PALQO: Physics-informed model for Accelerating Large-scale Quantum Optimization

We sincerely thank Reviewer 2QJF for the constructive feedback, which has been instrumental in improving the quality of our submission. To ensure clarity and better align with the points raised under “Weaknesses,” we have marked the Weakness as W1,W2,W3, the Questions as Q1–Q5 and Limitation as L1 accordingly. We hope our detailed responses offer clear clarification and assist in the continued evaluation of our work.

[W1, Q1 & Q2]: The reviewer questions whether PALQO truly preserves quantum advantage, noting that if the classical PINN scales with VQE and simulates parameter updates, it may undermine the need for a quantum VQE.

Response: We thank the reviewer for raising the important point regarding the potential impact of our method on quantum advantage.

The VQE remains a key focus in the field, as it currently lacks provable advantages on practical tasks but holds significant potential for near-term quantum computers [1,2]. While a definitive demonstration of quantum advantage is still on the horizon, recent there are still various effort to highlight the potential and progress of VQE [3,4]. Besides, recent work [5] shows that under certain type of ansatz and input state, there is the evidence of super-polynomial quantum speedups in quantum chemistry are theoretically achievable on near-term hardware [5].

$\bullet$ $\underline{\text{Preserves quantum advantage.}}$ We would like to clarify that PALQO does not operate independently of the VQE framework, nor does it diminish or negate the potential for quantum advantage. As explained in our response to W2, the PINN serves as a classical tool to analyze the training dynamics, but it does not replace the core quantum components that give VQE its power, such as the problem-agnostic ansatz and quantum state preparation.

The PALQO framework is broadly applicable to various quantum algorithms, including the Quantum Approximate Optimization Algorithm (QAOA). While methods like PALQO aim to predict model parameters, QAOA is still believed to offer quantum advantage for solving certain classically hard problems—such as the Low Autocorrelation Binary Sequences (LABS) problem [6,7].

$\bullet$ $\underline{\text{Need for a quantum VQE.}}$ Yes, it requires quantum VQE. The core misconception is that the PINN "simulates" the VQE in a way that replaces it; instead, it only models the high-level dynamics of the VQE's classical parameters, a task for which it is entirely dependent on the quantum processor. The crux of the matter lies in the calculation of the energy gradient, which dictates the optimization path.

[W2,Q3-Q5]: A related concern is whether VQE retains any inherent quantum advantage in this context. Additionally, it is important to understand how the size of the PINN model scales relative to that of the VQE—is the scaling exponential, or if not, what kind of scaling is necessary to maintain its effectiveness?

Response: We thank the reviewer for the insightful comment.

$\bullet$ $\underline{\text{Retain inherent quantum advantage.}}$ The VQE retains its inherent quantum advantage in this context because the PINN operates purely on the classical side of the hybrid algorithm. The PINN does not attempt to simulate this complex quantum process; instead, it simply provides a powerful classical framework to learn and predict the optimization process by leveraging the gradient data generated by the VQE.

$\bullet$ $\underline{\text{The scale of PALQO.}}$ We would like to clarify that the PINN used to model the training dynamics of VQE avoids exponential scaling, as it operates in the space of classical parameters rather than the exponentially large quantum Hilbert space.

To further support the above claims, we conducted scaling tests of PALQO on the transverse field Ising model (TFIM) using a hardware-efficient ansatz (HEA) across various system sizes, ranging from 4 to 36 qubits. The number of paramters $p$ scales linearly with the system size $n$, i.e. $p\sim O(n)$. In the following table, we present the speedup ratio of PALQO (with PINN) over vanilla VQE when converging to the ground state, under varying PINN width and depth. Specifically, we linearly increased the PINN width (from 5𝑝 to 50𝑝) and depth (from 2 to 6). In the left three columns of the figure, we fixed the depth at 2 and varied the width. We observed that even with linear increases in width, PINN consistently delivered stable speedup as the system size increased. Moreover, the speedup ratio steadily improved with wider networks. In the right three columns, we fixed the width at 20𝑝 and varied the depth. Again, PINN maintained stable acceleration across all tested depths. These scaling results suggest that PINN achieves strong performance with only linear increases in width and depth, avoiding the need for exponential scaling.

W3: There is a very high variance in speedup ratios of PALQO, which adversely affects its stability and reliability, especially as the number of qubits and number of ansatz layers (i.e., the size of the VQE) is varied. It is not exactly clear why this is happening.

Response: The observed high variance in the speedup ratio may be attributed to the following factors:

1. Like vanilla VQE, PALQO is sensitive to the choice of initialization. Because the VQE energy landscape is highly non-convex with many local minima, different initial parameters can lead to significantly different optimization paths, contributing to greater performance variance.

2. As shown in Experiment II (Fig. 2), PALQO generally requires significantly fewer optimization steps than vanilla VQE. For example, to achieve the same target accuracy of $\Delta \leq 0.1$, vanilla VQE may take up to 120 steps, while PALQO—depending on initialization—often converges in just 3 to 4 steps. This results in speedup ratio of 40 and 30, respectively, which largely accounts for the high variance observed.

