Title: PALQO: Physics-informed model for Accelerating  Large-scale Quantum Optimization
Venue: NeurIPS 2025
Accepted: yes
Keywords: quantum optimization

physics-informed machine learning

machine learning
Abstract: Variational Quantum Algorithms (VQAs) are emerging as leading strategies with the potential to unlock practical applications and deliver significant advantages in the investigation of many-body quantum systems and quantum chemistry. 
A key challenge hindering the application of VQAs to large-scale problems is rooted in the no-cloning theorem in quantum mechanics, precluding standard backpropagation and leading to prohibitive quantum resource expenditure such as measurement cost.
To address this challenge, we reformulate the training dynamics of VQAs as a non-linear partial differential equation and propose a novel protocol that leverages physics-informed neural networks (PINNs) to model this dynamical system efficiently. Given a small amount of training trajectory data collected from quantum devices, our protocol predicts the parameter updates of VQAs over multiple iterations on the classical side, dramatically reducing quantum resource costs. Through systematic numerical experiments, we demonstrate that our method achieves up to a 30x speedup compared to conventional methods and reduces quantum resource costs by as much as 90\% for tasks involving up to 40 qubits, including ground state preparation of different quantum systems, while maintaining competitive accuracy. Our approach complements existing techniques aimed at improving the efficiency of VQAs and further strengthens their potential for practical applications.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work explores the integration of Physics-Informed Neural Networks (PINNs) with Variational Quantum Eigensolvers (VQEs), as proposed in the PALQO framework. The key idea is to enhance the VQEs by reducing the number of measurements required by the parameter shift rule by predicting parameter values using PINNs trained and run on classical computers. PALQO achieves up to a 30x speedup compared to conventional methods and reduces quantum resource costs by up to 90% for quantum workloads involving up to 40 qubits.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nVQE currently requires a prohibitive number of measurements. Reduction in the number of measurements, as facilitated by PALQO, could be very beneficial.\n\nPALQO's PINN can achieve good performance even with a very limited number of training samples.\n\nWeaknesses:\n\nMy fundamental question concerns the extent to which quantum advantage is preserved in the proposed approach. Specifically, if the classical PINN model used in PALQO must be scaled in conjunction with VQE, it raises the issue of what precise quantum advantage is actually obtained. The idea behind VQE is to only use classical resources for optimization, but if parameters also get resolved classically -- with the PINN essentially simulating the parameter shift rule -- then what is the need for a quantum VQE?\n\nA related concern is whether VQE itself retains any inherent quantum advantage in this context. Furthermore, it is important to understand how the size of the PINN model scales relative to the size of the VQE. Is the scaling behavior exponential, or if not, what form of scaling is required to maintain efficacy?\n\nThere 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.\n\nQUESTIONS:\nMy fundamental question is about the quantum advantage retained with such a proposal.\n\n(1) If the classical PINN model proposed by PALQO has to be scaled with VQE, what exactly is the quantum advantage?\n\n(2) Would VQE retain any quantum advantage?\n\n(3) How does the size of the PINN model scale with the size of the VQE?\n\n(4) Is it exponential scaling?\n\n(5) If not, what is the scaling required?\n\nLIMITATIONS:\nThe 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.\n\nOn the other hand, the work does not have any direct negative societal impact implications.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nTo reduce the quantum computational resources required for training Variational Quantum Algorithms (VQAs), this paper proposes an approach that leverages a neural network model inspired by Physics-Informed Neural Networks (PINNs) to predict VQA training dynamics. The authors also derive theoretical generalization error bounds for the proposed dynamics prediction model. By applying this method to several quantum-mechanical tasks, the paper demonstrates its practical effectiveness in significantly reducing the necessary quantum computational resources.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength**\n- The paper proposes a novel approach that experimentally demonstrates significant improvement in addressing one of the critical challenges in training VQAs, specifically the increasing requirement for quantum computational shots.\n- Extensive experiments from multiple perspectives have been thoroughly conducted, providing detailed verification of improvements in training speed.\n- The manuscript is well-written and thoughtfully structured, making it accessible even to machine learning researchers who may not be familiar with quantum computing.\n\n**Weakness**\n- The paper lacks sufficient qualitative discussion about the proposed method. For instance, it does not adequately address how effectively the method tackles significant challenges in VQA, such as Barren Plateau (Q1). \n- It remains unclear how PALQO differs from existing methods and what specifically contributes to its advantages (Q2, Q4).\n- The validity of the proposed approach is unclear (Q3, Q5). Predicting learning dynamics in VQA appears to be as complex as performing the actual VQA training itself.\n\nQUESTIONS:\nTo recommend acceptance, several of the following major questions (though not necessarily all) need to be resolved.\n- [Q1] Does this approach help mitigate the Barren Plateau problem?\n\n  For researchers in quantum computing, the primary concern regarding the difficulty of training variational algorithms is the barren plateau problem [1], in which parameter gradients vanish exponentially with respect to the number of qubits. Please clarify whether this study is influenced by barren plateaus, and whether it could potentially contribute to resolving this issue.