MG-Net: Learn to Customize QAOA with Circuit Depth Awareness

We sincerely appreciate your detailed evaluation and thoughtful feedback on our manuscript. We are pleased to hear that you found MG-Net to be a novel, innovative, and sound contribution, with strong theoretical backing and comprehensive performance evaluation. In the following, we separately address your concerns:

Q1: The “Quantum ... about the circuit depth.

A1: As explained in the Section "Motivation" of our Global Response, we consider the allowable circuit depth as the primary hardware constraint, a common issue for both early fault-tolerant and NISQ devices. The allowable circuit depth of a quantum device is closely related to the metric of maximum coherence time. To make our title clearer, we have followed your suggestion and rewritten the title as "MG-Net: Learn to Customize QAOA with Circuit Depth Awareness" and clarified the hardware constraints in the main text of the revised manuscript.

Additionally, MG-Net can be flexibly extended to other hardware constraints. Refer to the Section "Impact" of our global response for further details.

Q2: The paper ... the noise in the device.

A2: We appreciate your recognition of the soundness and presentation of our work. We address your concerns regarding the broader interest to the NeurIPS community and the consideration of device noise as follows:

1. Broader Interest to the NeurIPS Community. While the primary focus of our manuscript is on improving the QAOA to tackle complicated combinatorial optimization problems, the proposed method provides a general framework for automatically learning a hardware-oriented circuit generator that can balance performance and hardware resources. The methodologies and insights derived from this work have broader implications for the field of machine learning for physical sciences, which aligns with NeurIPS's emphasis on interdisciplinary collaboration. The principles of dynamically optimizing quantum circuits using deep learning can be applied to other quantum algorithms and hybrid quantum-classical approaches, making this work relevant to a wide audience interested in quantum technologies and their integration with machine learning. There is a growing number of papers related to quantum computing have been published at top AI conferences like NeurIPS, ICML and ICLR [Gao et al., 2023; Sidford et al., 2023; Wu et al., 2023; Tang et al., 2024; Patel et al., 2024; Lei et al., 2024]. Among these, QAOA is particularly appealing for solving combinatorial optimization problems.

2. Noise Settings. We acknowledge that our current manuscript does not explicitly consider noise constraints in quantum devices, as discussed in the conclusion of our manuscript. We address your concerns from two dimensions: the importance of fault-tolerant quantum research and the extension to noisy devices.

   • Importance of fault-tolerant quantum research. Conducting research within a fault-tolerant, noiseless framework is crucial for unlocking the full potential of quantum computing and achieving quantum advantages over classical methods, such as works on quantum learning theory and optimization acceleration [Larocca et al., 2023; Liu et al., 2024]. Our theoretical analysis on QAOA is situated at the same fault-tolerant context to provide practical insights for improving QAOA's performance to achieve quantum advantage within limited circuit depths in idealized settings.

   • Suitability for noisy devices. Our algorithm implementation MG-Net can adapt to device noise. Refer to the Section "Impact" of our Global Response for further details. The success of our approach within this framework sets the stage for future exploration of our model's applicability to noisy systems and real-device experimentation.

Q3: The paper does not ... in Table 1.

A3: Since only MG-Net includes a training phase, we have followed your advice to add inference time for each method shown in Table 2 of the newly uploaded pdf.

Q4: “Intial” in Figure 1.

A4: We have corrected this typo in the revised manuscript.

Q5: Add $d_{eff}$ after "effective dimension" to make it clearer in Figure 1.

A5: We have followed your advice to add $d_{eff}$ after "effective dimension" in the caption of Figure 1 of the revised manuscript.

Q6: It is not clear ... one-hot encoding?

A6: Thank you for your insightful question. It is indeed an interesting idea to replace position embedding with integer encoding or one-hot encoding. As mentioned in the "Contribution" section of our Global Response, MG-Net acts as an initial protocol and provides a flexible circuit-generation framework where model components can be conveniently replaced by advanced techniques. To address your concerns, we are conducting additional experiments to replace position embedding with one-hot encoding or integer encoding. Due to the limited rebuttal period, we will update the results as soon as they become available.

