TITAN: A Trajectory-Informed Technique for Adaptive Parameter Freezing in Large-Scale VQE

We sincerely thank Reviewer cy7n for the constructive feedback. We have carefully addressed the concerns raised under Weaknesses [W1–W5] and the specific Questions [Q1-Q5]. We hope the following responses effectively address the reviewer’s concerns and aid in the reassessment of our submission.

W1: Report the raw energy values or the difference as a percentage of the ground-state energy.

Reply: We have followed the reviewer's advice to exhibit how far TITAN and the baseline models deviate from the ground-state energy. For convenience, Table R9 summarizes the corresponding key findings. Please refer to Table R3 (reply to W1 for Reviewer 8G53) for details.

Table R9: Energy differences between Molecules with three initialization strategies and TITAN Freezing.

The achieved results in Table R9 provide the following insights:

1. For most cases, the baseline model [1] and TITAN are very close to the exact results;
2. For the molecule CO, TITAN achieves an additional energy reduction of approximately 0.015 while halving the number of parameters.
3. The slight discrepancy between TITAN and the exact results stems from the choice of ansatz. For example, in the 30-qubit TFIM experiment, TITAN can freeze 30% of the parameters while still achieving convergence comparable to the exact result, as shown in Table R10.

Table R10: Statistical performance of TITAN on diverse VQE benchmarks (Simplified version of Table R5)

[1] Zhang et al. Escaping from the barren plateau via Gaussian initializations in deep variational quantum circuits. NeurIPS (2022).

W2: No error bars are provided in Figure 2.

Reply: We would like to remind the reviewer that Figure 2 is intended solely to illustrate the 'parameter-freezing' phenomenon that motivates our study. For ease of visualization in the 3D plot, uncertainty bands were omitted. To support the illustration with statistical rigor, we have included detailed results in the Appendix, based on ten independent runs with different random seeds.

In addition, we would like to emphasize that all performance figures already include statistical variation in the submission, i.e., Figs. 5-7, as well as newly added results (Table R1-R11).

Regarding the concern "the number of frozen parameters increases ..., if this is a novel insight", we believe this is indeed a novel observation. To the best of our knowledge, the freezing phenomenon has not been explicitly characterized in prior work. Notably, this novelty was also acknowledged by Reviewer 8G53. Moreover, this observation serves as the foundation for TITAN, whose effectiveness has been systematically validated through extensive simulations.

W3: Shouldn’t the Hamiltonian itself be an input to the TITAN model training?

Reply: We would like to emphasize that whether the Hamiltonian itself should be encoded as input to TITAN depends on the nature of the Hamiltonians under consideration. In our submission, we focus on Heisenberg and TFIM models, which share a common Hamiltonian structure and therefore offer limited additional information to encode. As such, we do not include the Hamiltonian structure as input.

However, we fully agree that for more complex lattice systems, e.g., triangular, hexagonal, or long-range interacting graphs, the Hamiltonian encodes rich structural information that may strongly influence parameter freezing behavior [1–2]. We have added a discussion of this direction to the updated version as a promising avenue for future work.

[1] Yang et al. Scalable variational Monte Carlo with graph neural ansatz.

[2] Roth et al. Group convolutional neural networks improve quantum state accuracy.

W4: Is it appropriate to freeze them statically at initialization?

Reply: We would like to emphasize that the choice between static and dynamic parameter freezing is not unique, as each strategy has its own merits and trade-offs. In particular, a dynamic freezing strategy may incur significant computational overhead, as updating the freeze mask at each iteration requires re-measuring gradients and re-synchronizing the optimizer. Accordingly, this cost scales linearly with both the number of parameters and the shot budget.

We consider TITAN's static approach a practical first step, supported by strong empirical results (e.g., Table R5). That said, dynamic or hybrid strategies remain promising directions for future work, now noted in the revised manuscript.

W5: In most cases, the number of parameters frozen seems to be very low.

Reply: We acknowledge that our current presentation may not have sufficiently highlighted the statistical advantages of TITAN over baseline methods, which might have contributed to the reviewer's impression that the improvements are marginal. To address the reviewer's concern, in the following, we briefly introduce the main reasons.

1. Regarding the marginal improvement in Table 1. The number of frozen parameters is highly dependent on $\tau$ and Hamiltonian. With the results in Table R9, it is notable that for CO and N2, more than 50% parameters are frozen.

2. As elucidated in Table R5, TITAN often attains an evident improvement for In-Distribution tasks. Even for domain-generalization tasks, TITAN can provide notable enhancement, enabling a broader applicability.

