the expect effects (but are not necessary to prove the theory is true).", "methods_and_evaluation_criteria": "The empirical evaluation benchmarks are fine. The artificial datasets highlight the points made by the theory and the later CIFAR/MNIST highlight more realistic datasets.", "theoretical_claims": "I did not check the correctness of proofs that were not in the main body.", "experimental_designs_or_analyses": "Yes. The experiments are generally valid. However, figure 4 it looks like the pre-training is missing/overlapping and is hard to identify. Also for Figure 5, it would be worth having the error bars (like in later figures) to give an estimate of if the variance is important.", "relation_to_broader_scientific_literature": "The results connect to the broader scientific body of literature in several ways. First, it builds upon the generally large body of literature on trainability of quantum circuits. This paper highlights the difficulties in training deep reuploaded circuits, as many other papers have shown the difficulty of training circuits like HEAs, or shown the difficulties in training circuits with circuit cost functions (e.g. local vs global). Second, it builds specifically upon the body of work analyzing the reuploading circuit. The reuploading approach is quite common in QML works because it is useful to encode data larger than the number of qubits (which is common in hardware, and all but ubiquitous in simulation), and has built quite a substantial influence in the QML community. Previous works have indicated difficulties in training/expressivity of reuploading circuits (as pointed out in the paper) and this expands in that direction.", "essential_references_not_discussed": "In general, the references are sufficient, one that would be worth adding is https://arxiv.org/abs/2501.16228 in which the effect of e.g. number of reuploading layers is considered in terms of generalization capability of the model.", "other_strengths_and_weaknesses": "The paper is overall clear and to the point. The theory is clearly stated, and the results highlight expected features of the theory. One point that could be clearer is the impact of the assumptions about the data. Specifically, in 3.1 data is drawn from a Gaussian for each point. Although this allows the Pauli decomposition, it would be beneficial to further discuss the implications/strength of this assumption.", "other_comments_or_suggestions": "A few minor typographic things could be improved (e.g. Fig G1 gets reference but the link when clicked on goes to Figure 1 not Figure 1 of appendix G). Additionally in Figure 7, I think it would benefit from more color scaled lines showing different repetitions as was done in figure 5. Figure 6 would benefit from more datapoints (adding more points to the line).", "questions_for_authors": "1. In 4.3, what is the need to keep entanglement effects out? Does this meaningfully change the trend/statistics?", "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 submission investigates the effectiveness of quantum machine learning models with data re-uploading, offering insights into which quantum circuit architectures may be more favorable for quantum learning tasks. All reviewers agree that this is an important and solid contribution. The authors are encouraged to consider the reviewers' comments on presentation in their revision.
Decision: Accept (poster)