Q7: Why $J_{ij}$ is ... frustration?

A7: The choice of $J_{ij}$ values in our study is influenced by several considerations related to TFIM. The TFIM undergoes a quantum phase transition at the critical point $h/J=1$. To more comprehensively study the behavior of the system in different phases, we set $J\in [0.5, 1.5]$ and $h\in [0.1, 2]$. This range ensures that the value of $h/J$ can be smaller than, equal to and greater than 1, covering the critical point and both phases of the transition. We do not consider the case where $J$ is negative because we mainly focus on ferromagnetic interactions to avoid frustration.

Q8: Can this approach be extended ... preparation?

A8: Yes, our approach can be flexibly extended to other problems in quantum computing that use parameterized quantum circuits. Please refer to the Section "Impact" of our global response for further details.

We sincerely appreciate your thorough evaluation and thoughtful feedback on our manuscript. We are pleased that you recognized the strengths of MG-Net, including its cost-effective two-stage training strategy and the extensive numerical results that highlight its advantages, enhancing the practicality of QAOAs. In the following, we separately address your concerns:

Q1: The parameter grouping ... sections.

A1: To address the reviewer's concern, we would like to emphasize that the theoretical results are not established to completely guarantee the performance of MG-Net. Instead, the proposed MG-Net in our work is theory-inspired to improve the convergence by tailoring ansatz design with optimal parameter grouping strategy. To explain this point, we elaborate on the relationship between the theoretical results and proposed MG-Net algorithms.

We first recall our theoretical contributions stated in Global Response that the Theorem 3.1 only establishes the qualitative analysis for the effect of parameter grouping strategy on the convergence in the overparameterization regime, showing that parameter grouping strategy could potentially reduce the effective dimension with the reduction amount determined by the specific Hamiltonian. Moreover, it is difficult to quantitatively analyze the reduction of effective dimension enabled by the parameter grouping strategy for various problem Hamiltonians, as this reduction depends on the detailed graph structure of problem Hamiltonians which is hard to analyze and is case-by-case. In this regard, there is no explicit optimal grouping strategy for significant reduction of the effective dimension for general problem Hamiltonians. To address this problem, we propose MG-Net to exploite the power of deep learning to automatically tailor the ansatz design with optimal parameter grouping strategy to achieve better convergence.

Q2: The details regarding ... Constraints."

A2: We acknowledge that further clarification is needed regarding the integration of hardware constraints in the training of MG-Net in our manuscript. As explained in the Section "Motivation" of our Global Response, we consider the allowable circuit depth as the primary hardware constraint, a common issue for both early fault-tolerant and NISQ devices. In the following, we address your concerns by providing a detailed explanation of the hardware constraints referred to in our manuscript and the implementation of the training of the mixer generator with hardware information.

• Quantum hardware constraints. There are many properties of a quantum device that affect its capacity, such as qubit connectivity, noise level, and maximum coherence time. In our manuscript, the term "hardware constraint" refers to the allowable circuit depth of a quantum device, which is closely related to the metric of maximum coherence time.

• Hardware-information-aware training. When constructing the labeled training dataset, we collect the achieved approximation ratio of a mixer Hamiltonian with varying circuit depths. After the first-stage training, the cost estimator can precisely capture the intrinsic correlation between the circuit depth and the achievable cost, as verified in Lines 282-288 of our manuscript. In the second-stage training, this circuit depth information is incorporated by encoding it into the input features of the mixer generator. Guided by the circuit-depth-oriented cost estimator, the mixer generator can be trained to predict the optimal mixer generator according to the given hardware constraints, i.e., circuit depth.