3. As shown in Table R6, the benefit of TITAN increases rapidly as problem instances grow larger.

Q1: How were 100-qubit circuits simulated—were they Clifford-only?

Reply: We would like to point out that efficient simulation of large-scale quantum circuits, those exceeding 100 qubits, is an active area of research. Recent advances in tensor-network simulators and Pauli propagator techniques [Sci. Adv.10,eadk4321(2024)] have enabled the simulation of specific classes of non-Clifford circuits. For instance, [1] demonstrates the simulation of quantum convolutional neural networks with up to 1,024 qubits.

[1] Bermejo et al. "Quantum convolutional neural networks are (effectively) classically simulable."

The main challenge in evaluating TITAN beyond 30 qubits lies in the demands of VQE optimization. While large-qubit circuits can be simulated, existing simulators are not yet efficient enough to support full optimization workflows, i.e., collecting the rich gradient trajectories required for training. Any progress in this direction would broaden TITAN’s applicability.

Q2: Would a different type of TITAN model need to be trained for different types of ansatz? An alternative to HEA?

Reply: TITAN demonstrates strong domain generalization capability. As shown in Table R5 (response to W1 for Reviewer 27vu) and Table R11, in domain generalization tasks, TITAN is trained on a family of Hamiltonians using HEA and then applied (without retraining) to structurally distinct ansätze such as SU2 and SEL by directly predicting their frozen parameters without increasing final energy. For an anisotropic Hamiltonian, which is more complex due to the coupling strengths differing between directions, TITAN consistently attains better energy estimations even when parameters are frozen, underscoring its domain generalization ability.

Table R11 (Simplified Table R2): Performance of $\Delta E$ on domain generalization tasks.

Q3: What is the overall overhead of training and all the measurements that need to be done for training?

Reply: We would like to accord Table R7 (Q2 for Reviewer 27vu) to address the reviewer's concern. Once TITAN is optimized, no further training is required. As the number of test samples increases and the scale becomes larger, TITAN's benefits become more significant.

Q4: Any interesting trends?

Reply: We observe that the freezing phenomenon exhibits problem-dependent patterns. In Figure 4 (main text) and Figures 6, 7, 9, 12, 16, and 17 (Appendix), the distribution of frozen parameters varies across different ansätze and Hamiltonians. To capture these variations, we propose TITAN as a data-driven solution that adaptively selects freeze candidates, enabling it to generalize effectively across HEA, SU2, and SEL circuits (see the reply of Q3 for Reviewer 27vu ).

Q5: Dataset & cross‑ansatz generalization

Reply: Let us address the reviewer's concerns in sequence.

1. TITAN does not require a large dataset as referred to in the response to W1 for Reviewer 27vu.
2. We further evaluated the HEA‑trained model outside the training distribution, as referred to in the response to W2 for Reviewer dadQ.

Dear Reviewer,

Thank you very much for your positive update and for taking the time to review our additional results. We appreciate your constructive feedback throughout the process!

Thank you!

We sincerely thank Reviewer 27vu for handling our submission and providing constructive feedback. We have carefully addressed the concerns outlined under Weaknesses [W1], specific Questions [Q1-Q3], and Limitations [L1-L2]. We hope the responses below clarify our contributions and support a more informed reassessment of our submission.

W1: (1) The added implementation complexity of TITAN does not seem to yield significant performance gains over simpler baselines. (2) TITAN’s benefits appear inconsistent across settings, making it unclear when it will be effective. (3) The results lack clarity in highlighting TITAN’s advantages, and there is insufficient discussion of training set size and runtime overhead.

Reply: To ensure clarity, we address the reviewers’ three concerns in sequence below.

(1) TITAN does not seem to yield significant performance gains over simpler baselines. We acknowledge that our current presentation may not have sufficiently highlighted the statistical advantages of TITAN over baseline methods, which might have contributed to the reviewer's impression that the improvements are marginal. To address this, we have reorganized the simulation results in the main text and now present the supporting statistics in Table R5.

Table R5: A summary of the statistical performance of TITAN. DG and ID refer to Domain Generalization and In-Distribution, respectively. Isotropic means equal couplings; anisotropic means direction-dependent couplings Hamiltonian. Energy difference $\Delta E$ is defined in Table R1.

Sub-Table R5: VQE benchmarks for molecules (i.e., H2 , HF, LiH , BeH2, H2O, N2, CO).