3. Additionally, as shown in Figure 3 of the original manuscript, the case with $J/h=2$ is more challenging because it has a smaller energy gap between the ground and first excited states compared to $J/h=0.5, J/h=1$. As a result, PALQO requires more optimization steps to remain competitive, which leads to a lower variance in the speedup ratio. Below, we present the iteration counts and variance of the speedup ratio for $J/h=0.5,1$ and $J/h=2$ in the following table.

L1: The paper includes one sentence about the limitations of this work in the conclusion, which discusses a potential future research direction of this work could be the reduce the high variance in speedup ratios of PALQO as the number of qubits and layers is varied. Other limitations discussed above are not mentioned in the work.

Response: We appreciate the reviewer's suggestion. In the revised manuscript, we expanded the discussion of our work's limitations beyond the high variance in speedup ratios, noting that achieving high accuracy often requires substantial effort in tuning the PINN.

[1] Tilly J, Chen H, Cao S, et al. The variational quantum eigensolver: a review of methods and best practices[J]. Physics Reports, 2022, 986: 1-128.

[2] Cerezo M, Arrasmith A, Babbush R, et al. Variational quantum algorithms[J]. Nature Reviews Physics, 2021, 3(9): 625-644.

[3] Google AI Quantum and Collaborators*†, Arute F, Arya K, et al. Hartree-Fock on a superconducting qubit quantum computer[J]. Science, 2020, 369(6507): 1084-1089.

[4] Kim Y, Eddins A, Anand S, et al. Evidence for the utility of quantum computing before fault tolerance[J]. Nature, 2023, 618(7965): 500-505.

[5] Leimkuhler O, Whaley K B. Exponential quantum speedups in quantum chemistry with linear depth[J]. arXiv preprint arXiv:2503.21041, 2025.

[6] Zhou L, Wang S T, Choi S, et al. Quantum approximate optimization algorithm: Performance, mechanism, and implementation on near-term devices[J]. Physical Review X, 2020, 10(2): 021067.

[7] Shaydulin R, Li C, Chakrabarti S, et al. Evidence of scaling advantage for the quantum approximate optimization algorithm on a classically intractable problem[J]. Science Advances, 2024, 10(22): eadm6761.

Dear Reviewer 2QJF,

We hope this message finds you well. Thanks for your detailed and constructive feedback which has been invaluable in shaping our work, and we deeply value your insights. We would like to kindly ask whether our latest responses adequately addressed your concerns and clarified the points raised in your previous question. If there are any remaining concerns or areas where further clarification might help, we would be more than happy to address them in the spirit of collaboration and continuous improvement.

Best regards, Authors

Q3: I'm not certain what to do with the LCE example.

Response: The reviewer mentioned that why we not optimize the loss function entirely classically, our answer is we hope but we can not.

The LCE is the example tell that even avoiding barren plateaus (might inherently lead to classical simulability, thus limiting the opportunities for quantum advantage which often hinders large-scale VQA), there is still quantum advantage.

The LCE is a classical technique that can modify the first order Clifford structure of Parameterized Quantum Circuits (PQCs), which ensures constant gradient scaling. However, in such loss landscape areas, they proved that there is no classical simulation technique is known to exist that can efficiently surrogate this landscape region.

Therefore, in such cases, no classical method can effectively optimize the loss function, making PALQO as a strong candidate. Unlike classical simulators that attempt to emulate VQAs directly, PALQO relies on quantum hardware to estimate gradients and operates in the parameter space rather than the full quantum Hilbert space for training and prediction. Besides, PALQO can substantially reduce the quantum resources required to optimize these models, making it feasible to experimentally explore their scalability and practical advantages, and enabling large-scale experimental validation of these methods.

Dear Authors,

Thank you for the crisp and clear answers.

(1) "Even though the introduction of PALQO can diminish the quantum advantage of VQE (reducing it from $T$-step to $T'$-step advantage), it does not necessarily eliminate it entirely." I agree with this statement completely. Thank you for the acknowledgement. It would be useful to include a brief discussion on this in the paper to clarify for the reader.

(2) Thank you for also providing some LCE results. They look good.

I have revised my score positively based on our discussion.

We sincerely appreciate the reviewer xXH5 for his/her constructive comments, which are invaluable to improve the quality of our submission. For clarity and to better align with the statement in Weaknesses, we have reordered the questions [Q1-Q5] accordingly. We hope our detailed responses provide helpful clarification and support the reviewer’s continued evaluation of our work.

[W1 & Q1] The paper lacks sufficient qualitative discussion of PALQO, i.e., how it addresses key challenges in VQAs, such as barren plateau (BP).

Response: Thanks for the comments. In the remainder of this response, we follow the reviewer’s suggestion to discuss the key challenges of VQAs and the connection between BP and PALQO. All explanations have been incorporated into Appendix B of the revised version.

