AiDE-Q: Synthetic Labeled Datasets Can Enhance Learning Models for Quantum Property Estimation
Authors: Xinbiao Wang
Yuxuan Du
Zihan Lou
Yang Qian
Kaining Zhang
Yong Luo
Bo Du
Dacheng Tao
Venue: NeurIPS 2025 poster ()
Keywords: Quantum Computing
Quantum Property Estimation
Data Engine
Quantum Many-body Problems
Deep Learning
Primary area: machine_learning_for_sciences
TL;DR: We propose an automatica data engine framework for quantum property estimation under limited quantum resources.

Abstract: Quantum many-body problems are central to various scientific disciplines, yet their ground-state properties are intrinsically challenging to estimate. Recent advances in deep learning (DL) offer potential solutions in this field, complementing prior purely classical and quantum approaches. However, existing DL-based models typically assume access to a large-scale and noiseless labeled dataset collected by infinite sampling. This idealization raises fundamental concerns about their practical utility, especially given the limited availability of quantum hardware in the near term. To unleash the power of these DL-based models, we propose AiDE-Q (\underline{a}utomat\underline{i}c \underline{d}ata \underline{e}ngine for \underline{q}uantum property estimation), an effective framework that addresses this challenge by iteratively generating high-quality synthetic labeled datasets. Specifically, AiDE-Q utilizes a confidence-check method to assess the quality of synthetic labels and continuously improves the employed DL models with the identified high-quality synthetic dataset. To verify the effectiveness of AiDE-Q, we conduct extensive numerical simulations on a diverse set of quantum many-body and molecular systems, with up to 50 qubits. The results show that AiDE-Q enhances prediction performance for various reference learning models, with improvements of up to $14.2\\%$. Moreover, we exhibit that a basic supervised learning model integrated with AiDE-Q outperforms advanced reference models, highlighting the importance of a synthetic dataset. Our work paves the way for more efficient and practical applications of DL for quantum property estimation.

AiDE-Q: Synthetic Labeled Datasets Can Enhance Learning Models for Quantum Property Estimation
Xinbiao Wang 1. Yuxuan 1,2,, Zihan Lou?, Yang Qian', Kaining Zhang',
Yong Luo Bo Du Dacheng Tao**
College of Computing and Data Science,
Nanyang Technological University, Singapore, Singapore School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore National Engineering Research Center for Multimedia Software, School of Computer Science, Wuhan University, Wuhan, China 'Department Data Science, City University of Hong Kong, Hong Kong, Hong Kong yuxuan. duentu. edu. sg, 1100mm00m com, dacheng. taoentu, edu sg
Abstract
Quantum many-body problems are central to various scientific disciplines, yet their ground-state properties are intrinsically challenging estimate. Recent advances in deep learning (DL) offer potential solutions in this field, complementing prior purely classical and quantum approaches. However, existing DL-based models typically assume access to a large-scale and noiseless labeled dataset collected by infinite sampling. This idealization raises fundamental concerns about their practical utility, especially given the limited availability of quantum hardware in the near term. To unleash the power of these Dl.-based models, we propose AiDE-Q (automatic data engine for quantum property estimation), an effective framework that addresses this challenge by iteratively generating high-quality synthetic labeled datasets. Specifically, AiDE-Q utilizes a consistency-chede method to assess the quality synthetic labels and continuously improves the employed DL models with the identified high-quality synthetic dataset. To verify the effectiveness of AiDE-Q, we conduct extensive numerical simulations on diverse set quantum many-body and molecular systems, with up to 50 qubits. The results show that AiDE-Q enhances prediction performance for various reference learning models, with improvements of up to 14.2%. Moreover, we exhibit that basic supervised learning model integrated with AiDE-Q outperforms advanced reference models, highlighting the importance of synthetic dataset. Our work paves the way for more efficient and practical applications of DL for quantum property estimation.
Introduction
Many fundamental problems across diverse scientific disciplines, from condensed matter physics to quantum chemistry and materials science, can be reduced solving quantum many-body problems [1-3]. A central challenge in this regime is characterizing ground-state properties, task known as quantum property estimation (QPE), which offers critical insight into the behavior of complex quantum systems [4-6]. Numerous methods have been proposed towards QPE, ranging from classical simulations [7-11] 1] to quantum algorithms for state learning [12-17], but their applicability remains limited. That is, classical simulations face exponential computational costs as the system size increases [18, 19], while quantum algorithms often require extensive measurements and complex
"Corresponding authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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operations to evaluate intricate properties such as entanglement entropy [20, 211. These difficulties are further compounded by the scarcity of quantum resources in the early stages of quantum computing.
Learning-baae baseed approaches, especially deep learning (DL) algorithms, have recently emerged promising solutions for QPE [22-39], particularly for systems with shared features. These methods involve training a deep neu- ral network on a large dataset of measurement data ob- tained from quantum many-body states with varying pa- rameters, then leveraging the trained model to predict the properties of previously unseen quantum states. Over the past years, huge efforts have been devoted to exploring different learning paradigms and neural architectures to achieve accurate property estimation while using fewer measurements. In this endeavor, existing DL models can be broadly categorized into three paradigms, which are supervised learning [40-50], semi-supervised learning [51], and self-supervised learning and fine-tuning [52, 53]. Despite their promise, all of these approaches often as- sume access to high-quality training datasets with exact labels, overlooking the exponentially increasing overhead of dataset construction the system size grows. This ignorance poses two severe issues: (i) The reported empirical results fail to reflect the true performance of these methods in real-world scenarios. (ii) How train these DL models with limited quantum resources remains unknown.
0.8
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shots: 2
shots: 2
0.0
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Ratio of Noisy Data
Figure 1: Coefficient of determination R² of DL. models for predicting entanglement entropy of 50-qubit Heisenberg models. The prediction performance decreases the ratio noisy labels in training datasets increases or the number of measurements decreases.
Compared to training on ideal datasets with exact labels, learning under limited quantum resources poses substantially greater challenges for DL models. Specifically, in this scenario, we have three choices for the dataset construction: (i) a dataset consisting of small samples with approximately accurate labels; (ii) dataset consisting of large number of samples with noisy labels; (iii) hybrid dataset consisting of a few samples with approximately accurate labels and numerous samples with noisy labels. Notably, the first two choices denied by the well-established principle in DL theory, where small-sized datasets lead to overfitting and poor generalization, and a dataset with noisy labels often results in incorrect learning [54-57]. As viable alternative, hybrid datasets offer a principled compromise, enabling a balance between label accuracy and dataset scale under realistic resource constraints. However, direct training the hybrid dataset could significantly degrade the predictive performance of DL models, as empirical evidence is given in Fig. 1. In this regard, a critical question is: How to maximally exploit the hybrid dataset to improve the performance of DL models in QPE?
To address this question, here we propose AiDE-Q (Automatic Data Engine for QPE), a simple but effective framework inspired by traditional data engines [58-64] that iteratively enhances DL models by acquiring data with high-quality synthetic labels for training. A notable feature of AiDE-Q ADDE-QIA is its compatibilitty most DL models for QPE. Without increasing any measurement overhead, AiDE-Q demonstrably enhances the performance of the underlying DL model. On the algorithmic level, our key technical contribution is introducing consistency-check method of AiDE-Q to assess the quality of the synthetic labels and select the high-quality data to further train the employed DL model. Unlike traditional data engines [58, 59, 63], which typically rely on extensive manual intervention for labeling and selecting high-quality data, AiDE-Q automates this process by employing a DL model for data labeling and consistency-check module for data selection. As an additional contribution, we systematically conduct extensive numerical experiments for integrating AiDE-Q into various DL paradigms on predicting the entanglement entropy and correlation of the Heisenberg XXZ model and cluster Ising model with up to 50 qubits. The results show that AiDE-Q could effectively improve the prediction performance of various reference DL models, with the highest improvement reaching 14.2%. A notable phenomenon is that a vanilla supervised learning model integrated with AiDE-Q Q outerforrs the state-of ofte--ofthheart reference learning models. This finding underscores the critical role of pioneering synthetic data in advancing the applicability of learning based methods to QPE tasks. We release the source code and dataset at Github to facilitate future research in this domain.
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2 Preliminaries
In this section, we outline the essential background on quantum computation, classical shadows, and quantum property estimation (QPE). Additional technical details are provided in Appendix A.
Basics of quantum computing. The elementary unit of quantum computation is the qubir (or quantum bit) [65], which is the quantum mechanical analog of classical bit. A qubit is a two-level quantum-mechanica system described by a unit vector in the Hilbert space C2. In Dirac notation,
qubit state is defined (6) = co (0) e1 [1) E C² where [0) [1, 11 and [1) [0, 1j specify two unit bases and the coefficients co, C1 E C yield lcol + |a = 1. Similarly, the quantum state unit vector in C²" i..., (&b) = Σ}= 9j le,), where ley) R² is the |egl² 1 with Cj E C.
of n qubits is defined
computational basis whose j-th entry is 1 and other entries are O, and Besides Dirac notation, the density matrix can be used to describe more general qubit states. For example, the density matrix of the state (1)) is p = (v)) (1b) C² x2". where (vbl 116) refers to the complex conjugate transpose of (w)) For a set of qubit states (pg, (y))
with Pj > 0, Eラ Pj = and (bb)) C² for j E [m]] its density matrix is p 23 PyPj with Pj (wy) (Wby and TT(p) 1.
The quantum measurement refers to the procedure of extracting classical information from the quantum state. It is mathematically specified by Hermitian matrix H called the observable. Applying the observable H to the quantum state (&)) yields a random variable whose expectation value {1) H (&) or Tr(Hp) for the density matrix p = Ib/) (vbl. For instance, applying computational basis measurement (0) (0) on a quantum state p for m times yields bit string b E {0, 13"". and the
estimated expectation values is given by b Σ b./m with B(b) Tr(p 0) (0l).
Classical shadow. The classical shadow [13] of a quantum state p is constructed using a set of randomized measurements. Given unitary operator U sampled from a unitary ensemble U. and the subsequent measurement outcome b under computational mcasurement, the classical shadow from one snapshot is represented by أ (Ut [b) (b) U), where N' is linear map associated with the measurement process. In this manuscript, we focus on the random Pauli measurement such that the classical shadow from m snapshots refers to P 1 m ΣØ=
(3
(1)
[b/0) (6)) ( 12),
(6)
where random unitaries ( are sampled from the Pauli ensembles U {12, X, Y, Z) @m " with /2 = (2=())) and = (9)). Y ).). Z = (b 1)) being Pauli-X. X. -Y, -Z operators.
Quantum properties estimation (QPE). We consider the QPE task for ground states of quantum many-body systems with physical parameters p. described by a Hermitian H (p). The ground state
((())) is defined as the eigenvector relating to the minimum eigenvalues of H (p). The task of QPE involves predicting properties (p(p)) with p(p) = (p)) (x)()) using the measurement outcomes obtained by applying measurement operators M to the quantum state p(p).
Two important QPE tasks are estimating the Renyi entanglement entropy and two-point correlations in quantum many-body systems, which are respectively defined as
Sp(P) = logg Tr(pa), and (p) Tr (pofog)
where a E {2, a) and of (af²) refers to the Pauli-Z (Pauli-X) operator acting on the i-th qubit. Here, A refers to the index set of the subsystem and PA is the reduced density matrix of p on the subsystem A. For two-point correlations, the two indices i, satisfy < + j S n. These properties are widely studied as standard tasks in many studies of quantum states learning [13, 48, 52], and are essential for understanding critical behavior like phase transitions in quantum many-body systems [66, 67].
Related work. We highlight that there are no comparative studics with our work, as AiDE-Q is compatible with various Da-based models for QPE. Refer to Appendix B for claborations.
3 Problem setup and implementation of AiDE-Q
For clarity, we recap the mechanism of DL models for QPE training on an ideal dataset in Sec. 3.1. Then, we formulate the learning-based QPE tasks with a limited measurement budget and present the implementation of AiDE-Q and exhibit its compatibility with various DL models in Sec. 3.2.
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3.1 Learning-based QPE models with an ideal dataset
Existing DL-based models for QPE can be broadly categorized into three learning paradigms: supervised learning (SL), semi-supervised learning (SSL), and self-supervised learning with fine- tuning (SSL-FT). Despite differences in label usage, all three follow common two-stage pipeline: (i) dataset construction and (ii) model implementation and optimization. In what follows, we first describe the data collection process specific to QPE, and then summarize the learning procedures in each paradigm, emphasizing their differences at each stage. We defer the details to Appendix C.
Data collection for QPE. We define the classical data representation by reviewing the process of obtaining classical data through the classical shadow method, as introduced in Sec. 2. Consider an --qubit quantum many- many-body state p(p) parameterized by d-dimensional physical parameter p R.d. the classical shadow approach involves performing random Pauli measurements, denoted by M = (M1,' , Mm) E 2 on the quantum state p(p) with U4 being the Pauli ensemble. The measurement outcomes are denoted by {0, 1INm where oij refers to the outcome of the Pauli operator M, on the j-th qubit. As such, we define the classical data description of the quantum state p(p) x<p) (p. M, o). Additionally, we denote the exact quantum properties of various subsystems by a label vector y. where each entry yi corresponds to a quantum property for a specific subsystem. For example, the label y; for entanglement entropy of subsystem A, in Eq. (1) given by SA, (p(p)). Formally, classical data point with the ideal label for explored properties for p(p) is
ve(p).y()) = (p)) o(p), y())).
