AiDE-Q: Synthetic Labeled Datasets Can Enhance Learning Models for Quantum Property Estimation

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.

[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?

A1: To provide a comprehensive response, we organize our reply around three key aspects.

$\bullet$ 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].

The 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.

$\bullet$ 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 $m$.

To 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.

$\bullet$ 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.

Assume 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 $||W^{\*}-W(t)|| \le [(1+c_1/\sqrt{n_l} + c_2 /\sqrt{n_u}) \cdot c(1-\hat{\lambda})]^t ||W^{\*}-W(0)\|\|,$ where $c_1,c_2,c$ are constants, and $\hat{\lambda}$ is a decreasing function of the variance $\tilde{\delta}$ of unlabeled data.

For 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.

[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

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.

$\bullet$ 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.

$\bullet$ 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.

$\bullet$ 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.

[W1 & Q4]: How should AiDE-Q scale with system size or data volume? Why can we expect it will perform well at scale?

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].

In 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.

[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?

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 $\tau$.

The 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.

These 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.

[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?

A5: Let us separately address the reviewer's concerns.

$\bullet$ 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.

$\bullet$ 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.

Furthermore, 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.

Lastly, 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.

[a] Phys. Rev. Lett. 130, 210601 (2023).

[b] Nat Commun 15, 8796 (2024).

[c] "Unsupervised pretraining of quantum property estimation and a benchmark" ICLR (2024).

[d] "Semi-supervised learning of quantum data with application to quantum state classification" ICML (2024).

[e] "Understanding deep learning requires rethinking generalization" ICLR (2017).

[f] Proc. Natl. Acad. Sci. U.S.A. 116 (32) 15849-15854 (2019).

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.

[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?

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].

Although 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].

$\bullet$ 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., $||W(t) - W^{\*}|| \le [(1+c _1/\sqrt{n _l}+c_2/\sqrt{n _u})\cdot c (1-\hat{\lambda})]^t \cdot ||W(0) - W^{\*} |\|,$ where $n_l,n_u$ 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.

$\bullet$ 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 $c(1-\hat{\lambda}) >1$.

By 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.

$\bullet$ 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.

[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

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.

$\bullet$ 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.

The 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.

$\bullet$ 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.

1. 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.

2. 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 $s=3,5,7$. | Method | s=3 | | | s=5 | | | s=7 | | | |--------|-----|-----|-----|-----|-----|-----|-----|-----|-----| | | 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 | | SL | 0.723 | 0.722 | 0.779 | 0.75 | 0.754 | 0.721 | 0.774 | 0.73 | 0.739 | | SL 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 |

3. 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 $s=3,5$.

The 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.

[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.

A3: To better address the reviewer's concerns, let us first recap AiDE-Q and then provide a detailed explanation.

$\bullet$ 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 $\mathcal{S} _h(t+1)$.

$\bullet$ 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.

In 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.

We 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.

[W4]: Is it a fixed increment, or does it depend on the magnitude of the performance drop?

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.

[a-f] Refer to the response for Reviewer Kvb1 [g] ArXiv:2201.08514. ICLR 2022

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.

[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?

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.

$\bullet$ 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 $\tilde{\rho}= \mathcal{N}_q[\rho] = (1-q)\rho + q \frac{\mathbb{I}}{2^N},$ where $0\le q\le 1$ 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 $\tilde{\mathcal{S}} _{\text{hyd}}=\tilde{\mathcal{S}} _{L}\cup \tilde{\mathcal{S}} _U, \text{with } \tilde{\mathcal{S}} _{L}=\\{ \tilde{\mathbf{x}}^{(i)}|\tilde{\mathbf{o}}^{(i)}\in\mathbb{R}^{Nm_l} \\} _{i=1}^{n_l} \text{ and } \tilde{\mathcal{S}} _{U}=\\{\tilde{\mathbf{x}}^{(i)}|\tilde{\mathbf{o}}^{(i)}\in\mathbb{R}^{Nm_u}\\} _{i=n_l+1}^{n_l+n_u},$ where $\tilde{\mathbf{x}}^{(i)}=(\mathbf{p}^{(i)},\mathbf{M}^{(i)}, \tilde{\mathbf{o}}^{(i)})$ 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.

$\bullet$ 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 $q\in\\{0.1,0.3,0.5\\}$.

The 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 $m_u \in \\{2^6, 2^7\\}$.

$\bullet$ 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'.

Note: Best results are shown in bold and second-best results in italics.

An 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 $m_u$.

[W2]: I didn't really spot many weaknesses outside of a typo or two. ''Basic of quantum computing'' should be ''Basics of quantum computing''

A2: Thank you for this feedback. We will conduct a thorough proofreading of the entire manuscript and correct all typos in the revised version.