PALQO: Physics-informed model for Accelerating  Large-scale Quantum Optimization
Authors: Yiming Huang
Yajie Hao
Yuxuan Du
Jing Zhou
Xiao Yuan
Xiaoting Wang
Venue: NeurIPS 2025 poster ()
Keywords: quantum optimization
physics-informed machine learning
machine learning
Primary area: machine_learning_for_sciences

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.

PALQO: Physics-informed Model for Accelerating Large-scale Quantum Optimization
Yiming Huang", Yajie Hao² Jing Zhou², Xiao Yuan Xiaoting Wang?t, Yuxuan Du
Center on Frontiers of Computing Studies,
and School of Computer Science, Peking University, Beijing, China
Institute of Fundamental and Frontier Sciences
University of Electronic Science and Technology of China, Chengdu, China Coollege of Computing and Data Science
Nanyang Technological University, Singapore, Singapore Sccooo of Physical and Mathematical Sciences
Nanyang Technological University, Singapore, Singapore
*Department of Physics, Fudan University, Shanghai, China
yiminghvang@gmail. com, yuxuan , du@ntu. edu. xiaoyuan@pku edu, cn edu. cn
xiaotingquestc. wew.geeuesttee..u..eee.
Abstract
Variational quantum algorithms (VQAs) are leading strategies to reach practi- cal utilities of near-term quantum devices. However, the no-cloning theorem in quantum mechanics precludes standard backpropagation, leading to prohibitive quantum resource costs when applying VQAs to large-scale tasks. To address this challenge, we reformulate the training dynamics of VQAs as nonlinear partial differential equation and propose novel protocol that leverages thysics-informer neural networks (PINNs) to model this dynamical system efficiently. Given 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 to a 30x speedup com- pared conventional methods and reduces quantum resource costs by as much 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.
1 Introduction
Modern quantum computers, with a steadily increasing number of high-quality qubits, are approaching the threshold of practical utility [ In this pursuit, variational quantum algorithms (VQAs) ННΣ have emerged as a leading strategy, attributed to their flexibility in accommodating circuit depth and qubit connectivity among different platforms. In recent years, wide range of theoretical and experimental studies have demonstrated the feasibility of VQAs across diverse applications, such quantum chemistry [3) optimization [16H18] and machine learning [19 23]. Despite the progress, they face critical challenges when applied to large-scale problems. In particular, the no-cloning theorem and the unitary constraints in the quantum universe prohibit the use of backpropagation techniques common in deep learning 24 requiring VQAs to update sequentially to
"Equal contribution. i Corresponding authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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This approach imposes substantial and even prohibitive quantum resource demands, especially in terms of the number of measurements required. Given the scarcity of quantum computers in the foreseeable future, enhancing the optimization efficiency of VQAs while minimizing resource consumption is crucial for enabling their practical deployment.
minimize a predefined cost function 2 26
To advance VQAs for large-scale problems, substantial efforts have been devoted to improving the optimization efficiency. Prior literature in this field can be broadly categorized into three primary classes (see Sec. 2. for details). The first aims to reduce measurement costs in quantum many-body and chemistry problems by grouping terms in the Hamiltonian to enable simultaneous measurements
27 29
The second harnesses classical simulators or learning models to identify well-initialized
parameters that are close to local minima of the cost landscape for a given VQA, thereby improving
convergence efficiency
E0 [2]. The third seeks to predict the dynamics of parameter updates by
revising past optimization trajectories for the given task [%), as exemplified by methods such as recurrent neural networks
34 35 and QuACK [36.. Despite significant advancements, no existing
approach effectively balances optimization efficiency and accuracy at scale. In this regard, a critical question arises: is it possible to achieve both for large-scale VQA systems?