(2)
Through sampling different physical parameters p from a specified distribution, the training dataset for DL models is constructed. Hereafter, for simplicity, omit the dependence of æ. M, o, and y on the parameters p when no ambiguity arises, unless otherwise specified.
Remark. The data collection from quantum systems can utilize other measurement operators M beyond random Pauli measurements, such as information-complet measurement operators [68, 69], to collect measurement outputs Additionally, not all elements in the triplet (p, M, o) are necessary for model training, depending on the specific DL model being used. Refer to Appendix for details.
SL paradigm. A large number of studies utilize the SL paradigm to address different QPE tasks, exploring various neural architectures such as multi-layer perceptrons [45, 48] and convolutional neural networks [29, 43, 47, 70J, along with different loss functions like mean squared error (MSE) and cross-entropy. In this paradigm, the dataset SSL. = {xx(),, y })]= is constructed to train the learning model fsL. under a loss function LSL. which for MSE defined as,
Lst. Σ
(x) (0))
where x: (0) = (p((), (0)) o0) are collected from n different quantum states p(p(0) with varying parameters p sampled from a specific distribution D. The exact labels ( are assumed to be available for all data in Sst. Finally, the trained model fst. is used predict the quantum properties y' based on the collected data x' collected from unseen quantum states p(p').
SSL paradigm. The SSL paradigm involves training learning model fSSL with dataset consisting
of both labeled and unlabeled data, i.e., SSsL = ((x=(), ,y {)) U (()()) 1i-my+11 where is the number of labeled data points. The only work in this paradigm is Ref. [51], which exploits
maccer-stuuentt architecture to train the learning model fsst.- Specifically, this architecture comprises
a student model fSSI. and a teacher model J'sL,. both sharing identical structural designs. The student
model learns through both a supervised loss and a consistency loss
LSSL- Σ
(as)( - 2
Σ (=
(x)) - SESL (ar (/)
2=1
where À refers to the consistency weight. The parameters of the teacher model are maintained as an exponential moving average of the student model's parameters throughout training [71]. Once trained, the student model fssI. is employed perform the prediction.
SSL-FT paradigm. The dataset Ssp considered in SSL-FT is similar to SSsI. in SSL., which consists of both labeled and unlabeled data. However, unlike SSL, SSL-FT first performs self-supervised

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(a) Labeling Collection Abya (t)
Learning Model Far
Consistency- Chessstenyy-Chekk
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(b) Model Training
(y
Model Evaluation
Learning Model (3
Update Learning Model forc
Syat
Prodiction
p(pll)
p(p)
M
*
O
Figure 2: Framewrok of the AiDE-Q. AiDE-Q follows an iterative pipeline consisting of three primary stages: (a) data labeling and collection: this stage first use the trained model foçe) the f-iteration to generate labels for the data in Sayd (,, and then using the consistency-check to collect the data (p. M, o) and its synthetic label ÿ with small variance among the s generated labels yı, of the masked data (Mz, Oz, ). as defined in Eq. (6); (b) model training: this stage further tine-tunes the DL model with the updated dataset Sx (t + 1) and obtain a new DL model fe(:4); (c) model evaluation: the updated DL model fec++1) is evaluated on a validation dataset Sval to examine whether the prediction performance is improved compared to fe(()-
learning on the unlabeled dataset to obtain a pre-trained model fsr that can extract generic and meaningful hidden features, and then fine-tunes fsr using the labeled data for specific QPE tasks [52, 53]. The loss function for model fine-tuning follows the SL paradigm, as defined in Eq. (3).
Despite the different learning paradigms and implementation details, almost all existing DL models utilize ideal labels {y during training, which are often unavailable due to the exponentially increasing computational overhead to acquire limitation makes it unclear whether DL algorithms can effectively tackle QPE tasks in practice, where the number of measurements is restricted and the accessible dataset is hybrid.
the size of the quantum system grows. This
3.2 Implementation of AiDE-Q towards the hybrid dataset
Before we present the implementation details of AiDE-Q, let us first formulate the objective of DL models given hybrid dataset, where the total number of measurements for data collection is limited.
DL-based models for QPE with hybrid dataset. To account for practical measurement overhead, the label of p(p) can only be derived from collected measurement data o(p) in Eq. (2). As a result, the label may be noisy, with the noise level determined by the number of measurements m. Let the noisy label of p(p) be ÿ. Without loss of generality, the hybrid dataset takes the form as
Shyd = Sy U with St (z(1)) E IR JRNNm Ji=1 and Sy (+) (+))) E p. Jimnpe (5)
where the training dataset Snyd consists of S₁ and Sy, in which the number of examples are respectively denoted as n, and n,, The number of measurements per example in St. and S is denoted by my and m. with mn > my. Specifically, each example in S,, contains large number of measurements, yielding small difference with the accurate label y. In contrast, Stj contains more data points nu, though at the cost of increased label noise between ý and y.
Overview of AiDE-Q. Our initial empirical findings, shown in Fig. 1, suggest that naively training existing DL models on the hybrid dataset Shyd can result in suboptimal predictive performance. This underscores the need for novel methods that allow DL-based models for QPE to efficiently learn from hybrid datasets and deliver accurate predictions.
To address this challenge, we propose the automatic data engine for QPE (AiDE-Q), an effective and scalable framework that automatically identifies and generates high-quality data labels through iterative interactions between various learning models and the hybrid dataset, thereby continually enhancing model performance. An overview of AiDE-Q shown in Fig. 2. Given a learning model fe and hybrid dataset Shyd in Eq. (5), AiDE-Q involves iteratively updating subset of
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the training dataset Sh. C. Shyd with high-quality labels and training fe on this updated dataset Sn. For clarity, denote (t) as the high-quality dataset after t updates, and fo(s) as the learning model trained on Sr (t). Initially, we set Sn (0) = Sz: the employed learning model is flexible, i.c. Foco) E [fsL. fSsL. fsF J. which is optimized under the corresponding loss as defined in Eq. (3) and Eq. (4). Once fero) and (0) are prepared, AiDE-Q follows an iterative pipeline consisting of three primary stages: data labeling, model training, and model evaluation. In what follows, we detail the procedure of AiDE-Q at the 1-th update with 0 < I < T and T being the total number of updates.
Stage I: data labeling. This stage aims to generate and identify more data with high-quality labels from Sy using the learning model fe. and collect the newly identified data {(x ý = fo(x)} into the updated high-quality dataset S (t 1). More specifically, we define the data quality of a data point (x. y) as the closeness between the synthetic label ÿ generated by the learning model fo and the ideal label y. However, y is generally unavailable due the prohibitive acquisition cost. In this regard, to evaluate the quality of data with a finite number of measurements, we propose a consistency-chet check method that examines the confidence of ÿ an alternative.
The core idea behind the consistency-check method to evaluate the variance of synthetic labels generated by so using partial measurement outputs, randomly selected from (M, o). As shown in Fig. 2(a), for a given data point (x, fo())) with x = (p, M, o) and M, O E Nm we randomly sample subset {æI, == (p, MI, oI, 0I,)] )= from the input data T. where Is C [m] refers to the index set uniformly sampled from the set of all column indices of the measurement outputs For each subset, we generate the synthetic labels with Dz. fo(z)), We call (a:1, yr,, ) the masked data corresponding to (x, fo())). Subsequently, we examine the consistency level of fo(s) (ax) by computing the variance over the masked estimations ÿt,, i.e.
E (0,1).NT.
=ll., with y = Σ UL.
Var ( = Ur.
(6)
The synthetic labels y = (x) estimated with a large number of measurements m or a powerful learning model so lead to an accurate approximation to y with a small variance Var(ğ), and hence could be regard as high-quality data.
Supported by the proposed consistency check method, AiDE-Q assess Var(ğ) of all training examples Shyd (t). Those training examples whose Var(g) is less than given threshold are identified as the high-quality labeled data and incorporated into the updated high-quality dataset St,(t + 1).
Stage II: model training. After Stage I, the learning model fa(:) will be further trained on the updated high-quality dataset (1++1)). yielding an updated learning model fe((+1)- In particular, when = O. any learning paradigm introduced in Sec. 3.1 could be employed train the learning model fo(o) on the whole dataset Shyd. As shown in Fig. 2(b), when & > 1, the learning model fo(1) is optimized on the updated high-quality dataset S (f + 1) by minimizing the loss function
1
((0)-
Σ ((+1)
= -
[s((++))
where (t)(t + 1)] refers to the size of dataset Sx (t 1) and ÿ denotes the the synthetic labels in Sp (t + 1). Here, the optimized parameters (t) in the model foca) serve as the initial parameters for model training of fo on the dataset S, + 1). Throughout T iterations, the neural architecture in fo remains unchanged, while only the parameters O are updated.
Stage III: model evaluation. At shown in Fig. 2(c). the updated fe((+1) obtained from Stage II is evaluated on a validation dataset Syal to assess whether its predictive performance has improved relative to the previous model fo(t)- This evaluation determines whether the updated model fo(e+1) should be adopted for subsequent iterations. The employed validation dataset consists of quantum data with the same number of measurements as the data in Sy, i.e., m m_,, while the corresponding label is estimated by a larger number of measurement outcomes with m = m,- If the performance of fo(t+1) decreases compared to fø(t)- the newly added high-quality data in (t) may contain erroneous or harmful samples. In this case, AiDE-Q raises the threshold 7 in the consistency-cher module, re-initiating the high-quality labeled data collection, model training, and evaluation process until the model's ss performance on the validation dataset Syal improves (Refer to Appendix C.3 for details). This ensures that only reliable and informative samples that positively contribute to model generalization are included in the updated high-qualit dataset St + 1).

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Table R² in predicting the entanglement entropy SA two-point correlations Cī₃ and Ci₃ of 10-qubit XXZ model, where A [J] and [N 1]. The number of measurements for low-quality data is set 176, 2 best results are emphasized in blue while second- best results are distinguished orange.
S
C
Ci
Method
0.8
0.6
SL
0.722
0.740
0.820
0.848
0.883
0.930
0.915
0.936
0,953
SSL4Q
0.823
0.804
0.814
0.938
0.951
0.951
0.958
0.966
0,958
LLM4QPE
0.745
0.814
0.857
0.851
0.891
0.937
0,923
0.933
0,961
NTK
0.864
CS
0.308
0.308
0.308
0.605
0.605
0.605
S1. DE
0.825
0.864
0.904
0.944
0.972
0.985
0.966
0.978
0.989
Remark. The AiDE-Q framework is flexible and can be integrated into other learning models, such as machine learning models [31, 36]. Refer Appendix D for the discussion.
4 Experiments
In this section, conduct extensive studies about the effectiveness of AiDE-Q two standard quantum systems: the Heisenberg XXZ model and the one-dimensional cluster-Ising model. Refer to Appendix F for the omitted details and more results.
4.1
Data constructions for the explored quantum systems
Heisenberg XXZ model. An N-qubit Heisenberg XXZ model defined by the Hamiltonian [72]
N/2 H _ 1 to 12 + -2.110511 (24) + J
N/2-1
Σ
(02,0%111 + +o 2+11 + 02.02.11. (8)
where J and JJ' refer the alternating nearest-neighhor spin couplings, and of (a (a( represent the Pauli-X (Pauli-Z) operator acting on the i-th qubit. The focus on the Heisenberg model is because it an important statistical mechanical model used to study the behaviors of magnetic systems [73]-
We consider a set of ground states corresponding to n uniformly sampled coupling parameters from the region of J/J E 20, The system size N is varied N {10, 20, 30, 40, 50), with the corre- sponding number of sampled parameters for each system size being E {720, 1280, 1860, 2740). For the hybrid dataset in Eq. (5), the ratio of the number of high-quality data nı the total dataset size n is defined as == nt/n E {0.2, 0.4, 0.6, 0.8). The number of measurements for the initial high-quality dataset St of measurements is m. E {25, 20,2,, 27, 28, 29). The validation dataset consists of nval = 120 data points, with the number of measurements set to myal ma.
to my 20 while for the initial low-quality dataset Sy the number
Cluster Ising model. The N-qubit cluster Ising model [74], parameterized by a two-dimensional vector (h,, h2), is defined by the Hamiltonian
N-2
N Σo h2 Σ
N-1
Hel Σ aio
h1
o of o
(9)
This system has been extensively applied to describe a variety of quantum systems and is closely related to combinatorial optimization problems [75-77]. For the hybrid dataset construction, uniformly sample n different parameter values in the region h2/ E (0, 2). The number of qubits N and 720, respectively. Other hyperparameters used
and the training dataset size to build the hybrid dataset in Eq. (2) are set to be the same as those used in Heisenberg XXZ models.
QPE tasks. We focus on the non-linear property, entanglement entropy SA. and linear properties, two- defined as [1, ... .j]
point correlations C Ci as defined in Eq. (I). Here, the subsystem and the index j ranges in (2, ..... N). this regard, the data label y or the synthetic label ý. which corresponds to the properties of interest for given input is (N dimmennsio vector.