• Extension to other hardware constraints. Although we only consider the allowable circuit depth as the hardware constraint in applying MG-Net to QAOA in our manuscript, our proposed method is a general framework to learn a hardware-oriented circuit generator. It can be flexibly extended to other hardware constraints. Refer to the discussion on impact in our global response for details.

To clarify, we have followed the reviewer's advice to clarify the hardware-constraint-aware training in the revised manuscript.

Q3: The numerical results comparison should include the Grover-Mixer method, as it also involves a unique design of the mixer Hamiltonian.

A3: The Grover-Mixer (GM) method uses Grover-like selective phase shift mixing operators based on the prepared equal superposition of all feasible states. GM is designed to perform particularly well for constraint optimization problems. However, in our manuscript, we utilize QAOA to solve Max-Cut and TFIM problems, which are both unconstrained optimization problems. When applying GM to these tasks, it is equivalent to the original QAOA. Therefore, we did not consider GM in our original manuscript.

To effectively compare our method with GM, we shifted our focus to a new constrained optimization problem. Due to the limited rebuttal period, we will update the results as soon as they become available.

Q4: Have the author considered and compared with other mixer Hamiltonian design, e.g. Grover-Mixer?

A4: Please refer to A3 for details.

Q5: Can MG-Net be adapted to other VQA problems?

A5: Yes, MG-Net can be flexibly adapted to handle other VQA problems. While the primary focus of our manuscript is on improving the QAOA, the proposed method provides a general framework for automatically learning a hardware-oriented circuit generator that can balance performance and hardware resources, based on the interdisciplinary collaboration of quantum science and artificial intelligence (AI). Many works that utilize deep learning techniques to assist the research of quantum algorithms have been successfully applied to tackle multiple VQA problems [Zhang et al, 2022; Wu et al, 2023; Fürrutter et al, 2024]. We kindly refer the reviewer to the discussion about extending our work to other problems in our Global Response.

Thank you for taking the time to review our manuscript. We value your insights and address your concerns as follows:

Q1: It is too simple ... candidate gate set.

A1: We respectfully disagree with the reviewer's perspective that designing the mixer Hamiltonian is too simple. As mentioned in the Section "Motivation" of our Global Response, We release all freedom of the mixer Hamiltonian, including operator type and parameter grouping. Directly designing the mixer Hamiltonian based on Pauli operators remains computationally challenging and our proposed method tries to overcome this challenge efficiently.

1. Structure of Mixer Hamiltonian: As described in Lines 176-189 and Eqn. (5) of our manuscript, designing a mixer Hamiltonian involves not only choosing appropriate operator types but also determining the parameter groups. This dual requirement adds complexity to the design process beyond simply selecting Pauli operators.

2. Challenges in Mixer Hamiltonian Design: As discussed in Lines 194-197 of our manuscript, the search space for both parameter correlation and operator types grows exponentially with the system size $N$ (i.e., scaling at $O(N^N)$ and $O(2^N)$, respectively). This exponential growth makes designing an effective learning method difficult, as directly training a model in a supervised learning paradigm may require an infeasible amount of training data to achieve high accuracy. MG-Net addresses these challenges by bypassing the need to directly seek the optimal parameter correlation strategy and operator type.

3. Additional Experiments on Extended Candidate Operator Type Set: To further support our claims, we have conducted additional experiments with an extended set of candidate operator types $\{X,Y,XX,YY\}$, achieving an approximation ratio $0.985\pm 0.003$. As demonstrated in Figure 1 of the uploaded pdf, MG-Net maintains its efficiency and high performance even when the complexity of the mixer Hamiltonian design is increased.

Q2: The ... overparameterization.

A2: We would like to emphasize that our primary contribution lies in the development of novel algorithms designed to tailor optimal ansatz. While our study includes theoretical results, they are not the central focus but rather provide the theoretical foundation for MG-Net. Please refer to the Global Response for details. Notably, the theoretical results presented are non-trivial in terms of both contributions and techniques.