We would like to clarify that the tasks in Table R5 fall into two broad categories: in-distribution and domain generalization tasks. The in-distribution tasks involve predicting frozen parameters for unseen Hamiltonians within the same family, using a fixed ansatz. In contrast, the domain generalization tasks evaluate TITAN’s ability to transfer across ansätze: the model is trained on a family of Hamiltonians using the HEA, and then applied to structurally distinct ansätze (i.e., SU2 and SEL) without retraining, by directly predicting their frozen parameters. While domain generalization is inherently more challenging, it demonstrates TITAN’s broad applicability better, enabling a single model to generalize across a wide range of VQE tasks.

The achieved results in Table R5 indicate that TITAN achieves substantial improvements over baseline methods on in-distribution tasks. Specifically, more than 4% parameters are saved while maintaining more accurate ground energy, and saved parameters are up to an average of 36% for molecule tasks in Sub-Table 5. Even for domain generalization tasks, it provides notable improvements over baseline models that lower the ground energy with an average of 3.5% parameters saved.

(2) It is unclear when TITAN will be effective. The results in Table R5 indicate that an optimized TITAN model can benefit a wide range of VQEs across diverse ansätze and Hamiltonians. This broad applicability allows TITAN to reduce measurement overhead, with its advantages compounding as the number of inference-time Hamiltonians increases.

A concrete example is advancing quantum phase classification [1]. For an $N$-qubit TFIM, we need to sweep $n_J$ couplings and $n_h$ field parameters to determine the critical point. As a result, the total number of TFIMs to be explored is $n_{grid} = n_J n_h$. For each TFIM, the number of Pauli terms is $K = 2N-1$. Assume the employed ansatz contains $P$ trainable parameters and the parameter-shift rule is used, which requires two circuit evaluations per parameter for each Pauli term. For each evaluation, $10^3$ shots are required. Taken together, the measurement cost for each TFIM is $M_0 = 2KP_\theta = 2\times 10^3 (2N-1) P$ $. For$n_{grid}$TFIMs, the vanilla VQE takes$ M_{tot} = n_{grid} M_0 = 2\times 10^3 (2N-1) n_J n_h P$ for a single iteration. We show the quantitative analysis in Table R6 below, which exhibits the benefit of TITAN compounds rapidly as problem instances grow larger.

Table R6: Baseline measurement cost for HEA ansatz in 1D-TFIM phase classification when layer number is $l = 6$ with 5% parameters freezing for TITAN. Here we only consider a single iteration.

[1] Herrmann et al. Nat Commun 13, 4144 (2022).

(3) What are the advantages of TITAN? How about its training set size and runtime overhead?

The required training dataset size and runtime overhead of TITAN are summarized in Table R7. Following explanations in W1 (2), the computation cost in the training is ignorable, particularly when amortized over a growing number of inference-time Hamiltonians. Moreover, TITAN’s advantages compound as the number of such inference tasks increases.

Table R7: Training details of TITAN. torch: 2.5.1+cu121| NVIDIA GeForce RTX 3060 12 GB

Q1: Provide a single plot summarizing the typical expected improvement.

Reply: Due to figure upload restrictions, we present the results in Table R5. As noted in our response to W1 (1), TITAN achieves an average parameter freeze of 3–5% on most in-distribution tasks while maintaining the accuracy of the minimum energy solution. Notably, for larger molecules or large-scale VQE instances, the freeze ratio can reach up to 60%. For domain generalization tasks, TITAN can still provide meaningful improvements.

Q2: (1) Specify training details used for TITAN. (2) Were the training models the same as those tested? If they were unrelated or smaller in scale, it would further support the generalizability of the approach.

Reply: Let us address the reviewer's two concerns in sequence.

Concern (1). As suggested, we have included additional implementation details of TITAN and have provided the code and dataset in the supplementary materials. For illustration, Table R7 outlines the data provenance, network hyperparameters, and training time used.

Concern (2). The reviewer's understanding is correct. As shown in Tables R2 and R5, TITAN exhibits strong domain generalization across ansätze. Specifically, the model is trained on a family of Hamiltonians using HEA, and is then successfully applied (without retraining) to predict frozen parameters for structurally distinct ansätze such as SU2 and SEL, achieving competitive performance.

Q3: How does TITAN compare to a heuristic that freezes early layers with higher probability?

Reply: We would like to clarify that the distribution of frozen parameters is problem-dependent. While Figure 4 illustrates a tendency for frozen parameters to concentrate in the shallow layers when using HEA on isotropic Hamiltonians, this pattern is not universal. As shown in Figures 6, 7, 16, and 17 of the Appendix, the distribution varies substantially across different ansätze and problem Hamiltonians.