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Coefficients determination R2
LLMAOIE
1.0-
0.8
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0.4
0.2
C
G
Figure 3: R2 of reference models with and without integrating AiDE-Q in predicting entanglement entropy SA- two-point correlations Cit and C of 10-qubit XXZ model, where A and j E [NV 1]. The initial ratio of high-quality data and the number of measurements for low-quality data is set = 0.4 and m. 2
4.2 Experimental settings
Reference models. The first reference model is classical shadow (CS) [13], a cearning-free protocol for efficiently predicting various properties of quantum states. For machine- -ew.-len-aaaseeee baselines, we include the neural tangent kernel (NTK) [31] ] as representative classical model. Among DL-based models for QPE tasks, we adopt SSL4Q the reference for the SSL paradigm [51] and L.LM4QPF for the SF paradigm [52]. To evaluate AiDE-Q's predictive capabilities under the SI. paradigm, we benchmark it as standalone SL model. All DL models employ identical neural network architectures for feature representations to ensure a fair comparison. Refer to Appendix F for details.
Evalutaion metrics. To assess the predictive performance of different learning models, coefficient determination (R²) as the evaluation metric. Specifically, given the test dataset with Bice of size Ntt: the coefficient of determination is defined as
the
ground-truth labels ((x(() (8) ))
Ra =1 Li=1 (g (z)
(10)
Mise Σ
where = Ent ())) 100/n /te refers to the mean of the true values of the property of interest in the training dataset, and (f) denotes the predicted values from the employed learning model. The quantity R² typically ranges from to 1, with larger R² indicating that the model achieves better predictive accuracy. Specifically, when the model's predictions perfectly match the true values, have >inn ()( f() and R = 1. This metric eliminates the influence of the magnitude of estimated properties, providing clearer assessment of a model's predictive ability.
Hyperparameter settings of AiDE-Q. The maximum number of iterations of AiDE-Q is set to T ===6 6. For consistnncy-checkie in each iteration, the measurement data is divided into s subsets, each containing 25% of the total number of measurements. Data points whose consistency levels for the synthetic labels fall within the top 10% are used to retrain the learning models.
4.3 Experimental results
Despite the wide range of possible hyperparameter configurations, we present numerical results for selected settings to demonstrate AiDE-Q's predictive capabilities when integrated with various learning models, with more numerical results provided in Appendix F.
Supervised learning models integrated with AiDE-Q outperform all reference models. Table presents the coefficient of determination R for predicting entanglement entropy and two-point correlations for the 10-qubit Heisenberg XXZ model. The results shown for both machine learning and DL models, with varying ratios of the initial high-quality dataset Sz. while the number of measurements is fixed at m. 26 It can be observed that although DL models trained under the SSL (SSL4Q) and SSL-FT (LLM4QPE) paradigm outperform those trained under the SL paradigm by utilizing the dataset Sy as unlabeled dataset during training, the SI model incorporated with AiDE-Q demonstrate significantly performance improvement, and consistently achieve superior prediction accuracy across different properties and various ratio settings, as evidenced by their higher R² The most substantial improvement occurs when predicting entanglement entropy with 0.6.

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(8)
1.0
Rd 0.8
R²
SL (n0.4)
8
SL+DE (n0.6)
0.2.2
SL-DE (c0.6)
0.7
(n0.6)
St. (n0.6)
SL+DE (:0.4)
0.5
SL-DE (-0.8)
SL (n004)
SL (n0.8)
10 20
30
40
0
2
2'
N
0
Figure 4: R² in entanglement entropy prediction for the Heisenberg XXZ model. (a) R² values of AiDE-Q integrated SL models across varying quantum system sizes N with different total training dataset sizes and fixed my 2 (b) Evolution R² across AiDE-Q's iterations for 50-qubit XXZ model and fixed mu 2". (c) R² values with a varying number of measurements my for low-quality data points in 50-qubit XXZ model.
Table 2: R² in predicting the entanglement entropy SA of 9-qubit cluster Ising models with = [j] and j E [N - 1]. The best results are emphasized in blue while the second-bes results are distinguished in orange.
2
=27
2
Method
0.4
r=
0.6
0.8
r= 0.4
y 0.6
r
0.5
r T= 0.4
r 0.6
1 0.8
SL
0.226
0.356
0.51
0,255
0.234
0.559
0.317
0.444
0.443
SSL4Q LLM4QPE
0,422
0.448
0.489
0.501
0.539
0.626
0.513
0,58
0.616
0,29
0.322
0,436
0,322
0.308
0,587
0.194
0.458
0.458
SL DE
0.563
0,791
0,873
0.697
0.788
0.908
0.686
0.896
SSL4Q w. DE
0.794
0.836
0.86
0.829
0.865
0.892
0.821
0,902
0.907
LLM4QPE w. DE
0.569
0.767
0.901
0.596
0.821
0.939
0.646
0.842
0.924
where the SL model incorporated with AiDE-Q increases the R2 from 0.74 to 0.864 compared to the vanilla SL model. Notably, this surpasses the prediction performance of the optimal reference model, LLM4QPE, which only achieves an R² of 0.857 even with a higher high-quality data ratio r = 0.8.
AiDE-Q enhances the prediction performance of various DL-based models. We further examine the R² values of various learning models integrated with AiDE-Q in predicting quantum properties. Fig. 3 presents numerical results with setting r = 0.4 and m. = 26. The results demonstrate that all learning models show improved R2 after integrating the AiDE-Q into their vanilla learning models. with lower
Notably, the improvement is inversely proportional to the original performance-most baseline R² values show more substantial gains after AiDE-Q incorporation. For example, when predicting two-point correlations ciy. the R² values for LLM4QPE, SSL4Q, and NTK methods improved by 0.042, 0.092, and 0.124 from their original R2 values 0.938, 0.851, and 0. 0.799, repspectively. These findings suggest that AiDE-Q is compatible with various learning paradigms, enhancing their prediction performance.
The scalability of AiDE-Q. We next evaluate the scalability of AiDE-Q by applying it to the task of predicting entanglement entropy across varying system sizes N E {10, 20, 30, 40, 50) and fixed measurement number m. = 26. The achieved results are shown in Fig. 4(a), demonstrating that incorporating AiDE-Q into the SL paradigm consistently enhances prediction performance across all quantum system sizes and initial high-quality data ratios, as evidenced by increased R2 values. These results confirm AiDE-Q's scalability larger many-body physics models.
AiDE-Q improves the prediction performance with each iteration. Fig. 4(b) shows the R values for entanglement entropy prediction actoss iterations of the AiDE-Q under the SL paradigm for the 50-qubit XXZ model. With varying initial ratios of high-quality data r E {0.2, 0.4, 0.6, 0.8} and fixed measurement number m., 26, the R² values consistently improve AiDE-Q iterations progress, demonstrating AiDE-Q's effectiveness in identifying high-quality data with a synthetic label enhance model generalization. Notably, smaller initial rations of high-quality data yield larger performance improvements. For instance, at r = 0.2, AiDE-Q Q constantly increases the R² from 0.76 to 0.87, achieving an improvement of 0.11, while the improvement is less than 0.07 for other ratios.
The effect of the number of measurements on the AiDE-Q. Fig. 4(c) shows the R2 values for entanglement entropy prediction, evaluated across different numbers of measurements for low-quality

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data, m E , using SL models for the 50-qubit XXZ model. The numerical results demonstrate that increasing the measurement count my for the low-quality dataset Sy effectively enhances the predictive performance of SI. models, both with and without AiDE-Q integration.
Experimental results the cluster Ising Model. Table 2 shows the coefficient of determination R of entanglement entropy prediction for the 9-qubit cluster Ising model. The results show that integrating AiDE-Q into different learning-based models could significantly improve their predictive performance for different ratios of high-quality data r E {0.4, 0.6, 0.8} and measurement counts ma E {2%, 2", 2*). This confirms AiDE-Q's effectiveness across diverse quantum systems.
Conclusion
In this study, we introduced AiDE-Q, simple but effective framework that can be integrated into various learning paradigms for quantum property estimation (QPE) to improve the prediction performance of deep learning (DL) models, with limited quantum resources for dataset construction. Numerical experiments on the Heisenberg XXZ and cluster Ising models demonstrate AiDE-Q's effectiveness in enhancing the performance of several learning-basse based models, highlighting AiDE-Q's potential to improve the practicality of DL for QPE. However, a key limitation of our work is the unknown optimal allocation of limited quantum resources for constructing hybrid datasets.
Acknowledgment
Y. D. acknowledges the support from the SUG grant of NTU. Y. L. acknowledges the support from National Natural Science Foundation China (Grant No. U23A20318 and 62276195).
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A More basics of quantum computing
This section provides more details about quantum computing and reviews the classical shadow algorithms for quantum properties estimation (QPE), along with their associated sample complexity.
Basics of quantum computation. The elementary unit of quantum computation is the qubit (or quantum bit), which the quantum mechanical analog of classical bit. A qubit is a two-level quantum-mechande mechanical system described by a unit vector in the Hilbert space C?. In Dirac notation,
a qubit state is defined as (6) en [0) + C1 [1) E C² where [0) = [1, 1, and [1} [o, 1J" specify two unit bases and the coefficients co, et E C yield |co| 2 + lei 12 = 1. Similarly, the quanium state of n qubits is defined as a unit vector in C²", i.e., (&b) = Σ}= Gj (ej), where (ez) E R²" is the
computational basis whose j-th entry is and other entries are O, and Σj 1e,12 = 1 with oj E C. Besides Dirac notation, the density matrix can be used to describe more general qubit states. For example, the density matrix of the state (ab) is p =
(b) (ubl E C x2", where {vb = | refers to the complex conjugate transpose of (v)). For a set of qubit states (oy, (v)) ym with pj Σ Pj = 1, and lxby) E C² for j E [m]] its density matrix is 2 with Pj = Web)) (19l and TT(p) = 1.
A quantum gate is unitary operator that can evolve a quantum state p to another quantum state p'. Namely, an n-qubit gate U e U(2") obeys UU = U = 12", where (2(2) refers to the unitary group in dimension 2n. Typical single-qubit quantum gates include the Pauli gates, which can be written as Pauli matrices:
0 1 =
[10]
0
-i 0
1
0
X
1 100
y=
==
0
-1
(11)
The more general quantum gates their corresponding rotation gates Rx (0) = - Ry
(0)
e-g and Rz(0)) = - with tunable parameter 0, which can be written in the matrix form as
(0) = -i sin
cos 2
i sin 2
Ry
cos sin
sin e cos 2
R2(0) = e 2 0
cos
0 2
0
(12)
They are equivalent to rotating tunable angle 9 around x, y, and 2 axes of the Bloch sphere, and recovering the Pauli gates X, Y. and when 0 = ग. Moreover, a multi-qubit gate can be either an individual gate (e.g.. CNOT gate) or tensor product of multiple single-qubit gates.
The quantum measurement refers to the procedure of extracting classical information from the quantum state. It is mathematically specified by a Hermitian matrix H called the observable. Applying the observable H to the quantum state {ı8) yields a random variable whose expectation value is (1b) H (zb).
Hamiltonian and ground state. In quantum computation, a Hamiltonian is a Hermitian matrix that is used to characterize the evolution of a quantum system or as an observable to extract the classical information from the quantum system. Specifically, under the Schrödinger equation, quantum gate has the mathematical form of U -tt where H is Hermitian matrix, called the Hamiltonian of the quantum system, and / refers to the evolution time of the Hamiltonian. Typical single-qubit Hamiltonians include the Pauli matrices defined in Eqn. (11). As a result, the evolution time t refers to the tunable parameter 9 in Eqn. (12). Any single-qubit Hamiltonian can be decomposed as the linear combination of Pauli matrices, i.c., H = aj I + a2X + agY + Z with aj E C. In the same way, a multi-qubit Hamiltonian is denoted by H = ΣE 4" a, where P3 E X,, Z) &m is the tensor product of Pauli matrices. In quantum chemistry and quantum many-body physics, the Hermitian matrix that describes the quantum system to be solved is denoted as the problem Hamiltonian lcc
When taking the problem Hamiltonian the observable, the quantum state (ı²)) is said to be the ground state of problem Hamiltonian H if the expectation value (x>") H [b²)) takes the minimum eigenvalue of H, which is called the ground energy. The ground states encode much essential information about the problem Hamiltonian, such as the critical behavior of quantum body systems, the optimal solution of an optimization problem related to the problem Hamiltonian.
Numerous classical and quantum algorithms have been developed to efficiently obtain the ground states of problem Hamiltonians. These algorithms leverage various techniques, including variational
17

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methods, quantum annealing, and the application of tensor networks, to approximate or directly compute the ground state. In particular, quantum algorithms such as the variational quantum eigensolver [78], quantum approximate optimization algorithm [79-82], and the adiabatic quantum algorithm [83] show promising results for solving combinatorial optimization problems by preparing and measuring the ground state of problem Hamiltonians. The efficiency and feasibility of these methods continue to be the subject of extensive research, particularly in the context near-term quantum devices.