Towards this question, we observe that prior efforts have primarily focused on hardware improvements and heuristic optimizations, while the potential of approximating training dynamics to alleviate the quantum resource burden remains underexplored. In response, we introduce fresh perspective by utilizing Taylor expansion to reformulate the parameter optimization process in VQAs as a nonlinear partial differential equation (PDE). In this way. the evolution of this nonlinear PDE corresponds to the trajectory of parameter updates during training. Building on this formulation, we devise a protocol, dubbed PALQQ., that employs a physics-informed neural network (PINN) 37 38 to approximate solutions to the PDE, where high-order terms in the Taylor expansion serve as boundary conditions. The proposed framework is generic, which encompasses the quantum neural tangent kernel (QNTK) as special case [1] Besides, we prove that polynomial number of training samples is sufficient to ensure that PALQO attains a satisfactory generalization ability.
We then conduct extensive numerical simulations on different ground state preparation tasks, including the transverse- Ising model. Heisenberg model, and multiple molecule systems, to investigate the effectiveness of PALQO. Simulation results up to 40 qubits indicate that by learning from a limited set of initial parameter updates obtained from quantum devices, PALQO, which deployed on classical hardware, can accurately predict future parameter updates. These results indicate the effectiveness of reducing the number of quantum measurements required during optimization. Numerical experiments on ground state preparation tasks across large-scale systems, involving up to 40 qubits, validate the effectiveness of PALQO. Moreover, we show that the proposed PALQO is complementary to existing approaches for improving the optimization efficiency of VQAs. To support the community, we release our source code at (42. These results open a new avenue for leveraging the power of PINNs to enhance the efficiency of VQAs and advance the frontier of practical quantum computing.
Contributions. For clarity, we summarize our main contributions below. (1) To our best knowledge, we establish the first general framework between the optimization trajectory of VQAs and PDE, thereby enabling the employment of various PINNs to advance the optimization efficiency of large- scale VQAs. (2) We propose PALQO, an effective PINN oriented to reduce the required number of measurements when training large-scale VQAs, and prove its generalization ability. (3) Unlike prior studies that mainly focus on small-scale tasks, we conduct extensive numerical studies to validate the advancements of PALQO up to 40 qubits, providing valuable insights to further improve the optimization efficiency of VQAs at scale.
2 Preliminaries
In this section, we provide a concise overview of the basic concepts of quantum computing, variational quantum algorithms and physics-informed neural networks to set the stage for their integration in accelerating the quantum optimization process. Please refer to Appendix A for more details.
Basics of Quantum Computing. Quantum state, quantum circuit, and quantum measurement are three key components in quantum computing [3 44 In particular, an n-qubit quantum state is mathematically represented as a unit vector u CN in Hilbert space, where N |u, = 1, =
2", Here we follow conventions to use Dirac notation to represent and its transpose conjugate ut
2

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i.e., |u) and (ul. For quantum circuit, it serves computational model consisting of a sequence of quantum gates that describes operations on the given input state. The most widely used quantum gates are Pauli gates, i.e., X = (f)). 7. (i
Y = ( )'). According to the Solovay- Kitaev theorem [X] arbitrary operation can be approximated by quantum circuit U = =II, U3 where each gate Uj is drawn from a finite universal gate set, such as (CNOT, H, R. (,, Concretely, CNOT (0) (0) & I+ [1X1] H 1/v2( (z) e e-i6x Rg(0) e 182 = vZ.
For quantum measurement, it is the process that collapses a quantum state into a definite classical outcome. In this study, we are interested in the expectation value of the measurement outcomes with given observable O, a Hermitian operator, on quantum state |u), i.e. (ul ()). Suppose the observable presents as an n--ubii Hamiltonian that characterizes energy structure of the target quantum system in the form of H شكا To experimentally estimate (u(O) u) within error 6, we typically perform M ~ (11/e) repeated measurements for each P3 on multiple copies of the state |u) and get the outcomes {MA1*) L...,,..M then approximate the expectation value by statistical averaging (u|Olu) 1/(MNH) Ej cg Ma*
Nu cyP3. where PJ is tensor product of Pauli matrices, i.e. Py E {T,X, 22)
Variational Quantum Algorithms. Variational quantum algorithms (VQAs) algorithms designed for machine learning tasks are called quantum neural networks (QNNs) A1 52, while those applied to many-body physics and quantum chemistry are typically known as variational quantum eigensolvers
(VQEs) 633660 The primary objective of VQE is optimize a parameterized state (v)(0)
(o) to minimize the energy function & detined by a given Hamiltonian H Σ)= CjPy. Mathematically, the energy function to be minimized in VQEs takes the form of
min {() = (vb(0) H|
(1)
A common and widely adopted approach to complete this optimization problem is utilizing a gradient- based optimizer, like gradient descent, to iteratively adjust the parameters O according the partial derivative OgE. Because there is no-clone theorem and no backpropagation without exponential classical overhead in general VQEs (24), we need to perform the parameter shift rule without involving other quantum resource overhead, such as ancillary qubits to estimate the partial derivative 25 Concretely, the calculation of the partial derivative with respect to O; takes the form as
3c 00, = 2 -0
00
π =)/) E (0. (0:-2)
π ]
~
Σ cs J,k
M
(2)
MN
where 4/+, ch correspond to state (v, + x/2)) and (0, n 2/2)), respectively.