\n\n- [Q2] How is the reduction in shot count theoretically justified?\n\n  According to Corollary 3.1, training a PALQO model to achieve a given error requires a polynomial-order number of samples with respect to the parameter size. On the other hand, page 3, line 111 states that training a vanilla VQA requires $O(pN_H/\\epsilon^2)$ measurements, which scales linearly with $p$. A direct comparison of these two results does not clearly indicate a significant improvement. Please provide a detailed discussion clarifying the extent of the improvement over vanilla VQA that can be inferred from Corollary 3.1.\n\n- [Q3] Can PALCO accurately predict learning dynamics?\n\n  Although sufficient experiments have been conducted regarding improvements in training speed in this study, there is a lack of experiments evaluating how accurately PALCO predicts the learning dynamics of VQA. Comparing PALCO's predicted results to actual learning dynamics, or comparing parameters obtained from TFIM or HQ experiments in this study with optimized parameters from vanilla VQA, would help determine whether predicting learning dynamics via PINN plays an essential role in VQA research.\n\n- [Q4] What contributes to the superiority of PALQO over the other methods?\n\n    In the benchmark comparisons, the LSTM-based method and QuACK were evaluated. However, the discussion is insufficient regarding what factors contribute to PALQO's superiority.\n\n- [Q5] Is this method applicable only to quantum optimization?\n\n  Describing learning dynamics using differential equations and constructing a predictive model via a PINN-like approach seems applicable to classical machine learning contexts as well. Please clarify which specific parts of this method necessitate focusing exclusively on quantum optimization.\n\n[1] McClean, Jarrod R., et al. \"Barren plateaus in quantum neural network training landscapes.\" Nature communications 9.1 (2018): 4812.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel perspective on the optimization dynamics of variational quantum algorithms (VQAs) by modeling them through partial differential equations (PDEs). It establishes a PDE-based framework that governs VQA parameter updates and enables efficient classical prediction via physics-informed neural networks (PINNs). Beyond the theoretical insight, the authors propose a practical protocol to reduce quantum measurement costs during training by leveraging this classical surrogate model. Importantly, unlike many prior works limited to small-scale simulations, this study validates its approach empirically on systems with up to 40 qubits, offering meaningful insights into the scalability of PDE-based optimization for VQAs. The principal contribution lies in showcasing a promising classical strategy to accelerate large-scale VQA training, addressing one of the critical bottlenecks in near-term quantum computing.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper makes a compelling theoretical contribution by connecting the optimization trajectories of VQAs with dynamical systems governed by PDEs. This analogy allows the use of PINNs to predict parameter evolution classically, thereby circumventing the need for repeated quantum measurements—a major constraint in training variational circuits—while sidestepping limitations such as the no-cloning theorem.\n- The proposed framework is general and applicable across VQA variants. It directly targets a practical bottleneck—quantum measurement overhead—and provides a protocol that can adapt to existing VQA pipelines.\n- The work is backed by extensive numerical studies, including simulations on systems with up to 40 qubits. These empirical results provide strong support for the scalability and practical utility of the approach, which is rare among works in this domain.\n\nWeakness:\n- The initialization strategy for the variational parameters, i.e., uniform sampling from the [0,1] range as used in the QuACK benchmark, may limit the generality of the results. If the performance of PALQO (the proposed method) is sensitive to this specific range, it raises concerns about robustness. Additional experiments with varied initialization schemes would strengthen the conclusions.\n- In Section 4.2, while the authors quantify the reduction in measurement overhead, the analysis is based on single instances for each system size. It is unclear whether these reductions persist across different Hamiltonians or circuit depths. A broader statistical analysis would be beneficial to evaluate the consistency of the advantage.\n\nQUESTIONS:\n1. 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?\n2. 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.\n3. 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. \n4. 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.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

comments: SUMMARY:
The paper proposes a learning-based method to predict the dynamics of parameter evolution in variational quantum circuits. It rewrites the dynamics of variational quantum circuits’ training as dynamics of a partially differentiable equation and employs the physics-informed neural network to predict the dynamics of PDE. This framework is scalable and helps accelerate the training of VQA. An extensive set of experiments demonstrates the efficacy of the proposed framework.

STRENGTHS AND WEAKNESSES:
Pros
1. The motivation of PALQO roots in the heavy resources needed in VQA training. PALQO effectively exploits the patterns in the VQA training dynamics to reduce the number of samples needed, which is clever.
2. The experiments are comprehensive and clearly demonstrate the advantages of PALQO.

Cons
1. From my observations of Figure 2., in many cases PALQO has not converged to zero ΔE. What are the causes of this?
2. I feel there is a lack of discussion on the experiment settings. How do you make sure that the comparison of PALQO and other methods is fair? I.e., what metrics are kept the same when you report the required number of measurements for different methods, especially when you cannot obtain ~0 ΔE for each experiment?

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

LIMITATIONS:
No.
TITLE: Official Review
CLARITY: 2
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-04