Classical shadow for QPE. We review the classical shadow algorithm [13] and the sample complexity for estimating the linear and nonlinear properties of quantum states. Given a unitary operator U sampled from a unitary ensemble U, and the subsequent measurement outcome b under computational measurement, the classical shadow of the N- N-qubit quantum state from m snapshots refers to
N
Ot (3U
1 (D)
(2)
U -12),
(13)
m
where random unitaries US are sampled from the Pauli ensembles u = XY, (12". has been shown that the sample complexity for estimating the expectation of observables O to e-precision O (4llccalty(O) ||1133 /²"), with locality(O). representing the number of non-identity operators acting on the qubits. For two-point correlation, the observable refers to O of of acting on the i-th and j-th qubits with locality (O) 2, leading the sample complexity (116/e2) for QPE.
The classical shadow could also be used to estimate the entanglement entropy of the quantum state on the subsystem A, which is given by
(p(p) = logg P(pA), with P(p) m(m - 1) jÆン
Tr(pA )2l")
(14)
where (Pp refers to the purity estimation of Tr())) 264) 8 A (3U refers to the local classical shadow on subsystems obtained from the j-th snapshot. It has been shown that the statistical error associated with P((A)) quantified by variance, which can be bounded as follows [84, 85].
(1) 10,0) (D) 1 12)
2 ((Aa) m
221 m
2
Var(P(pA)) $4
(15)
where |A| denotes the number of qubits in A. This bound is known to be essentially optimal [84]. It implies that estimating the entanglement entropy requires an exponentially large number of measurements as the size of subsystem A increases.
B Related work
In this section, we present a concise review of the literature on quantum property estimation (QPE), categorizing the approaches into three primary paradigms: conventional algorithms, machine learning (ML) algorithms, and deep learning (DL) algorithms. Our discussion emphasizes that the proposed AiDE-Q framework can be seamlessly integrated with various ML and DL algorithms.
Conventional algorithms for QPE. Conventional algorithms primarily focus on estimating the quantum properties of individual quantum states. A wide range of methods have been proposed, spanning classical simulation algorithms, quantum algorithms, and quantum state learning (QSL) algorithms [86].
Classical simulation algorithms achieve QPE entirely on classical computers, often utilizing tensor network techniques to simulate the whole quantum state [7-11], or employing Pauli-path simulation methods to estimate properties of quantum circuit generated states [87-91]. However, these classical simulation algorithms are typically limited to specific types of quantum states, such as low-entangled or low-magic states. Quantum algorithms for QPE involve the implementation of well-designed quantum circuits to evolve quantum states and extract the desired properties. Despite their potential, these quantum algorithms often require substantial quantum resources to construct the circuits, which can hinder their practical application, especially in the early stages of quantum computing [92, 93].
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Quantum state learning (QSL) algorithms for QPE operate by performing multiple measurements on a quantum state and post-procsssing the measurement results estimate the quantum properties of interest [13, 20, 16, 84, 85, 94]. Among these, the classical shadow algorithm has emerged as one of the most popular and efficient approaches, offering rigorous theoretical guarantees [13]. By using the measurement outputs from random measurement operators, this algorithm can simultaneously esti- mate multiple properties of quantum states, making it one of the most resource-efficint conventional algorithms for QPE.
ML algorithms for QPE. Unlike conventional algorithms that focus on QPE of individual states, machine learning algorithms address the QPE problem for classes of quantum states originating from the same quantum many-body systems with varying physical parameters. Notably, Huang et al. [95] proposed a kernel-based learning model that efficiently predicts linear properties of quantum many-body states with rigorous theoretical guarantees. This method eliminates the need for quantum devices during the prediction phase. Furthermore, Lewis et al. [36] introduced a Lasso regression model for N --qubit gapped local Hamiltonians that improves sample complexity from polynomial scaling N (where e is constant) achieved in Ref. [95] logarithmic scaling log(NV). Beyond the studies on the QPE problem for quantum many-body systems, Du et al. [96, 97] developed efficient classical learners for estimating the linear properties of parameterized quantum circuits under specific conditions.
DL algorithms for QPE. Deep learning algorithms provide enhanced capability for recognizing complex patterns in classical data collected from quantum many-body systems, enabling predictions of complex properties such as entanglement entropy with improved accuracy. Current DL algorithms can be categorized into three primary learning paradigms: supervised learning (SL), semi-supervised learning (SSL), and self-supervised learning with fine-tuning (SSL-FT).
The supervised learning paradigm explores various neural architectures for effectively constructing classical representations of the quantum states, including restricted Boltzmann machines [26, 28, 98], multi-layer perceptions [40, 41, 48], convolutional neural networks [29, 43, 47], and attention-based neural networks [35, 42, 49, 50]. The parameterized neural network for state restriction is optimized to approximate the target values of quantum properties in the training dataset under specific loss functions, such as mean square loss or cross-entroe cross-entropy loss.
The SSL and SSL-FT paradigms address the challenge of limited labeled data, allowing training with datasets comprising both labeled and unlabeled data. Tang et al. proposed a teacher-student model for semi-supervised training [51] and a large language model-style quantum task-agnostic pretraining and fine-tuning paradigm [52]. Both paradigms incorporate SL approaches for the labeled dataset, but differ in their sequence: SSL applies supervised learning before training with unlabeled data, whereas SSL-FT employs supervised fine-tuning after the self-supervised pretraining stage, which will be detailed in Appendix C.
C Implementation details of various learning paradigms for QPE
In this section, we present the implementation details of the three learning paradigms employed in the experiments: supervised learning (SL), semi-supervised learning (SSL), and self-supervised learning with fine-tuning (SSL-FT). All paradigms share common neural network architecture comprising three main components: the input (encoding) layer, the hidden layer, and the output layer. To ensure fair comparison of model performance, we adopt an identical architecture for the input and hidden layers across all learning paradigms as introduced in Appendix C.I, while the output layer is customized to suit the specific requirements of each paradigm as elucidated in Appendix C.2. Finally, we elucidate the adjustment strategy of the threshold in the model evaluation step of AiDE-Q in Appendix C.3.
C.1 Input layer and hidden layer
We follow the attention based neural architecture proposed in Ref. [52] to design the input and hidden layers for representing quantum states.
Input layer. To capture the hidden patterns of the quantum system, the input layer incorporates three types of embeddings: token embeddings, condition embeddings, and position embeddings.
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Each measurement outcome o, E {0.1} under the corresponding Pauli measurement M, E (X, Y, 2) on the j-th qubit yields six possible combinations: (X,00) 0). (X, 1). 0). (Y, (Z, 0). and (Z, 1). These pairs can be bijectively mapped to integers E 66. Consequently, the mea- surement outcomes (M, o) can be represented as a tokenized measurement string a [M, o), where reppees
each element o (M, 0,) [6] resembles a token in natural language processing (NLP). The token embedding layer applies a linear transformation to the measurement string, augmented with a start token s, producing a feature tensor E, E IP B, (N+1)xd, where B, denotes the batch size and is the feature dimension. A feed-forward network (FFN) with one hidden layer is then used to embed the physical condition p into a vector E, E R BBBidd The final input embedding is computed as the element-wise sum
Eont E, + E. + Epp
where Ep denotes the position embeddings. as described in Ref. [99]. The resulting tensor Evur is passed to the hidden layers for further processing.
Hidden Layer The hidden layer consists of a multi-layer Transformer decoder, following the architecture of Ref. [99]. It takes Eout as input and produces output F E BB which encodes high-level representations of both the measurement strings and the conditional physical parameters. For additional architectural details, refer to Ref. [52].
C.2 Output Layer and loss function
We now separately deseribe the neural architectures of the output layer and the associated loss functions under the explored three learning paradigms. Specifically, the design of the output layer is tailored to accommodate the specific loss function used in each paradigm.
SL paradigms. Recall that in the SL paradigm, the training dataset for batch is given by SSL (xx11, { where each input (+) (p()) (8) of() is derived from quantum states p(p())
with different parameters ( and y e R represents the target label vector of dimension K. The loss function for a training batch is defined as
LSL = B. Σ
(xx()) -000
(16)
To support this loss function, the output of the neural network for each input (66) must be a K- dimensional vector. Accordingly, the output layer consists of a feature aggregation module followed by linear projection. Specifically, hidden features F E (xN are aggregated along the second axis to produce feature representation F' TR. B.d for each training example. This is followed by a linear projection into F" E B with an optional task-specific activation function. For instance, tanh activation is used when predicting correlation functions.
SSL paradigms. In the SSL setting, the training dataset includes both labeled and unlabeled data, denoted by SSsI. = ((x(1) )² U (1 seacher-student framework [71] is adopted, consisting of a student model fSsi. and teacher model f'ss. both sharing the same architecture. The loss function for the student model in a given batch yields
where nu is the number of labeled samples.
Bi CSSL
B Σ B
((s( Bi 122l
1
g00l
(xs
fSsi (z))
(17)
B₁ B.-Bi
B
where À refers to the consistency weight, B, is the number of labeled samples in the batch, and B, is the total batch size, The proportion of labeled samples per batch kept consistent with their proportion in the overall training dataset, i.c., B B, n//n. The teacher model's parameters are updated an exponential moving average of the student model's parameters, following the approach in Ref. [71].
Since the outputs of the DL models in the SSL paradigm play the same role of K K-dimensional predicted labels as that in the SL paradigm, the architectures of the teacher and student networks are designed identically the ones used in the SL setting.
SSL-FT paradigms. The SSL.FFT paradigm adopts a two-stage training procedure. In the first stage, self-supervised pretraining is performed using the unlabeled dataset {a = (p, M, o)), where the
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goal is to model the classical probability distribution P(o o1).... , oMM, oN)) associated with a quantum state p(p) with fixed parameters p. This is achieved by minimizing the average negative log-likelihood loss
B. Σ
B
1
log (o(M[) of"). (0) (6) NN In()).
CSF
(18)
B Bi
which corresponds to maximizing the conditional likelihood of observed measurement outcomes.
To produce valid probability distributions, the output layer consists of a linear transformation followed by a softmax activation, ensuring normalization
Σ
Σ Oi), .MM,N oN)) 1. (My.oy)
(M1.01)
In the second stage, the pretrained model is fine-tuned on the labeled dataset { 3) using the same supervised loss and output layer design as in the SL paradigm, namely
CrT
Σ
(ss((r
(19)
Be
where the initial parameters of the input and hidden layers of fsr inherit the optimized parameters during the pretraining stage.
C.3
Adjustment strategy of the threshold in the model evaluation step of AiDE-Q
We first recall the step of model evaulation in AiDE-Q. As introduced in the maintext, given the hybrid dataset Snyd, AiDE-Q iteratively updates the learning model fa(t) and a high-quality dataset (t) C Snyd- Specifically, training examples whose variance Var(ū) is below given threshold TO are identified as high-quality data and added to the high-quality dataset Sx + 1).
The explanation of the strategy for adjusting the threshold is as follows. If the performance of fe(t+1) decreases compared to fo(r), AiDE-Q raises the threshold 7 in the consistency cceck module, re-initiating the high-quality labeled data collection, model training, and evaluation process until the model's 's performance on the validation dataset Syai improves. In the main text, the threshold raised to update the dataset (L + 1) by removing half of the newly added high-quality data in (A + (t))) that exhibit low consistency levels. Once the performance of fe(++1) improves, the threshold r is reset to the initial value To for the following high-quality data collection.
Remark. We remark that the threshold adjustment strategy in AiDE-Q is flexible. Alternative strategies, beyond those presented above, can be integrated into AiDE-Q.
D Integration of AiDE-Q with machine learning models
In this section, we briefly review the machine learning (ML) models for QPE [36, 95], and relate them to the supervised learning (ML) paradigm. In this regard, the integration of ML models with AIDE-Q could follow the same manner as that in the SL paradigm introduced in the main text.
We follow the conventions of Ref. [95] to introduce the ML model for QPE. In particular, the ML model considers the training dataset (() ()))) where ( are the sampled physical parameters, and (()()) refer to the classical shadow constructed with (M, o), as defined in Eq. 13. The classical ML models are trained on the size-n training data, such that when given the input (9), the ML model can produce a classical representation (() that approximates (0) During prediction, the classical ML produces h(p) for new values of different from those in the training data. particular, the predicted output of the trained classical ML models can be written as the extrapolation of the training data using learned kernel k(p, (9) R.
h(p) Σ "
(20)
The ground state properties are then estimated using these predicted classical representations h(p). Specifically, (p)(p) tr(Oh(p)) can be predicted efficiently whenever O is sum of few-body
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operators. In this regard, the trained classical ML for specific properties g( tr(Op(p())) could be written
fPL Σ x(p p 1 )8
(21)
Notably, the kernel function could be represented as the inner product of two feature vectors & (p) and (p)()) according to Mercer's 's theorem [100]. In this regard, the learning model fML(p) relates to the optimal solution of the following supervised learning problem
(xn.(D) = arg min LMI. (w) == arg min 1 Σ ( g(0)² ((p)= 7E
(22)
((p)=w+(pp)
where w refers to the optimized parameters. For the neural tangent kernel k(p. ()) [37], the feature vector (p) corresponds the output of deep neural network with large hidden layers [101].
To summarize, the ML models used in QPE naturally fit within the supervised learning paradigm described in Appendix C. Consequently, integrating these ML models with AIDE-Q can proceed in the same manner as outlined for the SL paradigm in the main text.
E Convergence analysis of AiDE-Q
In this section, we try to provide theoretical understanding of the convergence of AiDE-Q by employing the thcoretical results established for self-training algorithms in special case [102]. In the following, we first briefly recap the results from Ref. [102], and then introduce the convergence of AiDE-Q based on these results.