While such a method provides a closed-form expression for gradient estimation without requiring additional qubits and can be extended to general VQEs, it necessitates evaluating & twice with shifted parameter values at the same position to estimate the gradient of a single parameter. Hence, suppose the dimension of 0 is p. it requires to perform O(pN /c2) measurements to estimate the partial derivative Ögε, which becomes computationally prohibitive in large-scale tasks, especially for large molecules as whose n-qubits recond-quantize Hamiltonian has roughly NH O(n*) terms [ Therefore, it requires substantial resources for estimating updated parameters @ during the optimization, which is considered one major limitation of large-scale VQEs. Similarly, such scalability issue also arises in QNNs, where the number of measurements required per iteration scales linearly with p and the batch size.
Physical-Informedd Neural Network. Physics-informed neural networks (PINNs) have become a promising learning-based tool in approximating the solution of partial differential equations (PDFs) 61H33. With the advantages of computational efficiency for solving complex PDE, they have been widely employed in various practical scenarios such as fluid dynamics, battery degradation modeling, disease detection, and complex systems simulation [3 64-677. PINNS harness the core tool, automatic differentiation, of modern machine learning to efficiently enforce the physical constraints of the underlying PDE.
For PDE problem, it can be generally written as N[u(x, = = t) where x E D C Idd denotes variables, N' represents the differential operators, u(x, 1) stands for the solution, and (ax, () refers to input source function. The aim of PINNs is to build neural network fno with parameters to approximate the true solution u. Hence, the loss function of PINN for solving a general PDE based on residuals, including PDE residual and data residual. The PDE residual measures the difference
3

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between the neural network solution and the true solution, expressed as
Lp = M Sw ab
-
-2 (
LD - Σ fw (0 (2) (
6) ud
(3)
Here, (x/ to 13 used in Lp are selected collocation points for enforcing PDE structure. For Cp. the dataset {ad td ud J with un = u(x²d" :td x(()) denote the training data on u( t) 37. Thus, the total loss is putting all residuals together, i.e. Cr + Lp. By embedding physical principles into the learning process, PINNs serve versatile tool that only requires small amount of data to tackle the computationally complex problem.
Na
2.1 Related Works
Prior literature related to improving the optimization efficiency of VQAs can be classified into three main classes, i.c., measurement grouping, initializer design, and prediction of training dynamics. Since the first two classes are complementary to PALQO, we defer the explanations to Appendix B
The third class aims to harness learning models to approximate the training process. Some works inspired by meta-leamning utilize the recurrent neural network to learn a sequential update rule in a heuristic manner 64 B35 Nevertheless, the memory bottleneck and training instability of the recurrent neural network would lead to it being underwhelming 68 Recent work proposed QuACK, involving linear dynamics approximation and nonlinear neural embedding, to accelerate the optimization KG However, the prediction phase requires estimating the energy loss of each step to find the optimal parameters, which is not friendly for large-scale problems, To overcome these limitations, the proposed PALQO uniquely approximates VQA training dynamies using nonlinear PDE, embedding the dynamical laws directly into the learning process. In this way, it offers deeper physical insight and achieves superior performance through principled model-guided optimization.