To better address the reviewer's concern, we compare TITAN against the Early-Layer Freezing (ELF) heuristic, as suggested. The achieved results, as summarized in Table R8, indicate that the TITAN freezing strategy shows faster and more thorough gradient decay at all qubit scales (N=5–30). While ELF beats plain Random, it still trails TITAN, and the gap widens as circuit size increases, around 4 times higher than TITAN in gradient norm and 3.1% higher final energy.

Table R8: Comparison of gradient norm and final energy between TITAN and ELF for 30-qubit TFIMs.

L1 & L2: Discuss training cost and problems related to optimizing VQAs.

Reply: We have added a discussion on the training cost of TITAN; please refer to our response to Q1 for details. In addition, we have included more references in the updated manuscript related to VQA optimization, covering topics such as barren plateaus, measurement reduction, and the local minima landscape.

Dear Reviewer,

Thank you very much for your feedback and for updating your scores positively! Your thoughtful comments are important in refining our manuscript, and we greatly appreciate the time and care you have devoted to the review process.

Thank you again for your constructive support!

We sincerely appreciate Reviewer dadQ for the thoughtful and constructive feedback. We have carefully addressed the points raised under Weaknesses [W1-W3], and the specific Questions [Q1]. We hope the following responses help clarify our contributions and assist in the reviewer's reassessment of our submission.

W1: How does the threshold $\tau$ affect the performance of TITAN?

Reply: Thanks for the comments. We fully agree with the reviewer's perspective that exploring the varied threshold settings can offer deeper insights into TITAN’s effectiveness. For this reason, in the original manuscript, we reported how the varied thresholds (i.e., $\tau \in$ {50, 60, 70, 80, 90}) affect the performance of TITAN, as elaborated in Tables 3-5 (pages 19-23, Appendix). The key observation is that as the threshold $\tau$ decreases, a larger proportion of parameters are frozen; however, this comes at the cost of reduced energy estimation accuracy.

To better address the reviewer’s concern, we conducted additional simulations to support Table 1 in our manuscript. These experiments follow the same settings as in Table 1, with the only difference being the variation of the threshold values.

The achieved results are shown in Table R1. In particular, as $\tau$ is reduced from 90 to 50, the number of frozen parameters rises from at most $5$–$11$ per layer to more than half of the total, and higher ground state energy solution. These observations confirm that the proposed method is sensitive to $\tau$ in a predictable, monotonic fashion: lowering the threshold freezes more parameters and yields increasingly positive (i.e., worse) $\Delta E$, while excessively low thresholds offer limited additional improvement in single testing.

Table R1 (Supplementary Table 1) The performance of TITAN with the varied threshold $\tau$. The explored Hamiltonians are a family of 1‑D nearest‑neighbor Heisenberg models. The performance is evaluated by the energy difference, i.e., $\Delta E= E_{TITAN} - E_{Gaussian}$, where each $E$ denotes the estimated energy returned by TITAN-frozen Gaussian initialization and with standard Gaussian initialization, respectively.

In the updated version, we have incorporated the above clarification into Section 4 (“Sensitivity to $\tau$”) and reproduced Table 1 to make the relationship between threshold and performance explicit.

W2: The color in Figure 6 is faint for Random and TITAN.

Reply: We have followed the reviewer's advice to redraw Figure 6 using a higher‑contrast and colour‑blind‑safe palette, and increase both line weight and marker size to ensure clear differentiation in the revision.

W3: How does TITAN complement the zero initialization, and what does Figure 5 mean?

Reply: To fully address the reviewer's concerns, in this reply, we first show how Titan complements with other initialization methods, followed by elucidating the settings adopted in Figure 5 (page 8).

$\bullet$ Complementary with other initialization methods. We would like to clarify that TITAN complements Zero and other initialization strategies in a manner analogous to how Gaussian and Uniform initialization methods operate, as demonstrated in Figure 5 (Page 8). Specifically, TITAN predicts in advance which parameters of a given ansatz are likely to be frozen, before any initialization. After this step, any standard parameter initialization strategy (e.g., Zero, Gaussian, Uniform) can be applied to the remaining trainable parameters. We also showed how TITAN complements the Layer-wise Greedy Gradient Descent (LGGD) [1] method in Appendix Page 9. All explanations are appended to the revised manuscript.