Theoretical Contribution. Existing work on QNN convergence with symmetric ansatz does not address how specific ansatz design strategies affect symmetry or effective dimension. Our study qualitatively analyzes the relationship between the effective dimension and QNN convergence with various parameter grouping strategies in the context of QAOA.

Technical Aspect. Our theoretical techniques involve the novel concept of effective dimension and tools from representation theory, widely used in analyzing QNNs with symmetric ansatz in quantum optimal control and over-parameterization. While the notion of effective dimension is inspired by these fields, our techniques for analyzing its relationship with ansatz design are unique. We use representation theory to compare the effective dimension of QNNs with different parameter grouping strategies by examining their algebraic structures.

Q3: The experimental results are shaky ... parameters.

A3: We respectfully disagree with the reviewer's perspective that the experimental results are shaky. We address the reviewer's concerns one by one as follows:

1. Comparison with ma-QAOA: As demonstrated in Section 5 of our manuscript, our proposed MG-Net performs significantly better than ma-QAOA regarding convergence and approximation ratio. As the system scale grows from 6 to 64, the approximation ratio gap between our method and ma-QAOA increases from 1% to 96%. Besides, the mechanism for designing the parameter grouping strategy for MG-Net and ma-QAOA differs. MG-Net dynamically generates the optimal parameter grouping based on the given problem instance and allowable circuit depth, while ma-QAOA adopts a static non-grouping strategy to maximize the number of trainable parameters.

2. Number of Parameters: We kindly remind the reviewer to refer to Lines 295-303 and Figure 5(a) of our manuscript for a detailed discussion about the number of parameters of different methods.

3. Mixer Hamiltonian Design: We agree that the mixer Hamiltonian usually cannot be purely Pauli-Z due to the requirement of noncommutativity with the cost Hamiltonian. Therefore, we do not include Pauli-Z in the pool of operator types. However, the Pauli-Y operator is acceptable and can drive the quantum state toward the solution along a different and possibly more efficient path than the Pauli-X operator, as indicated in Figure 1(a) of our manuscript. This choice aligns with principles of counterdiabatic (CD) driving or shortcuts to adiabaticity (STA). Numerous works [Chandarana et al., 2022; Zhu et al., 2022] have verified the effectiveness of using Pauli-Y as a mixer Hamiltonian, as listed in Table 1 of the uploaded pdf. Additional experiments investigating the impact of randomly choosing mixer operator types from Pauli-X and Pauli-Y while maintaining the same number of tunable parameters indicate that mixer Hamiltonians with different operators and the same number of parameters behave differently, shown in Figure 2 of the uploaded pdf.

Q4: The whole ... mixer

A4: Our disagreement stems from the motivation and design of the mixer Hamiltonian in QAOA. Based on the elaboration in our Global Response and our response A1-3, we believe the review provides a clear understanding of our work. We are prepared to provide further explanations if necessary.

We thank the reviewer for recognizing the potential of the proposed method to enhance QAOA with practical utility. Your feedback is invaluable, and we address each of your concerns as follows:

Q1: In Section 3, there are several references mentioned that also study the convergence of VQAs, but the theoretical contributions of this work are somewhat unclear. This paper would benefit from a clear statement of how this work relates to the results of those existing papers.

A1: We have followed the reviewer's suggestions to state the theoretical contributions of our study in the main text and highlight the relation with existing papers in Appendix C. In the following, we will separately address these two concerns raised by the reviewer.

The relation with existing literature. As stated in the theoretical contribution of our Global Response, while the convergence of VQAs with symmetric ansatz has been established from various aspects, they did not consider how concrete strategies for ansatz design, specifically the parameter grouping, affect the symmetry degree or equivalently the effective dimension, and hence fail to apply to the problem setting in our manuscript.