3 PALQO: physics-informed model for accelerating quantum optimization
In this section, we first formally define the problem of learning the training dynamics of VQAs as nonlinear PDE problems in Sec. 3.1 Then, in Sec. 3.2 we introduce PALQO to solve this PDE via tailored PINN-based model, where the optimized solutions correspond the optimization trajectory of VQAs, followed by generalization error analysis.
3.1 Reformulating the optimization of VQA PDE problem
Recall the optimization of VQEs in Sec. 2 As it is costly in querying ((t) of cach step f, it is demanded to develop a protocol that only learns from a few trajectory data (e(') to classically predict future steps, thereby avoiding prohibitive resource costs without compromising accuracy. To achieve this goal, we start by revisiting the gradient descent dynamics of VQEs. The updating
rules of parameters 0 with learning rate 77 at step is given by 80 @ - g(t) = (O)
where (0)))/0 is estimated through phase shift rule shown in Eq. 2 Suppose is infinitesimally
small, the following ordinary differential equation, a.k.a., gradient flow, characterizes how parameters change in continuous time, i.e.,
00
3g
0L
80
Besides, we can similarly define the dynamics of E in a general form under Taylor expansion as
3g Øt
Σ
2
Σ
1
30 3g 00,00% 00, 00, + O
(5)
00,
2
j.k
where the first term of RHS of Eq. is termed as quantum neural tangent Kernel (QNTK) 39 40
39), which captures the sensitivity of outputs to parameter changes, shaping how the gradient flow evolves in parameter space. The second term involves the Hessian matrix of & that reflects the local curvature of the cost function in the optimization landscape. Thus, tackling the problem of learning optimization trajectory be recast to solve PDEs presented in Eqs. and This reformulation

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(b) training
[c]
step 1
step
stop
fe
fu
ge+-n
Lp
App
loss
()++1)
ein
gic+
(E()) ( m (1")), (l
geres)
Ground
masked output
7-steps sistepsopiimizaiionn trajectory dataset
git+1)
Predicteddate
Trained model
Figure 1: Framework of PALQO. (a) Dataset construction: the tunable angles 0 and corresponding loss & from VQE over optimization steps collected, forming sequential training dataset that captures the optimization trajectory over time. (b) Model training: PINN fw is trained by minimizing the loss
L= Xp, Lr, + ArgCrg + AnCp. (c) Prediction: starting with the last step 0 the trained f is used to recurrently predict parameters, mimicking the optimization process without access to the quantum device.
provides powerful framework and allows to leverage rich set of tools developed for PDEs to understand the underlying behavior of the optimization process in large-scale VQEs.
Remark. In Appendix C.I we elucidate the derivation of Eq. 5 and its relation to QNTK. Besides, the above reformulation can be effectively extended to broader VQAs such as QNNs.
3.2 Implementation of PALQO
In light of the above reformulation, we propose PALQO based on PINN introduced Eq. ( to learn the optimization trajectory of large-scale VQEs. Conceptually, once PALQO attains low training error, it can predict the optimization path, which substantially reduces the measurement cost as considered in large-scale VQEs. As such, it enhances scalability and resource efficiency by minimizing the need for extensive quantum circuit evaluations while maintaining high fidelity in modeling complex quantum optimization.
overview of our protocol depicted in Fig. J which consists of consists of three stages: Dataser construction, Model training, and Model prediction. In Fig. (a), it starts by generating dataset S (t H corresponding to trajectory data consisting of O and energy loss & over optimization steps collected from quantum device. Then, as shown in Fig. (b). PALQO utilizes the collected S to train a PINN-based model fw to capture the underlying optimization dynamics. Given the trained fw. as shown Fig. (c), it can recursively predict the parameters 0 of the future steps until convergence, i.e. (0))) 0 0++1) [gt0) g(++1)| S 5 with being small constant.