[1] Grimsley, Harper R., et al. "An adaptive variational algorithm for exact molecular simulations on a quantum computer." Nature Communications 10.1 (2019): 3007.

$\bullet$ Detailed settings in Figure 5. We would like to first remind the reviewer that the purpose of Figure 5 is to evaluate the domain generalization ability of TITAN. Specifically, TITAN is trained on a family of isotropic Hamiltonians using the hardware-efficient ansatz (HEA), and then tested on two structurally distinct ansätze, SU2 and SEL, by predicting their frozen parameters. This setup allows us to assess whether a single TITAN model, trained on one ansatz, can generalize to others, thereby demonstrating its broader applicability across different quantum circuit architectures.

The experiment setup adopted in Figure 5 is as follows. For each ansatz (SU2 and SEL), we evaluate three commonly used initialization strategies, i.e., Gaussian, zero, and uniform, and examine how TITAN refines their corresponding freezing parameters. A purely random freezing mask is also included as a baseline for comparison. Five experimental settings are considered: $(N,L)=(5,5),(5,15),(10,10),(15,5),(15,15)$, where $N$ and $L$ refer to the qubit count and the layer number, respectively.

To make the outcome transparent, we have extracted the raw data from Figure 5 and present it as the Table R2. An observation is that in almost every setting, TITAN yields lower ground energies than the random freezing, confirming its effectiveness. Moreover, the low estimation error of TITAN on both SEL and SU2 validates its domain generalization capability, which is crucial for enabling robust and transferable performance across diverse ansatz architectures.

Table R2: Evaluating TITAN’s domain generalization across distinct ansätze (restatement of Figure 5). The explored Hamiltonians refer to a family of 1‑D nearest‑neighbor Heisenberg. TITAN is trained with the HEA ansatz and tested on the SEL and SU2 ansätze. Gaussian initialization is adopted. The definition of $\Delta E$ follows the same meaning in Table R1.

Q1: Are shallow layers more prone to parameter freezing? Is this related to the gradient vanishing problem of classic NN?

Reply: For clarity, we address the reviewer's two concerns in turn in this response.

$\bullet$ Phenomenon of frozen parameters. We would like to emphasize that the distribution of frozen parameters is problem-dependent. While Figure 4 shows that frozen parameters tend to concentrate in the shallow layers when applying HEA to isotropic Hamiltonians, this behavior is not universal. As shown in Figures 6, 7, 9, 12, 16, and 17 of the Appendix, the distribution of frozen parameters varies significantly across different ansatzes and problem Hamiltonians.

Due to the problem-dependent nature of parameter freezing, we propose using a deep learning model to learn and capture this behavior. As demonstrated in Section 4 and the corresponding Appendix figures, TITAN generalizes well across diverse settings, significantly reducing measurement costs while preserving the final ground-state energy.

$\bullet$ Relation with vanishing gradients in classical neural networks. TITAN is conceptually related to certain phenomena in classical NN, but not to vanishing gradients. In classical NN, vanishing gradients refer to the issue where shallow layers receive very small gradients, making them hard to train. In contrast, the frozen parameters identified by TITAN, as evidenced in Figures 6, 7, 9, and 17 of the Appendix, do not consistently appear in shallow layers and are not caused by gradient flow issues.

A more relevant analogy is neural network pruning. In classical NN, some neurons may become inactive depending on the task or dataset. This is similar to our observation that certain VQE parameters remain inactive (i.e., frozen) depending on the Hamiltonian and ansatz. This connection may inspire further research into sparsity and parameter selection across classical and quantum learning models.

[1] Hochreiter, Sepp. "The vanishing gradient problem during learning recurrent neural nets and problem solutions."

[2] Pascanu R, Mikolov T, Bengio Y. On the difficulty of training recurrent neural networks International conference on machine learning. Pmlr, 2013: 1310-1318.

Dear Reviewer dadQ,

We sincerely thank the reviewer for the positive feedback and for confirming that all earlier concerns have been fully addressed!

We acknowledge the comment regarding information density relative to the page limit, and we will streamline the main text in the camera-ready version by relocating extended tables and explanations to the appendix.

These edits will preserve clarity while ensuring the core contributions remain easily accessible. We appreciate the reviewer’s time and constructive input!

We sincerely appreciate Reviewer 8G53 for the thoughtful and constructive feedback. We have carefully addressed the points raised under Weaknesses [W1-W3], and the specific Questions [Q1]. We hope the following responses help clarify our contributions and assist in the reviewer's reassessment of our submission.