Convergence performance of self-training algorithms. The self- telf-training algorithm follows a similar iterative training manner AiDE-Q on hybrid dataset Shyd Sr. U St by generating pseudo label of unlabeled data E St. while without considering the identification of high-quality data. Let the inputs of both the labeled and unlabeled data follow the normal distribution, x E St N(0, d and E E S. (O,, 82))). Let W E R be the trainable weights of nee-hidden-laeer neural network, and assume the existence of tround-truth model with weights W Let (())) be model trained the labeled data Sy only, and ((() be the model further trained on the unlabeled data St after t iterations. Ref. [102] established the convergence performance of self-training algorithms by deriving an upper bound of the norm of differences between (t)) and W i.e.
| W S [(1
C1
+
C2
c3 ÅJj² |W W I|.
Vni
where n4, nh, refer to the number of labeled and unlabeled data, C2, C3 could be regarded as constants, Ā 5/(6 + 8). It is observed that the prediction performance of ((()) achieves linear convergence regarding the iteration f, when Ā algorithms also apply to AiDE-Q, as AiDE-Q follows the same iterative training manner with the only difference of using different unlabeled data during the iteration of (()).
1/(cs). The results established for self-training
Convergence analysis of AiDE-Q. We now understand these convergence results in the context of using AiDE-Q for QPE with limited number of measurements. As discussed above, the model ((()) achieves linear convergence for À > 1 - as training on the unlabeled data proceeds. Moreover, the value À is determined by the variances d and for the labeled data x and unlabeled data æ. In this regard, the convergence of ( of AiDE-Q determined by the variance ò of the identified high-quality unlabeled data. For the problem of QPE, the variance of the input ã = (p, ō) varies with the different physical parameters p and the number of measurements m,, for obtaining o. In particular, a small number of measurements my could lead to large variance 5 for some specific parameters {p} such that < - 1/(cs K), and hence decrease the model's 'ss prediction performance. The theoretical results provide rationale for using AiDE-Q to improve the performance of deep learning models by identifying high-quality unlabeled data with low variance, as the variance of the input æ is generally unknown.
otherwise the model performance will decrease
22

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1.0
0.9
0.8
=
2
2
2
R
0.7
0.6
r: 0.3
0.0.3
0.5
0.1
0.4
r: 0.4
0.6
D.6.6
0.4
0.8
0.8
r. 0.8
0
5
Figure R² in entanglement entropy prediction for 50-qubit Heisenberg XXZ model without using the physical parameters for constructing the training dataset. The panels from left to right correspond to the prediction performance for the number of measurements E 22,,",,2 2*
F Experimental setting and more experimental results
In this section, we conduct extensive experiments to evaluate the effectiveness of AiDE-Q across various hyperparameter settings, as well different physical and chemical models. Furthermore, we also assess its robustness in the noisy settings, where the measurement data are obtained from noisy quantum processors.
F. Experimental setting
Hardware platform. All the generation of training datasets are implemented by PastaQ [103] and Ienssors [104] in the Julia language and run classical device with Intel(R) Xeon(R) Gold 6226R CPU @ 2.90GHz and 256 GB memory. All deep learning models in various learning paradigms are implemented by Pytorch [105] and are trained on single NVIDIA GeForce RTX 3090 with 24G graphics memory.
Hyperparameter setting. The deep learning (DL. models the attention-based neural architectures described in Appendix C. The embedding dimension is set to = 128, the number of attention heads is set to 4, and the number of hidden layers is set to L = 2. We optimize the DL model using the ADAM optimizer with a learning rate of = 10-3 and a batch size of B₁ = 64. The maximum number of training epochs is stopping strategy. During the initial training stage, early stopping is applied after 100 epochs. In the subsequent training stage, with the updated high-quality dataset, early stopping is initiated after 30 epochs. Each configuration is run 5 times to report the average prediction performance.
300. To mitigate overfitting, we employ an early
F.2 Experimental results for various hyperparameter settings AiDE-Q
Here, conduct extensive experiments examine the effect of the hyperparameters in AiDE-Q namely the number of subsets for the consistency check s, the size of these subsets [Z] and the variance threshold for data selection where drop the subscript of Ig for clarity. In particular, we focus on the 10-qubit Heisenberg XXZ model.
Hyperparameter setting. We set the number of subset as E {3, 5, 7}, the size of each subset as [Z] agm, with ag E {1/8, 1/4, 1/2} and m, being the number of measurements for the low-quality dataset Sy, and the threshold т E {95%, 90%, 80%} with 3% referring to taking the threshold as consistency level at the top 1 3%. The hyperparameters for constructing the hybrid dataset Shyd = S₁ U. are set to the same in manuscript. In particular, the total number of training data is set as n = 720, the ratio of the number high-quality data n, to the total dataset size nu/n = 0.4. The number of measurements for the initial high-quality dataset S, is set to m, = 20, while the number measurements for the initial low-quality dataset S is m., 26
is defined as
Experimental results. Table presents R² values in predicting the entanglement entropy SA for the supervised learning (SL) models with and without incorporating AiDE-Q. It is observed that
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Table 3: R2 predicting the entanglement entropy SA with various hyperparameters (s, d.,,)), where [ and j E [N - 1]. The initial ratio of high-quality data is set as = 0.4. The best results are emphasized in blue while the second-best results are distinguished in orange.
8=5
Method
a7
178
47
174
0g
1/2
90%6
901
80%
90%
90%
80%
907
90%
80%
0.745
0.75
0.722
0.717
0.754
0.741
0.726
0.721
0.788
SL
DE
0.834
0.801
0,805
0.853
0.854
0.809
0.82
0.815
0,857
Improvement
0.089
0.083
0.136
0.068
0.093
0.07
90%
Method
3
5
= 1/8
oz = 1/4
= 1/2
av = 1/8
0z = 1/4
= 1/2
1/8
0z = 1/4
= 1/2
SL
0.723
0.722
0.779
0.75
0,754
0.721
0.774
0.73
0.739
$1. w. DE
0.806
0.834
0.829
0.801
0.854
0.815
0.849
0.825
0.807
Improvement
0.083
0.112
0.05
0.051
0.1
0.094
0.075
0.095
0.068
1/4
Method
95%
90%
80%
SL
0.753
0.717
0.78
0.722
0.754
0.73
0.786
0.741
0.71
DE
0,851
0.836
0.834
0.825
0.822
0.785
Improvement
0.098
0.136
0.056
0.112
0.095
0.036
0.068
0.075
the R² of SL models is improved with incorporating AiDE-Q in all hyperparameter settings up to 0.136 at (s, (I, (5, m,, 0.05). We further separately explore the effect of varying the three hyperparameters |Z], on the improvement of incorporating AiDE-Q into SL models. In particular, the experimental results show that for fixed number of subsets E (3, 5]. Moreover, for the fixed threshold = 90%, the maximal improvement of R² is always achieved at [Z] = m.,/4 for all different setting of 8 E {3, 7). Additionally, for a fixed size of subset [Z] = m,.4 /4 and a large threshold 7 E (90%, 95%}. setting the number of subset 8 = 5 achieves the maximal improvement of R2. These results suggest that to maximize the power AiDE-Q. the size of subsets [Z] and the number of subsets should be set moderately to improve the effectiveness of the consistency- method for identifying high-quality data labels.
F.3 Experimental results for various data types
In this section, we examine the prediction performance of the deep learning (DL) model trained on dataset that does not include physical parameters as input. Specifically, instead of using the training dataset with inputs x (p)))) = (p()) (*)), (1) as defined in Eq. we construct the dataset with inputs (+) = (o (0), where the physical parameters p are assumed to be unknown during the training process. The experimental results for predicting the entanglement entropy of the 50-qubit Heisenberg XXZ model are presented in Fig. 5. It is observed that the prediction performance can be consistently improved by AiDE-Q when the dataset contains small ratio of high-quality data. For instance, with a large number of measurements (m,, 2º) or low-quality data and small ratio of high-quality data (r 0.3). the coefficient of determination R² improves from approximately 0.36 to 0.88. These experimental results demonstrate the effectiveness of AiDE-Q Q for handling various types of training data.
F.4 Experimental results for noisy quantum states
In this section, consider that the measurement data are collected from noisy quantum states. In the following, we separately introduce the construction of the training dataset for noisy quantum states, the hyperparameter setting, and the experimental results. All contents in this reply have been updated to the revised manuscript.
Training dataset construction from noisy quantum states. We first formulate the form of the training dataset when considering the quantum system noise. particular, we employ the depolariza- tion noise model Nr, to characterize quantum system noise in the worst-case scenario. The prepared
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Table R² in predicting the entanglement entropy SA, where = [i] and j E [N 1]. The initial ratio of
high-quality data is set as r = 0.4. In each setting of mh E 2"), the best results are emphasized in blue while the second-best results are distinguished in orange.
Method
Mu.
2
2
= 0
=
0.1
9= 0.3
0.5
9= 0
9=
0.1
9 y= 0.3
0.5
SL
0.754
0.783
0.701
0,402
0.734
0.725
0.74
0.462
SL w. DB
0.854
0.863
0.769
0.457
0.82
0.789
0.808
0.542
Improvement
0.10
0.086
0.064
0.035
0.085
0.062
0.068
0.08
Table 5: R² in predicting the entanglement entropy SA, two-point correlations Cī₁ and Ci₃ of 9-qubit cluster Ising model, where A = [s] and j E [N 1]. The number of measurements for low-quality data is set as my 2". The best results are emphasized in blue while the second-hest results are distinguished in orange.
Sa
CT
Ci
Method
0.4
0.6
0.8
0.4
0.6
0.8
0.4
0.6
0.8
SL
0.226
0,356
0.51
0.902
0.928
0.938
0.903
0.925
SSL4Q
0:422
0.448
0.489
0.948
0.932
0.952
0.958
0.964
0,952
LLMOQE
0.29
0,322
0.436
0.902
0.922
0.949
0.909
0.948
0.954
NTK
0.962
0.966
0.97
0.291
0.31
0,341
CS
0.829
0.829
0.829
0.
0.
SL w. DE
0.563
0.791
0.873
0.956
0.965
0.976
0,965
0.98
0,984
ground state under depolarization noise model N refers to
T e N = (1 - q)p + 4 22N
where 0 < 9 S 1 refers to the collected classical input about the noisy ground state p(p(1) (o) with physical parameters (0), and ((+) is the measurement output of the Pauli operator (0) Here, my, m, refer to the number of measurements with my < my. Let g()) be the noisy label of a00 with the noise coming from the statistical measurement error and the quantum system noise.
Hyperparameter setting. The experiment setup for evaluating the performance of AiDB-Q is as follows. The hybrid dataset Skyd is collected from noisy quantum states, focusing on the 10- qubit Heisenberg XXZ model. We set the noise rate of the depolarization noise model NN, as q E (0..1,03, 0.5]. The other hyperparameters are kept the same those in our manuscript. In particular, the total number of training data is set as n = 720, the ratio of the number of high-quality data ny to the total dataset size n is defined as r == T4/ /n 0.4. The number of measurements for the initial high-qualit, dataset St is set to my = 200, while for the initial low-quality dataset Sy, the number of measurements is m, E (2 }
Experimental results. Table 4 presents the R² values in predicting the entanglement entropy SA for the supervised learning models with and without incorporating AiDE-Q. is observed that while the prediction performance of supervised learning models significantly decreases for a large noise rate q == 0.5, AiDE-Q could still effectively improve the prediction performance of learning models for various noise levels q E {0.1, 0.3, 0.5). The improvement in R² decreases from 0.1 for the noiseless case of q 0 to 0.035 for the noisy case with q = 0.5, when the number of measurements small for the low-quality data, namely my 2 On the other hand, for a larger number of measurements my 2?, the AiDE-Q could remain a relatively large improvement of R namely 0.08, even when the noise rate reaches 0.5. This indicates that the improvement of prediction performance raised from AiDE-Q is robust to the quantum system noise for a large number of measurements my.
E,5 More experimental results for the cluster-Ising model
In this section, we present additional experimental results for the 9-qubit cluster Ising model, following the same format the Heisenberg XXZ model results discussed in the main text. The hyperparameter settings used to construct the training dataset are consistent with those described in the main text.
Supervised learning models integrated with AiDE-Q outperform all reference models. Table 5 presents the coefficient of determination R2 for predicting entanglement entropy and two-point correlations in the 9-qubit cluster Ising model. The results show similar trends to those observed
25

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LUNAQEE LLMARRE
NIK
Coefficlent: of determination
1.0
0.8
0.6
0.2
0.0
C2
Figure 6: R² reference models with and without integrating AiDE-Q in predicting entanglement entropy SA. two-point correlations сí₁ and Ci2 of 9-qubit cluster Ising model, where [j] and j E [N - 1]. The initial ratio of high-quality data and number of measurements for low-quality data set as r = 0.4 and my 2
Table 6: R² predicting the entanglement entropy SA of H4 molccular systems with A = [3] and [N - The best results are emphasized in blue while the sccond- results are distinguished in orange.