$\bullet~\underline{\text{Key challenges of VQAs}}$. Optimization efficiency remains the central obstacle to scaling VQAs. On one hand, increasing the number of qubits and circuit depth can lead to the emergence of BP, making gradient-based optimization increasingly difficult. On the other hand, due to the no-cloning theorem (Lines 27-28), VQAs must update parameters sequentially to minimize a predefined cost function, which limits the ability to parallelize training. In both cases, the number of measurements required during optimization becomes prohibitively expensive for large-scale VQAs.

To address this, a variety of approaches have been developed in recent years to improve the optimization efficiency of VQAs at scale. In tackling BP problems, common strategies include ansatz design [1] and informed initialization [2]. To reduce measurement overhead, typical methods, as reviewed in Appendix B, include measurement grouping, warm-start techniques, and learning-based optimization.

This work primarily focuses on the latter category. For this reason, the original version does not elaborate extensively on BP issues, we have append BP review in the updated version. Notably, these approaches are often complementary and can be integrated to improve overall performance.

$\bullet~\underline{\text{Relation between BP and PALQO}}$. PALQO is complementary to approaches aimed at mitigating barren plateaus, as both seek to improve the optimization efficiency of VQAs at scale.

To be concrete, consider an example where a BP-free ansatz is applied to estimate the ground state energy of a large molecule such as $Fe_2S_2$. In such cases, the VQE cost function typically decomposes into over 22300 Pauli terms that must be measured individually. To obtain an accurate energy estimate, each of these terms must be sampled 10³ to10⁴ shots, to suppress quantum statistical noise. Even in the absence of BP, assume optimization runs for 1000 steps, the total number of required measurements becomes prohibitively large ($22300×10^3×1000=2.2×10^{10}$). In this context, PALQO plays a crucial role in significantly reducing the measurement cost and shortening the overall runtime, making the optimization process more tractable for large-scale systems.

[W2 & Q2] It remains unclear how PALQO differs from existing methods. How is the reduction in shot count theoretically justified?

Response: Let us address the reviewer's concern in sequence.

As for the concern "How PALQO differs from existing methods", we follow established conventions in the field [3,4,8,9] by conducting systematic numerical simulations across varying problem sizes and types, comparing the performance of PALQO against several baseline models. Notably, unlike prior studies, our work demonstrates heuristic results for systems with up to 40 qubits, marking, to our knowledge, the first such demonstration at this scale.

As for the concern "How is the reduction in shot count theoretically justified?", we address this from two perspectives.

$\bullet$ Recall that it remains a long-standing challenge to provide strong theoretical guarantees for deep learning–enhanced VQEs, as most prior works [3,4,5,6] offer primarily heuristic results. While Corollary 3.1 in the submission does not fully resolve this issue, it represents a meaningful first step toward theory-driven algorithm design in this emerging direction.

$\bullet$ Corollary 3.1 and the measurement cost $O(pN_H/\epsilon^2)$ characterize different aspects of the algorithm and are not directly comparable. In particular, the results presented in Corollary 3.1 offer theoretical insight into how the number of training examples influences the generalization error of PALQO. As is common in deep learning theory, such bounds are often loose, and the amount of training data required in practice tends to be significantly smaller [7]. As illustrated in Figure 6, PALQO achieves competitive performance even with a limited number of training samples, which is the $\tau$ time-step trajectory data where we set $\tau$ equal to $2$ and $4$, demonstrating its advantage over vanilla VQAs in the measurement cost.

In contrast, the cost $O(pN_H/\epsilon^2)$ refers to the measurement overhead incurred during each iteration of VQE optimization and is independent of the use of PALQO. Typically, the number of the Pauli terms $N_H$ is much larger than $p$, which makes the total measurement cost more strongly influenced by $N_H$ rather than $p$ in quantum chemistry. Thus, PALQO is built for reducing such tremendous measurement costs.

[W2 & Q4] The paper lacks sufficient discussion on the key factors driving PALQO's superior performance.

Response: PALQO advances baseline models by leveraging PINN to emulate the optimization dynamics of VQEs, incorporating partial derivative information as constraints during training. In the following, we illustrate the advantages of PALQO by integrating results from the original submission with additional experiments conducted in response to the review.

We further evaluate PALOQ with variants where specific loss terms are removed, and compare its performance with LSTM and QUACK, as follows:

As can be seen, when constraints $L_{p1}$ and $L_{p2}$ are removed, PALQO degrades into a basic model that merely includes time series information in its input. Its performance is inferior to specialized time series models such as LSTM, and clearly lags behind models like QUACK, which are designed for fitting sequences. These outcomes underscore the critical importance of the PDE residual term as a loss function within the model.

[W3 & Q3] The validity of the proposed approach is unclear. Can PALCO accurately predict learning dynamics?

Response: To address the reviewer's concern, we have conducted the following experiments to illustrate that PALCO can indeed accurately predict learning dynamics.

We conducted simulations on H2, BeH2, and TFIM models. The parameter initialization of the PALQO algorithm is consistent with the main text. In particular, we sampled ten sets of initial parameters for both the VQE and PALQO. After optimization, we compared the deviation of the parameters obtained by VQE and PALQO by calculating their Euclidean norm as shown in the following Table.