Theoretical contributions. We qualitatively analyze the relation between the effective dimension and convergence rate of QNNs equipped with various parameters grouping strategies in the context of QAOA. The techniques used in our theoretical results are mainly the effective dimension and the tools of representation theory [Ragone et al., 2023], which have been widely used in existing literature to perform theoretical analysis for QNNs with symmetric ansatz. Please refer to the contributions discussed in Global Response for details.

Q2: Throughout this paper, $p$ is referred to as the circuit depth. Shouldn't it be the number of layers, which is not quite the same as the usually defined circuit depth (although they are proportional)?

A2: We agree that the terms "circuit depth" and "number of layers" are often used interchangeably but can have distinct meanings in different contexts. Circuit depth typically measures how many "layers" of quantum gates are executed in parallel, whereas the number of layers can either share this meaning or refer to the number of blocks with the same structure. In QAOA, $p$ specifically denotes the number of blocks, each consisting of a cost Hamiltonian and a mixer Hamiltonian, as defined in Eqn.~(1) of the manuscript. Following the reviewer's advice, we have revised the manuscript to ensure consistent and precise terminology.

Q3: For a given $H_C$, is the ansatz with partial grouping unique? I suppose not since there may be different generators of Per${H_C}$. If this is correct and the ansatz is not unique, then does Theorem 3.1 hold regardless of the choice of generators?

A3: For the reviewer's supposition, the answer is right. Namely, the ansatz with partial grouping for given $H_C$ is not unique. For the second concern, Theorem 3.1 indeed holds regardless of the choice of generators. In particular, Theorem 3.1 is established to perform the qualitative analysis about the effects of parameter grouping strategies on convergence. Multiple partial grouping strategies could exist to achieve the same effective dimension, which depends on the specific problem Hamiltonian. However, it is difficult to quantitatively determine the effective dimension and explicitly adopt the correct parameter grouping strategy for better convergence, as achieving this requires analyzing the complicated graph structure of the problem Hamiltonian case-by-case. In this regard, we delve into exploiting the power of deep learning and propose the MG-Net for automatically generating the ansatz design with optimal parameter grouping strategy.

Q4: It is stated that ... If not, what are the main limitations of QAOA that need to be overcome?

A4: As explained in the Section "Motivation" of our Global Response, we consider **the allowable circuit depth as the primary hardware constraint, a common issue for both early fault-tolerant and NISQ devices. To further address your concerns, we provide a detailed explanation of the rationale behind researching noiseless cases and discuss the suitability of our approach for noisy devices. Finally, we clarify the resource requirements of our approach.

• Importance of Fault-Tolerant Quantum Research: Conducting research within a fault-tolerant framework is crucial for unlocking the full potential of quantum computing and achieving quantum advantages over classical methods, such as works on quantum learning theory and optimization acceleration [Larocca et al., 2023; Liu et al., 2024]. Our theoretical analysis of QAOA is situated in the same fault-tolerant context to provide practical insights for improving QAOA's performance to achieve quantum advantage within limited circuit depths in idealized settings.

• Suitability for Noisy Devices: Although our theoretical analysis is based on noiseless settings, our algorithm implementation, MG-Net, does not impose limitations on device noise. Refer to the discussion on impact in our Global Response for details.

• Resource Efficiency: By dynamically generating mixer Hamiltonians tailored to specific problems and allowable circuit depths in practical hardware, MG-Net can improve the achievable approximation ratio of QAOA with small number of layers, making it more feasible for NISQ devices. As shown in Table 1 of our manuscript, our method achieves an approximation ratio of 0.96 for 64-qubit Max-Cut problems, which is much higher than other QAOAs with the same number of circuit layers.

We hope that this clarification will address the reviewer's concerns. We are prepared to offer more comprehensive responses if there are any further questions.

Dear Reviewers (cc Authors),

This is the message sent by the authors for global response with an uploaded PDF file. The readers contain "reviewers submitted", so you should be able to see the response message and the PDF file. Please let me know if you cannot see them.

Best wishes, AC