2
2
2
Method
y= 0.4
r= 0.6
r=
0.8
r= 0.4
== 0.6
r = 0.8
r= 0.4
0.6
r == 0.8
SL SSL4Q LLM4QPE
0.085
0.178
0,178
0.196
0.254
0,275
0.127
0,167
0.225
0,251
0.109
0.404
0.203
0.224
0.408
0.082
0.165
0.305
0.033
0.204
0,238
0,098
0.144
0,205
0.077
0,186
0.243
DE
0.405
0.751
SSL4Q DE
0.495
0,508
0,709
0.507
0.546
0.704
0,543
0,57
0.694
LLM4QPE w. DE
0.293
0.58
0.618
0.473
0.572
0.638
0.502
0.584
0.64
for the Heisenberg XXZ model, where the baseline SL model integrated with AiDE-Q achieves a significant performance improvement compared to the standalone SL model, outperforming all reference learning models across different properties and various ratio settings. Additionally, when the ratio of high-quality data is large 0.8, the baseline SL model even surasses the advanced SSL4Q and LLM4QPE models in predicting entanglement entropy. This suggests that directly training DL models with low-que low-quality data can harm prediction performance, even when using advanced learning models. contrast, the AiDE-Q-enhancer SI. model, trained with the identified high-quality synthetic labeled data, significantly improves the best R value across the reference models from 0.51 to 0.873. These results emphasize the importance of constructing high-quality synthetic labeled data when training DL models for quantum property estimation.
AiDE-Q enhances the prediction performance of various DL-based models. Fig. 6 presents numerical results of predicting quantum properties when integrating AiDE-Q into various learning models. The ratio of high-quality data is set to r 0.4, and the number of measurements for low-quality data is m, = 26. The results show that all learning models exhibit an improvement in R² after incorporating AiDE-Q into their baseline models, particularly in predicting the complex non-linear properties of entanglement entropy.
F.6 Experimental results for the chemical molecular model
We conduct numerical simulations for predicting the properties of chemical molecular models, focusing on the H,, molecule with varied inter-atomic length.
Dataset Generation. The Hamiltonian of a molecular system under Jordan- Jordan-Wigner transformation is given by
N H(p) - = Σ 1= aE(
N
(p)o + Σ Σ
(p)of
(23)
YJ=1 0= B.
where of (af) is the Pauli-Z (Pauli-X) operator acting on the i-th qubit, (p), (a,0) (1)))(p))))
(p) refer to the
Pauli coefficients dependent on the inter-atomic length p. This Hamiltonian could be constructed with the Pennylane package [106]. For the H4 molecule, this system is characterized by 8-qubit
26

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Hamiltonian H (p). Here, the onter-atomin length p is sampled from the range 0.7À to 1.3A. The number of sampled inter-atomic lengths is set as n 1280. Other hyperparameters used to build the hybrid dataset in Eq. (5) are set to be the same as those used in quantum many-body systems.
Experimental results. Table 6 shows the coefficient of determination R of entanglement entropy prediction for the H4 molecule model. The results demonstrate that integrating AiDE-Q into various learning-based models significantly enhances their predictive performance across different ratios of high-quality data r E {0.4, 0.6, 0.8} and measurement counts m, E {29, 2", 2 j. In particular, the mean R² values for the SL, SSL4Q. and LLM4QPE models are improved from 0.187, 0.239, 0.158 to 0.604, 0.586, 0.544, respectively, after integrating AiDE-Q. These results confirm the effectiveness of AiDE-Q in enhancing the prediction ability of DL models for variety of quantum systems.
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Reviews:
[{"review_id": "hz62rffEr7", "reviewer": "Reviewer_nKML", "summary": "The paper presents AiDE-Q, a method for enhancing the datasets used to train deep learning (DL) models to estimate the ground state properties of quantum systems. Exactly calculating the ground state properties of a quantum system is exponentially expensive on classical hardware due to theoretical constraints, and practically challenging on quantum hardware due to hardware noise and the limited availability of quantum hardware. As a result, DL models for ground state property estimation are typically trained on hybrid datasets consisting of a mix of high-quality data with highly accurate labels and low-quality data with very approximate labels. The high ratio of low to high quality data naturally leads to poor DL model performance.\n\nTo address this issue, the paper proposes the integration of high-quality synthetic data into hybrid datasets. AiDE-Q iteratively improves DL models by generating synthetic labels through a trained model and employing a consistency-check method to assess the quality of these labels. Experimental results demonstrate that AiDE-Q achieves up to a 14.2% improvement in model performance when predicting entanglement entropy and correlation in the Heisenberg XXZ model and cluster Ising model with up to 50 qubits.", "questions": "What happens when you add in noise?", "limitations": "No analysis is done in the presence of noise.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "This paper is overall a strong paper. Using quantum data to help train machine learning models is a popular idea. However, the effectiveness of these models will likely be limited even with the advent of fault-tolerant quantum computers. That's because access to quantum hardware will likely remain limited for the foreseeable future as building and operating quantum computers will continue to be very expensive. Having methods for improving the training sets of DL models for estimating the ground state properties will be critical.\n\nThe authors provide strong (given the length constraints of NeurIPS) evidence that AiDE-Q does improve model training. The authors ran comparative studies using a number of state-of-the-art DL approaches to ground state property estimation, showing improvement in most cases with AiDE-Q. \n\nAiDE-Q is conceptually simple, which helps streamline the paper's presentation. The authors do a good job at contextualizing their work and surveying the subfield of DL-based approaches to ground state property estimation.\n\n**Weaknesses:** It would be nice to see simulations done on noisy quantum computers. The noise models can be reflective of early fault-tolerant devices (<10^-3), Pauli stochastic errors, some very good qubits, etc. Otherwise it is hard to evaluate AiDE-Q's short-to-medium-term relevance to the community.\n\nI didn't really spot many weaknesses outside of a typo or two. \"Basic of quantum computing\" should be \"Basics of quantum computing.\"", "was_revised": true, "final_justification": "**Rating:** 5. \n\n**Justification:** This paper is a technically sound paper with the potential to have a medium-to-high impact on multiple subfields of AI (hybrid quantum-classical machine learning, synthetic dataset generation). The paper does not merit a higher grade as it is unlikely to have a groundbreaking impact on any subfield of AI.", "rebuttal": "We deeply thank Reviewer nKML for their affirmative comments and constructive feedback on our work. Since the concerns raised in the **Weakness** and **Question** sections both focus on AiDE-Q's performance in noisy scenarios, we have merged them under the index **W1 & Q1**, while the minor weakness is indexed as **W2**. All concerns have been carefully considered and are addressed in detail below. We hope our responses adequately address your concerns.\n\n**[W1 & Q1]: It would be nice to see simulations done on noisy quantum computers. The noise models can be reflective of early fault-tolerant devices ($<10^{-3}$), Pauli stochastic errors, some very good qubits, etc. Otherwise it is hard to evaluate AiDE-Q's short-to-medium-term relevance to the community. What happens when you add in noise?**\n\n**A1**: We thank the reviewer for their thoughtful and constructive comments. We have followed the reviewer's suggestion to evaluate the prediction performance of AIDE-Q on the measurement data collected from noisy numerical simulations. The key finding is that **AiDE-Q demonstrates strong performance even in the presence of quantum noise, highlighting its robustness to low noise levels**. In the remainder of this reply, we separately introduce the construction of the training dataset for noisy quantum states, the hyperparameter setting, and the experimental results. All contents in this reply have been updated to the revised manuscript. \n\n$\\bullet$ *Training dataset construction from noisy quantum states.* We first formulate the form of the training dataset when considering the quantum system noise. In particular, we employ the depolarization noise model $\\mathcal{N}_q$ to characterize quantum system noise in the worst-case scenario. The prepared ground state under depolarization noise model $\\mathcal{N}_q$ refers to \n$$\\tilde{\\rho}= \\mathcal{N}_q[\\rho] = (1-q)\\rho + q \\frac{\\mathbb{I}}{2^N},$$\nwhere $0\\le q\\le 1$ refers to the noise rate, $\\rho$ is the noiseless ground state corresponding to the case of $q=0$, and $\\mathbb{I}/2^N$ represents the $N$-qubit maximally mixed state. In this regard, we denote the hybrid dataset for quantum property estimation (QPE) as\n$$\\tilde{\\mathcal{S}} _{\\text{hyd}}=\\tilde{\\mathcal{S}} _{L}\\cup \\tilde{\\mathcal{S}} _U, \\text{with } \\tilde{\\mathcal{S}} _{L}=\\\\{ \\tilde{\\mathbf{x}}^{(i)}|\\tilde{\\mathbf{o}}^{(i)}\\in\\mathbb{R}^{Nm_l} \\\\} _{i=1}^{n_l}  \\text{ and } \\tilde{\\mathcal{S}} _{U}=\\\\{\\tilde{\\mathbf{x}}^{(i)}|\\tilde{\\mathbf{o}}^{(i)}\\in\\mathbb{R}^{Nm_u}\\\\} _{i=n_l+1}^{n_l+n_u},$$\nwhere $\\tilde{\\mathbf{x}}^{(i)}=(\\mathbf{p}^{(i)},\\mathbf{M}^{(i)}, \\tilde{\\mathbf{o}}^{(i)})$ refers to the collected classical input about the noisy ground state $\\tilde{\\rho}(\\mathbf{p}^{(i)})=\\mathcal{N}_q[\\rho(\\mathbf{p}^{(i)})]$ with physical parameters $\\mathbf{p}^{(i)}$, and $\\tilde{\\mathbf{o}}^{(i)}$ is the measurement output of the Pauli operator $\\mathbf{M}^{(i)}$. Here, $m_l,m_u$ refer to the number of measurements with $m_u<m_l$. Let $\\tilde{\\mathbf{y}}^{(i)}$ be the noisy label of $\\tilde{\\mathbf{x}}^{(i)}$ with the noise coming from the statistical measurement error and the quantum system noise.\n\n\n$\\bullet$ *Hyperparameter setting.* The experiment setup for evaluating the performance of AiDE-Q is as follows. The hybrid dataset $\\tilde{S}_{\\text{hyd}}$ is collected from noisy quantum states, focusing on the 10-qubit Heisenberg XXZ model. We set the noise rate of the depolarization noise model $\\mathcal{N}_q$ as $q\\in\\\\{0.1,0.3,0.5\\\\}$. \n\nThe other hyperparameters are kept the same as those in our manuscript. In particular, the total number of training data is set as $n=720$, the ratio of the number of high-quality data $n_l$ to the total dataset size $n$ is defined as $r := n_l/n = 0.4$. The number of measurements for the initial high-quality dataset $\\mathcal{S} _L$ is set to $m_l = 2^{10}$, while for the initial low-quality dataset $\\mathcal{S} _U$, the number of measurements is $m_u \\in \\\\{2^6, 2^7\\\\}$.\n\n$\\bullet$ *Experimental results.* The table below presents the $\\mathrm{R}^2$ values in predicting the entanglement entropy $S_A$ for the supervised learning (SL) models with and without incorporating AiDE-Q, corresponding to the labels 'SL' and 'SL w. DE'. \n\n| Method |      $m_u=2^6$ | | | | $m_u=2^7$ | | | |\n|:-------|:----------|:----------|:----------|:----------|:----------|:----------|:----------|:----------|\n| | $q=0$ | $q=0.1$ | $q=0.3$ | $q=0.5$ | $q=0$ | $q=0.1$ | $q=0.3$ | $q=0.5$ |\n| SL | 0.754 | 0.783 | 0.701 | 0.402 | 0.734 | 0.725 | 0.74 | 0.462 |\n| SL w. DE | 0.854 | 0.863 | 0.769 | 0.457 | 0.82 | 0.789 | 0.808 | 0.542 |\n| **Improvement** | **0.10** | *0.086* | 0.064 | 0.035 | **0.085** | 0.062 | 0.068 | *0.08* |\n\n**Note**: Best results are shown in **bold** and second-best results in *italics*.  \n\nAn immediate observation is that while the prediction performance of supervised learning models significantly decreases for a large noise rate $q=0.5$, **AiDE-Q could still effectively improve the prediction performance of learning models for various noise levels** $q\\in \\\\{0.1,0.3,0.5\\\\}$. The improvement in $\\mathrm{R}^2$ decreases from 0.1 for the noiseless case of $q=0$ to 0.035 for the noisy case with $q=0.5$, when the number of measurements is small for the low-quality data, namely $m_u=2^6$. On the other hand, for a larger number of measurements $m_u = 2^7$, the AiDE-Q could remain a relatively large improvement of $\\mathrm{R}^2$, namely 0.08, even when the noise rate reaches $q=0.5$. This indicates that the improvement of prediction performance raised from AiDE-Q is **robust to the quantum system noise** for a large number of measurements $m_u$.\n\n**[W2]: I didn't really spot many weaknesses outside of a typo or two. ''Basic of quantum computing'' should be ''Basics of quantum computing''**\n\n**A2:** Thank you for this feedback. We will conduct a thorough proofreading of the entire manuscript and correct all typos in the revised version."}, {"review_id": "869Yh9fvro", "reviewer": "Reviewer_bVXi", "summary": "This paper introduces AiDE-Q, a framework designed to improve the performance of deep learning (DL) models for quantum property estimation (QPE) when faced with practical limitations on data acquisition, such as a finite number of quantum measurements. The core challenge addressed is that real-world datasets for QPE are often \"hybrid,\" containing a small amount of high-quality data (many measurements, accurate labels) and a large amount of low-quality data (few measurements, noisy labels).", "questions": "Section 3.2 mentions that if the validation performance decreases, the variance threshold is raised to be more selective. Could you elaborate on the mechanism for adjusting this threshold?", "limitations": "Addressed", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\nThe paper addresses a critical and practical bottleneck in the application of DL to quantum many-body physics: the high cost and inherent noise of data acquisition from quantum systems. The problem of learning from hybrid-quality datasets is relevant for near-term quantum devices. The proposed AiDE-Q framework offers a practical solution to leverage large but noisy datasets effectively.