These results demonstrate that PALQO can accurately predict learning dynamics, achieving a low average deviation. Except for BeH2, PALQO shows very small parameter deviations across models, strongly confirming its accuracy and reliability in predicting optimized parameters.

[W3 & Q5] Is this method applicable only to quantum optimization?

Response: Thanks for the comment. While using PINNs to learn training dynamics can be extended to deep learning models, we would like to emphasize that PALCO is particularly well-suited for VQE optimization tasks. The reasons are as follows:

1. In deep learning, gradients can be efficiently computed via backpropagation. By contrast, the optimization of large-scale VQAs is very challenging, as explained in the reply of [W1 & Q1]. As a result, PALQO can provide substantial advancement to improve optimization efficiency.
2. The number of parameters $p$ in VQAs is typically much smaller than that in deep neural networks. In PALQO, the width of the PINN is set to $O(cp)$, where $c$ grows linearly with the number of qubits. Accordingly, PALQO remains reasonably sized, and the training complexity is significantly lower than that of applying the same framework to conventional deep learning models.

[1] Park C Y, et al. Hamiltonian variational ansatz without barren plateaus[J]. Quantum, 2024

[2] Zhang K, et al. Escaping from the barren plateau via gaussian initializations in deep variational quantum circuits[J]. NeurIPS, 2022

[3] Di L, et al. Quack: accelerating gradient-based quantum optimization with koopman operator learning. NeurIPS, 36, 2024.

[4] Guillaume V, et al. Learning to learn with quantum neural networks via classical neural networks. arXiv preprint arXiv:1907.05415

[5] Cervera-Lierta A, et al. Meta-variational quantum eigensolver: Learning energy profiles of parameterized hamiltonians for quantum simulation[J]. PRX Quantum, 2021

[6] Karim A, et al. Fast and Noise-aware Machine Learning Variational Quantum Eigensolver Optimiser[J]. arXiv:2503.20210.

[7] Jiang Y, et al. Fantastic generalization measures and where to find them[J]. arXiv:1912.02178, 2019.

[8] Meng F, et al. Conditional Diffusion-based Parameter Generation for Quantum Approximate Optimization Algorithm[J]. arXiv:2407.12242

[9] Khairy S et al. Learning to optimize variational quantum circuits to solve combinatorial problems[C]//AAAI. 2020

[Q6&Q7] In Figure 6, PALQO appears to achieve better results when trained on fewer data points. Could you explain why this happens? Is it common practice within the PINN community to train PINNs using only 2 or 4 data points? From a conventional machine-learning perspective, that sample size seems extremely small.

Response: Thank you for the comments. We address the reviewer’s concerns from two perspectives. First, we clarify the number of training examples used in each setting of Figure 6. Then, we provide intuition for why PALQO is able to perform well with a limited number of training examples.

The number of training examples in Figure 6. We follow the definition of $\tau$ and the implementation details of PALQO provided in our response to [W2 & Q2] to clarify the number of training examples used by PALQO in Figure 6.

Recall that for PALQO, the training and data recollection process is repeated $m$ times (typically $m<10$ in practice) before the PALQO loss function converges. The lengend in Figure 6 only exhibit the adopted number training examples of at each iteration. The employed number of iterations and the total number of training examples are summarized in the following table. We appologize for this misleading and have revised the manuscript to explain the total number of training examples in the caption of Figure 6.

Why is PALQO able to perform well with fewer training examples?. We emphasize that the number of training examples required at each iteration (repeated $m$ times) depends on the complexity of the optimization trajectory associated with the specific Hamiltonian–ansatz pair, which determines how difficult it is for PALQO to learn the underlying dynamics.

To provide some intuition, we refer to Ref. [1], which shows that with a well-chosen initialization, the local loss landscape can be approximately convex. This structure allows PALQO to accurately learn the optimization trajectory using only a small number of training examples. As further supported by the numerical results appended in our response to [W2 & Q4], the loss constraints $L_{P1}$ and $L_{P2}$ guide PALQO’s predicted trajectory to closely match the ground-truth optimization path.

[1] Puig, Ricard, et al. "Variational quantum simulation: a case study for understanding warm starts." PRX Quantum 6.1 (2025): 010317.

[W1 & Q1] I understand that this study focuses on reducing the number of measurements, which complements efforts to address the barren plateau problem. However, given that training difficulties and scalability are intrinsic challenges in VQE, a significant limitation of this work is that it does not contribute to resolving these fundamental issues.

Response: We appreciate the reviewer’s observation regarding the scope of our contribution in relation to the broader challenges in VQE, particularly the barren plateau (BP) problem. Below, we address this from two complementary perspectives:

Scopre and Positioning

• We fully acknowledge that the BP phenomenon represents a fundamental obstacle to the scalability of VQE, and we agree it remains one of the most critical open problems in the field. A recent Nature Reviews Physics article [1], provides a comprehensive survey of existing progress and highlights the importance of exploring both algorithmic opportunities and hardware-aware constraints in VQA design.

• While our work does not directly resolve the core difficulty of BP, we believe it still offers meaningful value. As emphasized in Ref. [1], reducing the need for large-scale heuristic implementations is itself an important goal for near-term quantum computing. By significantly lowering the number of measurements required during training, PALQO enables more practical and resource-efficient evaluation of VQE models.