W1: Can this method efficiently achieve accurate results for larger molecules?

Reply: We agree with the reviewer's viewpoint to understand the scalability of TITAN. To address this issue, we have conducted extensive experiments to study the scalability of Titan in the original submission. In particular, the results in Figure 7 (Page 9) showed that TITAN could be transferred to models other than those trained up to 30 qubits and still maintain the same performance level as the baseline.

Table R3: Energy differences with the ground truth across different molecules over three initialization strategies and TITAN Freezing.

To further address this issue, we explore the capability of Titan in seven quantum chemistry molecules under frozen threshold $\tau = 80$ in Table R3. For CO, which involves 12 qubits, TITAN succeeds in freezing more than $60\%$ of those parameters while maintaining or improving the energy estimate.

The challenge towards larger molecules is the lack of efficient simulators for large-qubit circuits with optimization. A detailed explanation in Reply to Q1 for Reviewer cy7n. In words, with the development of tensor network [7] and CUDA-Q [8], computing resource limitations will be better resolved, and the applicability of TITAN can be further broadened.

We appended the discussion to the revised manuscript.

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[2] Lathiotakis N N et al. A functional of the one-body-reduced density matrix derived from the homogeneous electron gas: Performance for finite systems[J]. The Journal of chemical physics, 2009, 130(6).

[3] Goldberg M C, Riter Jr J R. Confirmation of the predicted Hartree-Fock limit in BeH2[J]. The Journal of Physical Chemistry, 1967, 71(9): 3111-3112.

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[5] Chachiyo T, Chachiyo H. Understanding electron correlation energy through density functional theory[J]. Computational and Theoretical Chemistry, 2020, 1172: 112669.

[6] Zhang K et al. Escaping from the barren plateau via gaussian initializations in deep variational quantum circuits[J]. Advances in Neural Information Processing Systems, 2022, 35: 18612-18627.

[7] Kulshrestha et al. Qadaprune: Adaptive parameter pruning for training variational quantum circuits.

[8] Schieffer et al. Harnessing CUDA-Q’s MPS for Tensor Network Simulations of Large-Scale Quantum Circuits.

W2: The frozen parameter is innovative, but the model architecture shows limited novelty.

Reply: We appreciate the reviewer for the positive affirmation about TITAN. In fact, we have systematically explored the effect of different architectures. The main conclusion is that: Any reasonably expressive backbone can be plugged into the pipeline and will learn the same gradient‑suppression pattern, confirming that the framework is genuinely model‑free. In order to help readers better understand this work, we weakened the discussion about neural architecture.

Table R4: $\Delta E$ (Frozen Parameters) for a 1‑D nearest‑neighbor Heisenberg chain using the HEA ansatz, comparing different TITAN backbone architectures under frozen‑parameter evaluation.

To better address the reviewer's concerns, we retrained all substitutes—Transformer [1], EfficientNetV2‑S [2], and U‑Net [3] using the Adam optimizer with a fixed learning rate of $1\times10^{-3}$ for $1,000$ epochs, keeping batch size, loss function, and data augmentations identical to the default convolutional backbone (ResNet-18 in Table R4) in Table R4, where ResNet-18 based TITAN (model size 46 MB), Transformer based (345 MB), EfficientNetV2‑S based (90 MB), and U‑Net based (124 MB).

Table R4 shows that every backbone yields energy differences $\Delta E$ that are comparable to the baseline TITAN model (ResNet-18), while freezing a similar fraction of parameters. The same trend holds across all qubit counts: swapping the backbone changes the numeric value by at most $2\times10^{-1}$, yet the qualitative behavior—larger $\tau$, more frozen parameters, and lower measurement cost with minimal loss in accuracy remains intact.

We have added the new ablation study and accompanying discussion (Section 4) to highlight this architectural agnosticism and to clarify that the contribution of the paper is orthogonal to the choice of neural backbone in the revised manuscript.

[1] Vaswani A, Shazeer N, Parmar N, et al. Attention is all you need[J]. 2017, 30.

[2] Tan M, Le Q. Efficientnetv2: Smaller models and faster training[C]. PMLR, 2021: 10096-10106.

[3] Ronneberger O, Fischer P, Brox T. U-net: Convolutional networks for biomedical image segmentation[C], 2015: 234-241.

Dear Reviewer,

Thank you very much for your encouraging feedback and for maintaining your positive score! We appreciate the time you took to review our additional experiments. We are grateful for your constructive engagement throughout the review process!