\n\nWhile the concept of a \"data engine\" or self-training with pseudo-labels exists in the broader machine learning literature, its application to QPE is novel. The specific implementation, particularly the \"consistency-check\" method to gauge the quality of synthetic labels based on measurement data variance, is an original and clever contribution tailored to the problem domain.\n\nThe paper is well-written, clearly structured, and easy to follow. The problem setup is well-motivated (Figure 1), and the AiDE-Q framework is explained lucidly with the help of a detailed diagram (Figure 2). The experimental settings and results are presented clearly, making the contributions accessible.\n\n**Weaknesses**\n\nThe paper is primarily empirical. there is no theoretical analysis to formalize this or provide guarantees on convergence or performance improvement. For instance, under what conditions does the iterative process guarantee not to accumulate errors from incorrectly labeled data?\n\nThe framework introduces several new hyperparameters, such as the variance threshold for data selection (set to the top 10%), the number of subsets for the consistency check, and the size of these subsets. The paper uses fixed values for these and provides limited analysis of their sensitivity. A more detailed study on how these choices impact performance would strengthen the paper's claims of robustness.\n\nThe method of raising the variance threshold when validation performance drops is mentioned but not detailed. This seems like a crucial heuristic to prevent model degradation. A more principled or detailed explanation of this adaptive mechanism would be beneficial. Is it a fixed increment, or does it depend on the magnitude of the performance drop?", "was_revised": true, "final_justification": "Keeping the score.", "rebuttal": "We thank Reviewer bVXi for their affirmative comments and valuable feedback. For clarity, we index the points mentioned under **Weaknesses** as **W1-W4**, and **Questions** as **Q1**. In particular, we have addressed the points raised in **W3** and **Q1** together, as they are similar. All concerns have been carefully considered and are addressed in detail below. \n\n**[W1]: There is no theoretical analysis to guarantee convergence or performance improvement. Under what conditions does the iterative process guarantee not accumulate errors from incorrectly labeled data?** \n\n**A1:** We appreciate the reviewer’s concern regarding the lack of theoretical guarantees for the proposed method. We would like to respectfully point out that in the domain of deep learning–based quantum property estimation (**DL-QPE**), it is standard practice to validate new approaches through **extensive empirical studies** rather than formal theoretical analysis [Refs. a–d]. This is largely due to the fact that current deep learning theory often yields only loose bounds and offers limited explanatory power for the observed success of deep neural networks [e–f].\n\nAlthough theoretical analysis is *not the primary focus* of our study, we aim to best address the reviewer's concerns by bridging the gap between theory and model design. In the following, we provide a theoretical explanation of the convergence of AiDE-Q by using established results for self-training algorithms in a special case [g].\n\n$\\bullet$ *Convergence performance of self-training algorithms.* Let us briefly recap the results of Ref. [g].  Assume the existence of a ground-truth model $W^{\\*}$. Let $W(0)$ be a model trained on the labeled data only, and $W(t)$ be the model further trained on the unlabeled data after $t$ iterations. Ref. [g] established the convergence performance of $W(t)$ by deriving an upper bound of the norm of differences between $W(t)$ and $W^{\\*}$, i.e.,\n $$ ||W(t) - W^{\\*}|| \\le [(1+c _1/\\sqrt{n _l}+c_2/\\sqrt{n _u})\\cdot c (1-\\hat{\\lambda})]^t \\cdot ||W(0) - W^{\\*} |\\|, $$\nwhere $n_l,n_u$ are the number of labeled and unlabeled data, $c_1,c_2,c$ could be regarded as constants, $\\hat{\\lambda}$ is a decreasing function of the variance $\\tilde{\\delta}$ of unlabeled data.\n\n$\\bullet$ *Convergence analysis of AiDE-Q.* As AiDE-Q shares a similar training manner with self-training algorithms, the established results in [g] also apply to AiDE-Q. For QPE, the variance $\\tilde{\\delta}$ is often large due to a small number of measurements $m_u$ in the unlabeled data, which could harm the performance with $c(1-\\hat{\\lambda}) >1$. \n\nBy contrast, AiDE-Q can well suppress $\\tilde{\\delta}$ by identifying the high-quality data, where $c(1-\\hat{\\lambda}) <1$ leads to linear convergence. These results undermine the importance of using AiDE-Q to advance DL-QPE.\n\n$\\bullet$ *Error accumulation from incorrectly labeled data.* To prevent the accumulated errors caused by incorrectly labeled data, a module has been designed to remove harmful unlabeled data by increasing the variance threshold for identifying high-quality data when the model’s performance drops.\n\n**[W2]: The paper uses fixed values for these and provides limited analysis of their sensitivity. Conduct a detailed study on how these choices impact performance**\n\n**A2:** We appreciate the reviewer’s suggestion and have *conducted additional experiments* to examine how varying hyperparameters affect the performance of AiDE-Q. Specifically, we investigate the impact of three key hyperparameters: *the number of subsets $s$,  the size of these subsets $|\\mathcal{I}|$, and variance threshold $\\tau$*. For clarity, we first describe the experimental setup, followed by a presentation of the corresponding results. All content below reflects updates made in the revised manuscript. \n\n$\\bullet$ *Hyperparameter settings.*  We apply AiDE-Q to predict the entanglement entropy $S_A$ of the $10$-qubit Heisenberg XXZ models. We adopt the supervised learning (SL) model introduced in our manuscript as the baseline for comparison. \n\nThe experiment setup for evaluating the performance of AiDE-Q is as follows. The number of subsets is set to $s=3,5,7$.  The size of each subset is set to $|\\mathcal{I}|=\\alpha_{\\mathcal{I}} m _u$ with $\\alpha _{\\mathcal{I}} = 1/8, 1/4, 1/2$ and $m_u$ being the number of measurements for the low-quality dataset $\\mathcal{S}_U$. The threshold $\\tau =$ 95\\%, 90\\%, 80\\% with $\\beta$% referring to taking the threshold as a consistency level at the top $1-\\beta$%. The other hyperparameters, including $n,r,m_u,m_l$, are kept the same as those in our manuscript, and as those introduced in the response **A1** for Reviewer nKML. \n\n$\\bullet$ *Experimental results.* We present the results in the following three tables to separately explore the effect of varying hyperparameters $s, |\\mathcal{I}|, \\tau$. The presented results include $\\mathrm{R}^2$ values obtained from the SL models with and without incorporating AiDE-Q, as well as the difference between them, which are highlighted by labels 'SL',  'SL w. DE', and 'Improvement', respectively. \n\n1) The table below shows the results of varying $\\tau$ and $|\\mathcal{I}|$ with a fixed $s=5$. An observation is that the $\\mathrm{R}^2$ of SL models is improved with incorporating AiDE-Q in all hyperparameter settings up to $0.136$ at $(s,|\\mathcal{I}|,\\tau)=(5,m_u/4,95$%$)$. Moreover, setting a large threshold $\\tau$ leads to a large improvement of $\\mathrm{R}^2$ in most cases.\n\n| Method | α_I=1/8 | | | α_I=1/4 | | | α_I=1/2 | | |\n|--------|---------|---------|---------|---------|---------|---------|---------|---------|---------|\n| | τ=95% | τ=90% | τ=80% | τ=95% | τ=90% | τ=80% | τ=95% | τ=90% | τ=80% |\n| SL | 0.745 | 0.75 | 0.722 | 0.717 | 0.754 | 0.741 | 0.727 | 0.721 | 0.788 |\n| SL w. DE | 0.834 | 0.801 | 0.805 | 0.853 | 0.854 | 0.809 | 0.82 | 0.815 | 0.857 |\n| Improvement | **0.089** | 0.051 | *0.083* | **0.136** | *0.1* | 0.068 | *0.093* | **0.094** | 0.07 |\n\n2) The table below presents the results of varying $|\\mathcal{I}|$ and $s$ with a fixed $\\tau=$90%. The results show that the maximal improvement of $\\mathrm{R}^2$ is always achieved at $|\\mathcal{I}|=m_u/4$ for $s=3,5,7$.\n| Method | s=3 | | | s=5 | | | s=7 | | |\n|--------|-----|-----|-----|-----|-----|-----|-----|-----|-----|\n| | α_I=1/8 | α_I=1/4 | α_I=1/2 | α_I=1/8 | α_I=1/4 | α_I=1/2 | α_I=1/8 | α_I=1/4 | α_I=1/2 |\n| SL | 0.723 | 0.722 | 0.779 | 0.75 | 0.754 | 0.721 | 0.774 | 0.73 | 0.739 |\n| SL w. DE | 0.806 | 0.834 | 0.829 | 0.801 | 0.854 | 0.815 | 0.849 | 0.825 | 0.807 |\n| Improvement | *0.083* | **0.112** | 0.05 | 0.051 | **0.1** | *0.094* | *0.075* | **0.095** | 0.068 |\n\n3) The table below presents the results of varying $\\tau$ and $s$ with a fixed $|\\mathcal{I}|=m_u/4$. Namely, for a large threshold $\\tau=$ 90%, 95%, the maximal improvement of $\\mathrm{R}^2$ is consistently achieved at a small $s=3,5$.\n\n| Method | τ=95% | | | τ=90% | | | τ=80% | | |\n|--------|-------|-------|-------|-------|-------|-------|-------|-------|-------|\n| | s=3 | s=5 | s=7 | s=3 | s=5 | s=7 | s=3 | s=5 | s=7 |\n| SL | 0.753 | 0.717 | 0.78 | 0.722 | 0.754 | 0.73 | 0.786 | 0.741 | 0.71 |\n| SL w. DE | 0.851 | 0.853 | 0.836 | 0.834 | 0.854 | 0.825 | 0.822 | 0.809 | 0.785 |\n| Improvement | *0.098* | **0.136** | 0.056 | **0.112** | *0.1* | 0.095 | 0.036 | *0.068* | **0.075** |\n\nThe achieved results suggest that to maximize the power of AiDE-Q, the size of subsets $s$ and the number of subsets $|\\mathcal{I}|$ should be set moderately, and the threshold $\\tau$ should be set to be relatively large.\n\n**[W3 & Q1]: The method of raising the variance threshold when validation performance drops is mentioned but not detailed. Elaborate on the mechanism for adjusting this threshold.**\n\n**A3:** To better address the reviewer's concerns, let us first recap AiDE-Q and then provide a detailed explanation.\n\n $\\bullet$ *Overview of AiDE-Q.* As introduced in Sec. 3.2, given a hybrid dataset $\\mathcal{S} _{hyd}$, AiDE-Q iteratively updates the learning model $f _{\\theta(t)}$ and a high-quality dataset $\\mathcal{S} _{h}(t)\\subset \\mathcal{S} _{hyd}$. Specifically, training examples whose $\\text{Var} (\\hat{y})$ is below a given threshold $\\tau=\\tau _0$ are identified as *high-quality data* and added to the high-quality dataset $\\mathcal{S} _h(t+1)$.\n\n$\\bullet$ *Explanation of the strategy for adjusting the threshold $\\tau$.* If the performance of $f _{\\theta(t+1)}$ decreases compared to $f _{\\theta(t)}$, AiDE-Q raises the threshold $\\tau$ in the *consistency-check module*, re-initiating the high-quality labeled data collection, model training, and evaluation process until the model’s performance on the validation dataset $\\mathcal{S} _{val}$ improves. \n\nIn the main text, the threshold $\\tau$ is raised to update the dataset $\\mathcal{S} _h(t+1)$ by **removing half of the newly added high-quality data** in $\\mathcal{S}_h(t+1)  \\backslash \\mathcal{S} _h(t)$ that exhibit low consistency levels. Once the performance of $f _{\\theta(t+1)}$ improves, the threshold $\\tau$ is reset to the initial value $\\tau_0$ for the following high-quality data collection.\n\nWe remark that **the threshold adjustment strategy in AiDE-Q is flexible**. Alternative strategies, beyond those presented above, can be integrated into AiDE-Q. We have appended the discussion in the updated version.\n\n**[W4]: Is it a fixed increment, or does it depend on the magnitude of the performance drop?**\n\n**A4:** The increment of the variance threshold $\\tau$ is **not fixed and does not depend on the magnitude of the performance drop**. As explained in the response to **A3**, the purpose of adjusting $\\tau$ is to control the proportion of identified high-quality data based on their variances. Therefore, the increment of $\\tau$ is **determined by the magnitude of the variances computed over the identified high-quality data**.\n\n[a-f] Refer to the response for Reviewer Kvb1\n[g] ArXiv:2201.08514. ICLR 2022"}, {"review_id": "EvlOmhFanq", "reviewer": "Reviewer_Kvb1", "summary": "The paper introduces AiDE-Q, a data framework engine designed for machine learning tasks in so-called quantum property estimation, particularly in the context of quantum many-body systems. It enables the generation of “high-quality” training data by assigning reliable synthetic labels even to noisy data derived from a limited number of quantum measurements. The core technical innovation is a consistency-check mechanism that assesses the confidence of synthetic labels by evaluating their stability across randomly masked subsets of measurement data. The method is compatible with various deep learning paradigms and the authors provide experimental results on Heisenberg XXZ and cluster Ising systems.", "questions": "-Can the authors provide an estimate of the computational cost of running AiDE-Q for different problem sizes? Also, how should the key parameters—like the number of iterations— be chosen in order to obtain a target accuracy?\n-Related to this, how should one choose the threshold used in the consistency check? Does it have a strong impact on performance and running time?