• Furthermore, recent studies have proposed VQA variants that are provably BP-free and may exhibit quantum advantages. PALQO reduces the experimental burden needed to validate and scale such promising approaches, thereby expanding the scope of what can be feasibly tested on near-term devices.

Complementarity to BP-Free Variational Models

To further clarify the contributions of PALQO, we offer concrete examples showing how it can strengthen the practicality and scalability of recent BP-free approaches:

• Smart initialization Methods: The recent two papers [2,3] ensure that the variational landscape exhibits substantial gradients near a good initialization point. PALQO can dramatically reduce the measurement cost required to exploit this local structure, thereby enabling large-scale experimental validation of these methods.

• Trainability Beyond Classical Simulability: The work [4] introduces a Linear Clifford Encoder (LCE) that guarantees constant-gradient scaling within near-Clifford regions. As in previous examples, PALQO can reduce the quantum resource demands required to optimize such models, making it feasible to experimentally explore their scalability and practical advantages.

In summary, while PALQO does not directly solve the barren plateau problem, it addresses a key complementary bottleneck: measurement overhead. As with prior AI conference papers on AI-enhanced VQEs [5-10], while the results may not justify a strong acceptance on their own, they offer novel insights for both the quantum machine learning and variational quantum computing communities. We hope the reviewer will find that despite some limitations, PALQO offers a meaningful contribution, making it a valuable addition to the conference program.

[1] Larocca, M., Thanasilp, S., Wang, S. et al. Barren plateaus in variational quantum computing. Nat Rev Phys 7, 174–189 (2025).

[2] Zhang, Kaining, et al. "Escaping from the barren plateau via gaussian initializations in deep variational quantum circuits." Advances in Neural Information Processing Systems 35 (2022): 18612-18627.

[3] Puig, Ricard, et al. "Variational quantum simulation: a case study for understanding warm starts." PRX Quantum 6.1 (2025): 010317.

[4] Meyer, Sabri, et al. "Trainability of Quantum Models Beyond Known Classical Simulability." arXiv preprint arXiv:2507.06344 (2025).

[5] Luo, Di, et al. "Quack: Accelerating gradient-based quantum optimization with koopman operator learning." Advances in Neural Information Processing Systems 36 (2023): 25662-25692.

[6] Dai, Zhongxiang, et al. "Quantum bayesian optimization." Advances in Neural Information Processing Systems 36 (2023): 20179-20207.

[7] Nicoli, Kim, et al. "Physics-informed bayesian optimization of variational quantum circuits." Advances in Neural Information Processing Systems 36 (2023): 18341-18376.

[8] Qian, Yang, et al. "MG-Net: Learn to Customize QAOA with Circuit Depth Awareness." Advances in Neural Information Processing Systems 37 (2024): 33691-33725.

[9] Wu, Huanjin, Xinyu Ye, and Junchi Yan. "Qvae-mole: The quantum vae with spherical latent variable learning for 3-d molecule generation." Advances in Neural Information Processing Systems 37 (2024): 22745-22771.

[10] Ostaszewski, Mateusz, et al. "Reinforcement learning for optimization of variational quantum circuit architectures." Advances in neural information processing systems 34 (2021): 18182-18194.

[W2 & Q2]： Whether PALQO offers a true measurement reduction, given that collecting the required $O(p)$ training examples entails $O(p) \times (measurements per example)$, which may be comparable to the O(pN_H) measurements needed by vanilla VQE to achieve the same error tolerance—potentially eliminating any asymptotic advantage.

Response: Thanks for the comments. We would like to clarify that PALQO indeed offers a measurement reduction. The potential confusion may stem from the fact that the measurement cost for PALQO, both in the theoretical result of Corollary 3.1 and in its implementation, is expressed over $T$ optimization steps. In contrast, the measurement cost of vanilla VQE is typically reported per iteration. To clarify this distinction, we summarize the measurement reduction offered by PALQO in the table below, where the notation $m$ is the number of the training and data recollection of PINN and $\tau$ refers to the number of samples used in each training session of the PINN. (Recap: in practice, PALQO adopts a sliding-window approach thus it will recollect the data and retrain before finding ground state)

We next elucidate the required number of measurements for each approach.

Vanilla VQE. The reviewer's analysis of the required number of measurements for vanilla VQE is correct. That is, the measurement cost yields $O(TpN_H/\epsilon^2)$, as each optimization step involves estimating gradients using $O(pN_H/\epsilon^2)$ measurements.

PALQO. For clarity, let us first recap the goal of PALQO. Given a specified Hamiltonian and ansatz, PALQO aims to predict the optimization dynamics of VQE using physics-informed neural networks (PINNs). Training the PINN amounts to learning and mimicking the trajectory of parameter updates for the corresponding vanilla VQE. Once trained, PALQO can predict future optimization steps entirely on the classical side, without requiring additional quantum measurements. Consequently, quantum measurements are only needed during the training phase, while inference proceeds fully classically.