Dataset construction. To mimic the dynamic of VQE optimization, we need first perform 7 steps optimization via gradient descent to collect sequential trajectory the training data, including
0 (1)) E IRP is the parameters at j-th epoch. Notably, assume VQE with total T optimization steps, PALQO only needs t < T steps to construct the training dataset S. This is because the PINN-based model leverages strong inductive biases from the dynamical laws, such that it does not need to infer fundamental principles from scratch, allowing the model to generalize well from limited data.
loss (E ei {(T)} and {0(1), O(2) O2)).... ((7)} where ( = =(
Model training, Once the dataset is collected, PALQO employs deep neural network fwr that takes {(t, (())) comprising t-th time step and trajectory data ((() as the input, and predict the loss and parameters for the (1 + 1) 1-th step, (.e..(E(+++)) ( ((++)).. Refer to Appendix [ for the details about the neural architecture of fw. Denote the prediction of fw for the f-th step as (()). Through leveraging the automatic differentiation capabilities neural networks, the derivatives outputs with respect to inputs, i.c., aه /0t /t and dĚ /00, can be efficiently computed on classical devices. This efficiency enables the direct incorporation of dynamical law constraints, described in Eqs. and (). into the loss function. this regard, we devise the loss function of PALQO as
(6) where (AD, Ap, ) denote hyperparameters to balance the data-driven loss Ln and two PDE
ApCD Ap, Lp. + AP:CP.
residual losses Lp, and Lpy, In particular, the data residual loss is defined as Lp Σ -t= (g(0)
5

---

(()) + Ei= ( )( aiming to capture the temporal changes of & and 0 among steps. Meanwhile, the two PDE residual losses aim to enforce the PINN to capture the evolution of VQE optimization dynamics through the underlying derivative structure of the loss landscape. Following p -=1 (00 / 1810/800 ))²
Eq. the explicit form of the first PDE loss is Lp. Σ
By focusing the first two orders of the derivatives in Eq. the second PDE loss yields
OE(O)
OE(1)
2 a6 ae())
Pp
Ōt
00
2 j.k=1 0 0000
0
00 a0
J=1
The model fw is optimized by minimizing C in Eq. (G) via gradient- gradient-based optimizer Adam 77
Model prediction. As the trained fu can not only approximate the solution of the underlying PDE but also capture temporal dependencies of the trajectory data, we are able to recurrently predict the upcoming updates of the 0. As shown in Fig. [ (c), by passing (r, (())) through the trained fw and
masking the g(r) node, we can obtain the predicted data @(++1), and then fed (T + 1. ((+ the input back to the network to make the followed prediction. is worth noting that directly making
long-tern forecasts to reach the optimal solution of the target problem is exceedingly challenging. To that end, we employ the non-overlapping sliding windows to enhance the network for long-term prediction. More details on the prediction process be found in Appendix
Remark. The second PDE residual loss Lp, can be extended to arbitrary higher orders. Empirical results indicate that a second-order approximation offers a sufficient balance between accuracy and computational cost for modeling the optimization dynamics of the VQEs studied herein. Moreover, while we mainly focus on learning the training process of VQE, our model can be efficiently extended to more general tasks such as quantum machine learning [19 20 quantum simulation [7] 73
and quantum optimization 14 7 details.
by slightly modifying the Eqs. and (5). See Appendix F for
76 we conduct the analysis of the Lipschitz Building upon prior work on the error analysis of PINNs constant bound for PALQO and derive corollary to establish a generalization bound for PALQO applied to VQEs. An informal statement of the derived generalization bound is provided below, where the formal statement and the related proof are deferred to Appendix D
Corollary 3.1. (Informal) When utilizing PALQO, whose PINN is constructed by a L layer tanh neural network with most W width of each layer and trained over T data samples, to approximate the solution of PDE that describes training dynamics of a VQE with p tunable parameters O, with probability at least - 7, its the generalization error is upper bounded by
ò 0 pL?W?
In (2)
+ In
(8)
The achieved results indicate that for any c > 0, the number of training examples scales at most polynomially in L, and W is sufficient to guarantee a well-bounded generalization error. This warrants the applicability of PALQO in large-scale scenarios.
Experiments
To evaluate the practical performance of the proposed PALQO framework, we apply it to two representative quantum applications: finding ground state energy of many-body quantum system and molecules in quantum chemistry, which have broad applicability in understanding many-bodh physical phase transitions |77 [ and simulation of complex electronic structures of molecular
systems in drug design and discovery 15
80
81
The concrete settings are elucidated below.