\n-Why can we expect the algorithm will perform well at scale?\n-Finally, can the authors compare AiDE-Q with existing methods for improving data quality, such as [1], [2], [3] and explain in what ways their method is better or different?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strenghts:\n-The work introduces a novel framework to construct a dataset of accurate labeled data from a dataset containing inaccurate samples. The algorithm ensures that the quality of the training set does not degrade, as each iteration includes a validation step where the performance of the model is evaluated, and newly generated synthetic data are only accepted if they do not reduce performance.\n\n-It addresses a very important problem.\n\n-The consistency-check used to assess whether a synthetic label is sufficiently accurate does not require additional measurements on the training data, meaning it introduces no measurement overhead.\n\n-The authors validate their framework through experiments predicting physical quantities on systems such as the Heisenberg XXZ model and the one-dimensional cluster-Ising model.\n\n-The paper is well organized and the exposition is clear.\n\n\nWeaknesses:\n\n-The main weakness of the paper is that it does not include any theoretical analysis of the proposed algorithm. In particular, the authors do not discuss the computational cost of running AiDE-Q—for example, is a fixed number of iterations expected to be sufficient in more general cases? How should the algorithm scale with system size or data volume? Moreover, the effect of choosing different threshold values in the consistency check on the performance of the algorithm is not explored.\nWhy can we expect the algorithm will perform well at scale?\n\n-It is also not clear how the method is particularly suited for quantum property estimation problems. Specifically, why should the proposed method outperform existing techniques in the classical machine learning literature that also aim to enhance data quality? For example references [1,2,3] are not discussed or compared against in the paper.", "was_revised": true, "final_justification": "Upon discussion with the reviewers, I have changed my opinion for the necessity of further theoretical analyses. I was also encoureged by the additional experiments by the team. Overall my opinion of the paper has improved and I am raising the score to boarderline accept.\nHowever, given the high bar of NeurIPS, I believe the paper would have been significantly strenghtened with any attempt for a theoretical justification of the results. In particular it is not easy to assess the scaling of the method beyond still rather small sizes, and this critically impacts the real world value of this method.\nConsequently my grade is increased to boarderline accept.", "rebuttal": "We thank Reviewer Kvb1 for the valuable comments. As many of the points raised under **Weaknesses** and **Questions** overlap, we have consolidated similar concerns under unified headings for clarity, indexed as **W1–W5** and **Q1–Q5**. All concerns have been carefully considered and are addressed in detail below. We hope our responses adequately clarify the issues and help convey the merits of our work.\n\n**[W1 & Q1]: A key weakness is the lack of theoretical analysis, particularly regarding the computational cost of AiDE-Q. Can the authors estimate its cost for varying problem sizes?**\n\n**A1:** To provide a comprehensive response, we organize our reply around three key aspects.\n\n$\\bullet$ *Why does the submission lack theoretical analysis?* We respectfully note that in the context of *deep learning–based quantum property estimation*, it is conventional to validate proposed methods through extensive and systematic **empirical studies**, rather than through formal **theoretical guarantees** about computation cost. This practice is consistent with prior works, including Refs. [a–d].\n\nThe primary reason for this convention is that *deep learning theory typically offers only loose bounds and limited explanatory power regarding the superior performance of deep neural networks* [Refs. [e-f]]. Consequently, following the practices established in the deep learning community, research on deep learning–based quantum property estimation (**DL-QPE**) has predominantly focused on empirical results. In this regard, in the following response, we mainly focus on providing **empirical evidence** to address the reviewer's concerns.\n\n$\\bullet$ *Computational cost of AiDE-Q*. We would like to clarify that the computational cost considered in this field is primarily concerned with **the total number of queries to quantum computers**, given the limited availability of quantum resources compared to classical computing resources. In addition, as outlined in Lines 265-270 of our manuscript, the query cost is jointly determined by the size of the training dataset $n$ and the number of measurements $m$.\n\nTo better address the reviewer's concern,  the table below summarizes the **execution time** of AiDE-Q for various system sizes. Notably, the longest running time observed for the setting with N=50 is 3.6 hours, which indicates the efficiency of AiDE-Q.\n| N | 10 | 20 | 30 | 40 | 50 |\n|---|----|----|----|----|----| \n| Time (h) | 0.8 | 0.9 | 1.8 | 2.4 | 3.6 |\n\n$\\bullet$ *Theoretical analysis of AiDE-Q*. While providing a theoretical analysis is not central to this study, we nonetheless seek to address the reviewer’s concern by presenting a theoretical analysis of the **convergence of AiDE-Q**. We refer the reviewer to the response **A1** to Reviewer bVXi for details. Here we briefly summarize the achieved results. \n\nAssume the existence of a ground-truth model  $W^{\\*}$. Let $n_l$ and $n_u$ be the number of labeled and unlabeled data, and $W(0)$ be the model trained on labeled data. The trained model $W(t)$ after $t$ iterations of AiDE-Q yields \n$$||W^{\\*}-W(t)|| \\le [(1+c_1/\\sqrt{n_l} + c_2 /\\sqrt{n_u}) \\cdot c(1-\\hat{\\lambda})]^t ||W^{\\*}-W(0)\\|\\|,$$\nwhere $c_1,c_2,c$ are constants, and $\\hat{\\lambda}$ is a decreasing function of the variance $\\tilde{\\delta}$ of unlabeled data.\n\nFor QPE, the variance $\\tilde{\\delta}$ is often large due to a small number of measurements $m_u$ in the unlabeled data. Notably, a large variance of $\\tilde{\\delta}$ with $c(1-\\hat{\\lambda}) >1 $ could harm the prediction perfromance. By contrast, AiDE-Q can well suppress $\\tilde{\\delta}$,  where $c(1-\\hat{\\lambda}) <1$ leads to linear convergence. These results undermine the importance of using AiDE-Q to advance DL-QPE.\n\n**[W1 & Q2]: Is a fixed number of iterations sufficient in general? How should key parameters (such as iteration count) be selected to achieve a desired accuracy**\n\n**A2:**  To better address the reviewer’s concerns, we begin by briefly restating the main objective of our submission, followed by a detailed response to the two questions raised.\n\n$\\bullet$ *Main objective of our work*. We respectfully clarify that the primary focus of our study is to validate how AiDE-Q improves diverse DL-QPE models under **limited quantum query budgets**, rather than to analyze the runtime or iteration cost associated with its implementation.\n \n$\\bullet$  *Is a fixed number of iterations ... more general cases?* Our experimental results (**Table 4**) show that *AiDE-Q consistently improves the performance of nearly all DL-QPE models under the chosen hyperparameter settings*.  Moreover, the number of iterations can be **flexibly adjusted** to ensure reliable convergence in general cases, provided that the hyperparameters are applied consistently across different DL-QPE models.\n\n$\\bullet$ *How should the key parameters ... target accuracy?* We respectfully note that our key objective is not to identify optimal parameters to enable DL-QPE models to reach a specific target accuracy. As mentioned earlier, we aim to evaluate the relative performance gains brought by AiDE-Q under consistent training settings and limited quantum query budgets. To ensure a fair comparison, the hyperparameters used, including the number of iterations, are kept consistent for different backbone models. \n\n**[W1 & Q4]: How should AiDE-Q scale with system size or data volume? Why can we expect it will perform well at scale?**\n\n**A3:** As discussed in Response **A1**, we would like to re-emphasize that the conventions for studying the scalability in DP-QPE are through **systematic experiments** with varied qubit count $N$ and training dataset sizes $n$, as exemplified by Refs. [a-d]. \n\nIn line with these conventions, the scalability with respect to $N$ **has been studied** in our original manuscript, as illustrated in Lines 330-335 with experimental results shown in Fig. 4(a). The table below summarizes the main results, demonstrating that AiDE-Q achieves significant performance improvement with **linearly increasing $n$ and model parameters.**, thereby **validating its scalability**. \n\n| N | 10 | 20 | 30 | 40 | 50 |\n|---|----|----|----|----|----| \n| n| 720 | 1280 | 1860 | 2360 | 2740 |\n| Improvement of $\\mathrm{R}^2$ by AiDE-Q | 0.103 | 0.078 | 0.072 | 0.058 | 0.056 |\n\n**[W1 & Q2]: The effect of choosing different threshold values in the consistency check is not explored. Does it have a strong impact on performance and running time?**\n\n**A4:** To address the reviewer's concerns, we conduct additional experiments to show the effect of various hyperparameters of AiDE-Q on its performance, including the *the number of subsets for the consistency check $s$, the size of these subsets $|\\mathcal{I}|$, and the variance threshold for data selection $\\tau$*. \n\nThe table below presents the $\\mathrm{R}^2$ for varying $|\\mathcal{I}|$ and $\\tau$, and we refer the reviewer to Response **A2** for reviewer bVXi for a detailed introduction of the hyperparameter settings and experimental results.\n\n| $\\|\\mathcal{I}\\|=α_I m_u$ | α_I=1/8 | | | α_I=1/4 | | | α_I=1/2 | | |\n|--------|---------|---------|---------|---------|---------|---------|---------|---------|---------|\n| Threshold| τ=95% | τ=90% | τ=80% | τ=95% | τ=90% | τ=80% | τ=95% | τ=90% | τ=80% |\n| Improvement by AiDE-Q| **0.089** | 0.051 | 0.083 | **0.136** | 0.1 | 0.068 | 0.093 | **0.094** | 0.07 |\n\nThese results suggest that to maximize the power of AiDE-Q, the size of subsets $|\\mathcal{I}|$ and the number of subsets $s$ should be set moderately, and the threshold should be set to be relatively large.\n\n**[W2 & Q5]: It is also not clear how the method is particularly suited for QPE problems. Why should AiDE-Q outperform existing techniques in the classical ML literature, such as references [1,2,3]? Can the authors compare AiDE-Q with them?**\n\n**A5:** Let us separately address the reviewer's concerns. \n\n$\\bullet$ *The applicability of AiDE-Q to QPE.* To address the reviewer's concerns,  we begin by recalling that the proposed AiDE-Q is a **simple but effective** framework that iteratively enhances DL models. AiDE-Q's applicability to QPE stems from its **compatibility with existing DL models developed for QPE**, such as those in Refs [a-d]. Furthermore, AiDE-Q’s ability to enhance data quality is driven by the proposed consistency-check method, which is designed to identify high-quality data. The conducted experiments have **validated the effectiveness of AiDE-Q** in enhancing the prediction performance of various DL models for QPE.\n\n$\\bullet$  *The comparison with other classical ML methods for improving data quality.* Before addressing this concern, we would like to emphasize that AiDE-Q is the **first framework** specifically proposed to improve data quality for the task of QPE. This unique contribution is also highlighted in Reviewer bVXi's comments, where it is noted: *While the concept of a `data engine' or self-training with pseudo-labels exists in the broader machine learning literature, **its application to QPE is novel.*** \n\nFurthermore, since the reviewer **did not provide references [1-3]**, we are unable to clearly compare AiDE-Q with these mentioned methods. We would be happy to incorporate them and **conduct corresponding comparative experiments** in the discussion period, should the reviewer kindly provide them.\n\nLastly, we would like to highlight that a significant advantage of AiDE-Q is its **simplicity**, which allows it to be highly compatible with most existing DL models, thereby enhancing their performance.\n\n[a] Phys. Rev. Lett. 130, 210601 (2023).\n\n[b] Nat Commun 15, 8796 (2024).\n\n[c] \"Unsupervised pretraining of quantum property estimation and a benchmark\" ICLR (2024).\n\n[d] \"Semi-supervised learning of quantum data with application to quantum state classification\" ICML (2024).\n\n[e] \"Understanding deep learning requires rethinking generalization\" ICLR (2017).\n\n[f] Proc. Natl. Acad. Sci. U.S.A. 116 (32) 15849-15854 (2019)."}]

Author rebuttal:

Meta-review: The manuscript addresses the problem of limited datasets when training deep learning models to estimate the ground state properties of quantum systems. The proposed approach is based on synthetic data with the main technical contribution being a method for consistency-check. The resulting algorithm has been evaluated in a series of experiments with up to 50 qubits. The main strengths are the high relevance, the thorough evaluation, and the good presentation, while a lack of theoretical analysis is the main weakness. 
Only three reviews were provided, after rebuttal and discussion are all reviewers rating accept or BA. The reviewers basically agree on the strengths and weakness above, except for nKML who criticized a lack of noisy experiments instead of theoretical analysis..
Decision: Accept (poster)