Building on this understanding, we next elucidate the meaning of Corollary 3.1 and clarify the measurement requirements for implementing PALQO.

$\underline{\text{Interpretation of Corollary 3.1}}$. Recall that Corollary 3.1 provides a generalization bound of PALQO in terms of the number of training examples $\tau$. More specifically, $\tau$ corresponds to the optimization trajectory, i.e., the number of VQE iterations from which gradient information is collected via quantum hardware. Each training example in $\tau$, corresponding to the acquisition of gradient information at a single iteration, requires $O(p N_H / \epsilon^2)$ measurements.

The objective of Corollary 3.1 is to show that the generalization error of PALQO can be bounded. The derived bound depends on the number of parameters $p$, the number of training samples $\tau$, as well as the depth $L$ and width $W$ of the PINN. This corollary is not directly related to measurement reduction.

$\underline{\text{Measurement requirements for implementing PALQO}}$. Let us briefly recap the implementation of PALQO. To improve practical performance, PALQO adopts a sliding-window approach, similar to the strategy used in QuACK. Once the predicted optimization trajectory converges, additional data samples are collected from the quantum hardware, and PALQO is retrained using the updated dataset. This procedure is repeated iteratively $m$ times until the loss function of PALQO converges.

We now analyze the required number of measurements of PALQO. Suppose the training and data recollection process is repeated $m$ times (typically $m<10$ in practice) before the PALQO loss function converges. Now for each time, we assume $\tau=O(p)$ training examples are employed to train PALQO, where $O(pN_H/\epsilon^2)$ measurements are adopted per example. Then, the total number of measurements for PALQO is $O(mp^2N_H/\epsilon^2)$.

Consequently, the reduction in total measurement cost over $T$ steps is approximately $O((T-mp)pN_H/\epsilon^2)$. In practice, since $T\gg m*p$, PALQO provides a substantial measurement reduction.

[1] Luo, Di, et al. "Quack: Accelerating gradient-based quantum optimization with koopman operator learning." Advances in Neural Information Processing Systems 36 (2023): 25662-25692.

Q8: The primary concern continues to be scalability. Specifically, the theoretical justification provided for the reduction in the required number of measurements remains insufficiently strong, and the empirical validation presented in the manuscript is limited to relatively small-scale training data. Consequently, it is still unclear how the proposed methodology could be effectively scaled to handle larger and more complex circuits (beyond classical limitations) in practical scenarios.

Response: Thank you for the comments.

1. We agree the reviewer's comments that the previous theoretical analysis of measurement reduction provided an estimate of the scaling behavior, rather than a tight bound. To obtain a more precise estimation, we need to analyze the number of iterations $T$ required for VQE optimization. However, this remains a challenging unsolved open problem, as $T$ depends on various factors, including the choice of initialization, the optimization algorithm, and the specific ansatz etc. Thus, it is hard to have an accurate measurement reduction estimation in current, unfortunately.

2. In the original manuscript, we evaluated the scalability of PALQO through experiments on quantum systems ranging from 12 to 36 qubits. Specifically, we assessed the speedup of PALQO compared to vanilla VQE in reaching the ground state. The results, presented in the table below, show that PALQO consistently achieves satisfactory speedup performance.

From the results shown in the table, it is evident that the number of training samples required by PALQO does not grow exponentially with system size. Moreover, we observe that the introduction of physical constraints allows PALQO to accurately predict the optimization trajectory over the local loss landscape using only a small number of training samples.

3. For cases that cannot be classically simulated, we would like to clarify that PALQO does not compute the gradient information in VQA through classical simulation. Instead, this gradient information is obtained directly from quantum hardware. As a result, the dimension and sample size of PALQO’s training data are significantly smaller than those required for classically simulating a quantum system that scales exponentially as $2^n$. In contrast, the dimension of the training data depends only on the number of parameters $p$ in the variational quantum circuit. Typically, this number grows linearly or polynomially with the number of qubits, meaning that the size of PALQO’s training data also scales linearly or polynomially with system size. As PALQO does not operate in the full quantum state space; rather, it learns a trajectory in the parameter space, allowing it to bypass the exponential complexity associated with simulating the entire quantum system. Therefore, PALQO can effectively scaled to handle larger systems.

We sincerely thank Reviewer xXH5 for your time and for providing constructive feedback that has helped us improve our manuscript. We have carefully considered all comments and provide a point-by-point response. In light of these detailed responses and our planned revisions, we respectfully ask if the reviewer would be willing to reconsider your evaluation of our work.

We sincerely thank Reviewer KJmn for your constructive feedback and valuable suggestions. We have indexed the points mentioned under Weaknesses as W1 and W2, and Questions as Q1 to Q4. We hope our responses clarify the contributions and assist the reviewer in reassessing our submission.

W1: Whether the robustness of PALQO is due to its fixed [0,1] parameter initialization and suggests testing with varied schemes to validate generality.

Response: Thanks for the comment. To address the reviewer's concern, we perform extra experiments on different initialization strategy. Here we change the sampling from (0, 1) to ($-\pi$, $\pi$), and the result is as follows.

The results show that PALQO can still achieve good performance under various initialization schemes.

W2: The suggestion about broader statistical analysis to verify whether measurement reductions hold across various Hamiltonians and circuit depths.

Response: Thank you for your insightful suggestion to extend the statistical analysis of measurement reduction across different system sizes. We have followed the suggestion to add related experiments as blow.

This result shows that even in systems of different sizes and depths, PALQO can still effectively reduce measurement cost.

Q1: For the quantum chemistry benchmarks, do all cases achieve chemical accuracy (i.e., ~1e-3 kcal/mol)? If not, which systems fall short, and why?

Thanks for your comment. We apologize for not presenting the final model accuracy achieved after utilizing PALQO to accelerate optimization in the original manuscript. In all cases, the PALQO can achieve the chemical accuracy.

Q2: While the experiments extend up to 40 qubits, they are restricted to shallow circuits (e.g., 3-layer HEA). Given that such circuits may contain relatively few parameters within the effective light cone, how strong is the evidence for scalability? More discussion on this point would be helpful.

Response: We appreciate the reviewer's comment and suggestion. Here, we would like to clarify that even quantum circuits with shallow depth, when constructed from Clifford and T gates, can still be classically intractable. This is because such circuits evolve quantum states into regions of Hilbert space that lack the structure necessary for efficient classical simulation. We also followed the suggestion to add such discussion in revised manuscript.

Q3: In Line 111, the framework is evaluated using standard measurement protocols. Have the authors considered integrating classical shadows or other advanced measurement schemes? I am happy to see more discussions about this direction.

Response: Thanks for the suggestion. We agree that it would be beneficial to integrating the classical shadows or advanced measurement schemes to further enhance the performance. This aligns with our experiment on measurement grouping to reduce the number of distinct measurements, as presented in Section F.4 of the Appendix. We have followed this suggestion and extend the discussion of integrating other advanced measurement schemes to further improve the performance in F.4.

Q4: In Appendix F.3, while the noisy simulations are appreciated, it remains unclear whether the method maintains its predictive performance across varying noise levels. Evaluations under different noise profiles are necessary.

Response: Thanks for the comment. To address the reviewer's concern about the performance of PALQO under various noise level, we have added related experiments as follows.

In the main text, we only considered 5% depolarization noise. Here, we added 10% and 20% depolarization noise, as well as an Amplitude Damping Channel. The final results show that PALQO has strong robustness under different noise models.

We sincerely thank Reviewer KUyt for their constructive feedback and valuable suggestions. We have carefully addressed all concerns raised. For clarity, we index the points mentioned under Weaknesses as W1 and W2, and Questions as Q1. We hope our responses clarify the contributions and assist the reviewer in reassessing our submission.

W1: From my observations of Figure 2, in many cases PALQO has not converged to zero $\Delta E$. What are the causes of this?

Response: Thanks for the comments. To address the reviewer’s concern, we first clarify the purpose of the experiments in Fig. 2 and then provide additional simulation results demonstrating PALQO’s ability to converge to zero $\Delta E$.

$\bullet$ $\underline{\text{Purpose related to Fig. 2}}$. The primary goal of Fig. 2 is to show the performance advantage of PALQO. In particular, the evaluation metric is speedup ratio, which amounts to quantifying the ratio of the number of iterations required by vanilla VQE to that of the baseline method, rather than the energy difference $\Delta E$ (as the reviewer noted). Under this metric, the results in Fig. 2 indicate that PALQO consistently outperforms vanilla VQE, achieving up to a 30× speedup.

It is noteworthy that the evaluation of PALQO’s advantage over baseline models is not limited to a single metric. The experiments in Fig. 2 focus on performance comparison under a limited quantum resource budget (i.e., a fixed number of iterations), which reflects a common practical constraint.

$\bullet$ \underline{\text{Ability to converge to zero \Delta E}}. To better address the reviewer's concerns, we conducted additional simulations to demonstrate PALQO's ability to achieve zero $\Delta E$. In this setting, the evaluation metric is the energy difference from the exact ground-state energy after convergence, with no constraint on the number of iterations.

The experimental settings are as follows. We conducted simulations on a 12-qubit TFIM with HEA and a 14-qubit BeH$2$ with UCCSD ansatz. The hyperparameter settings for these are consistent with those detailed in the main text.

As shown in the above Table, once vanilla VQE is capable of converging to a small $\Delta$, then PALQO can do this either with the extra measurement reduction.

W2: The submission lacks the details regarding the experimental settings and fairness of the comparisons. Specifically, when comparing PALQO with other methods, what metrics are kept the same when reporting the required number of measurements for different methods?

Response: We highly agree with the reviewer's viewpoint, such that it is crucial to ensure fair comparisons. To this end, in the original submission, we adopt different metrics to systematically evaluate the capabilities and limitations of PALQO compared to other baseline models. For convenience, we summarize the adopted metrics and experimental settings of each figure in the following Table.

We use the above metrics under specific conditions to evaluate the performance of PALQO and other baseline models separately. The detailed settings are as follows:

1. Since both PALQO and the baselines can reach $\Delta E \sim 10^{-3}$, we fix the number of iterations to be 20 and compare the lowest $\Delta E$ each model can achieve within this limit. (left subfigure in each column of Figure 2).
2. By fixing $\Delta E \sim 10^{-3}$, we use the speedup ratio as a metric to evaluate PALQO’s improvement in training efficiency relative to the baselines (right subfigure in each column of Figure 2).
3. We calculate the total number of measurements throughout the entire optimization process until $\Delta E$ of the model approach $10^{-3}$ (Table 1).

Q1: I wonder if the PALQO will always converge for VQE. Can you provide a theorem guaranteeing the convergence on VQEs of the proposed method?

Response: We acknowledge the reviewer’s concern regarding the lack of strong theoretical guarantees for deep learning–enhanced VQEs, particularly in ensuring robustness and interpretability. Indeed, this remains a long-standing challenge in the field, as most prior works [1][5] offer primarily heuristic results. While Corollary 3 in our submission does not fully resolve this issue, we believe our work represents a meaningful first step toward theory-driven algorithm design in this emerging direction.

To better address the reviewer's concern, we briefly summarize the main results and provide a proof sketch showing that, in the extreme case, PALQO is guaranteed to converge for VQE. We would be happy to share additional proof details during the discussion session.

Assume that PALQO is applied to learn the optimization dynamics of a VQE model in the overparameterized regime with a sufficiently small learning rate $\eta$. Suppose further that the training dataset is sufficiently large and the employed PINN is also overparameterized. Under these conditions, deep learning theory guarantees that PALQO will exhibit a small expected risk $\epsilon_0$, and is therefore theoretically capable of accurately tracking the training dynamics of the overparameterized VQE.

The sketch of proof is as follows. According to the result of Theorem 1 (performance guarantee of optimization within the quantum neural tangent kernel (QNTK) approximation) in Ref [4], we have the $E(t)=(1-\eta K)^tE(0),$ where the scalar $E(t)$ refers to the loss at time $t$. Intuitively, the QNTK $K$ amounts to the inner product of the average gradient vector of VQE, i.e., $K=\sum_j(\partial_E/\partial_{\theta_j})^2$.

As the PINN is trained to emulate the optimization dynamics of VQE, the resulting gradient vector is expected to deviate only slightly from the ideal case. As a result, the QNTK related to PALQO, denoted by $K'$, has the relation $K'=K+\delta K,$ where $\delta K$ depends on $\epsilon_0$ and the number of trainable parameters. Meanwhile, the training dynamics of PALQO-enabled VQE yields $E'(t)=(1-\eta K')^tE(0).$ In this regard, we can bound the difference between estimated and ideal dynamics, i.e., $E'(t)$ and $E(t)$. That is, we have $\|E'(t) - E(t)\|\leq t\eta\|\delta K\|(\max(|1-\eta K|,|1-\eta K'|))^{t-1}|E(0)|.$ According to Theorem 3.5 in Ref [3] and our assumptions, $\delta K$ can be arbitrarily small by increasing the PINN width and depth, and $\max(|1-\eta K|,|1-\eta K'|)=1-\eta \min(K,K')<1$.

Taken together, the achieved results suggest that, given a sufficiently large number of iterations $t$ and ample training data, the optimization process guided by overparameterized PALQO can converge to the global minimum of the overparameterized VQE model.

[1] Di Luo, Jiayu Shen, Rumen Dangovski, and Marin Soljacic. Quack: accelerating gradient- based quantum optimization with koopman operator learning. Advances in Neural Information Processing Systems, 36, 2024.

[2] Guillaume Verdon, Michael Broughton, Jarrod R McClean, Kevin J Sung, Ryan Babbush, Zhang Jiang, Hartmut Neven, and Masoud Mohseni. Learning to learn with quantum neural networks via classical neural networks. arXiv preprint arXiv:1907.05415, 2019.

[3] Zeinhofer M, Masri R, Mardal K A. A unified framework for the error analysis of physics-informed neural networks[J]. IMA Journal of Numerical Analysis, 2024: drae081.

[4] Junyu Liu, Francesco Tacchino, Jennifer R Glick, Liang Jiang, and Antonio Mezzacapo. Representation learning via quantum neural tangent kernels. PRX Quantum, 3(3):030323, 2022.

[5] Guillaume Verdon, Michael Broughton, Jarrod R McClean, Kevin J Sung, Ryan Babbush, Zhang Jiang, Hartmut Neven, and Masoud Mohseni. Learning to learn with quantum neural networks via classical neural networks. arXiv preprint arXiv:1907.05415, 2019.

Dear Reviewer KUyt,

We hope this message finds you well. As the rebuttal deadline is approaching, we would greatly appreciate your valuable feedback. Your insights are highly important to us, and we are eager to address your comments thoroughly. We would like to kindly ask whether our latest responses have sufficiently addressed your concerns and clarified the points raised in your previous question. If there are any remaining issues or if further clarification would be helpful, we would be glad to provide additional explanations to support constructive dialogue and continued improvement.

Best regards, Authors