Many-body quantum system. A many-body quantum system consists of interacting quantum particles whose collective behavior and correlations lead to complex phenomena beyond single- particle descriptions. Here, we consider three typical many-body quantum systems. 1) Transverse- field Ising model (TFIM) describes spin particles a lattice interacting via nearest-neighhor coupling and subject to a transverse magnetic field, whose Hamiltonian is typically HIFM -J E Z, & ZJ+1 h E, Xy. where Z, and X, refer to the Pauli matrices Z and X applied on the j-th qubit, respectively. 2) Quantum Heisenberg model also describes the spin
form of

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particles on lattice. but spin-spin interactions occur along all spatial directions. Its Hamiltonian can be represented as Нон - (J, (J,X, + 12 + J,Z,Zj+ + hZz). 3) Bond- alternating XXZ model is an anisotropic variant of the Heisenberg model with unequal coupling strengths in the transverse and longitudinal directions. The Hamiltonian is given by Hxxz Ej=odd +1Xxx+++,Jyy- + (21 Ej=even J'(x + Yjy,+1 + 82, 2)+1). The
coefficients J, h, Jg. Jy. Je, 6 within the explored Hamiltonians represent the coupling strength that determines the ground state properties and phase transitions.
Quantum chemistry. The Hamiltonian of molecule describes the total energy of its electrons and nuclei and serves as the fundamental operator for determining the molecule's electronic structure in quantum chemistry. The general form of the molecule Hamiltonian can be presented as H = E, h, Pj where Pj represents tensor products Pauli matrices, and h are the associated real coefficients. Here, we select widely studied molecule- LiH, and relatively large and challenging BeH, molecule as We generate these molecule Hamiltonians with Openfermion [83 Refer to Appendix for more details about the molecule experiments.
the target molecules 8 53 82
4.1 Experimental Setup
We employ three standard ansatzes, i.c., hardware-efficient ansatz (HEA) 84 86.. Hamiltonian variable ansatz (HVA)
[8 (8).89), unitary coupled cluster with single and double excitations ansatz (UCCSD) 990 92 to implement VQE for the different Hamiltonians mentioned above. These ansatzes adopt layered architecture. Refer to Appendix 2.2 for the implementation of these ansatzes. For all ansatzes, their initial parameters ((0) are uniformly sampled from [0, 1]. following the strategy adopted in QuACK (36). The gradient descent is set as the default optimizer.
For implementing PALQO, we randomly initialize the parameters w of the neural network fw from [-1.1] and employ the Adam as optimizer, where the architectures of fw are listed in Appendix To improve the training stability and convergence, we utilize linear decay strategy to adaptively adjust the learning rate during training. Besides, the weight hyperparameters in loss function Ap, Ap, ÀD. are set as 1.0, 1.0 and 10 respectively.
2
Benchmark models. To show the outperformance of PALQO against the state-of-the-art methods, we introduce the following baseline and benchmark. First, we use a vanilla VQE as the baseline since provides a well-established reference point to evaluate improvements in accuracy, convergence, and efficiency. Second, we select a LSTM- based model [44] as a benchmark since it provides a strong reference for evaluating methods in modeling the temporal dependencies and iterative dynamics of optimization trajectories. Third, we pick QuACK 3d as another benchmark as it represents an advanced approach that learns surrogate dynamics of VQAs by embedding Koopman operator- based linear representations into nonlinear neural networks. The implementation of these reference models deferred to Appendix
Evaluation metrics. To quantify the performance of PALQO in accuracy and efficiency, we consider the following metrics, 1) Accuracy, we define the accuracy as how close the estimated energy É is to given target energy E of quantum system, i.e. AE É E 2) Efficiency. we define the speedup ratio as Ip/Iv, where Ip and Iy refer the number of iterations required by the baseline method (vanilla VQE) and PALQO or other benchmark models, respectively, to achieve an acceptable accuracy a. Specifically, we set a S 10-3
4.2 Experimental Results
We next evaluate the performance of PALQO and other reference models when applied to the aforementioned Hamiltonians under different settings.
PALQO significantly reduces the measurement overhead. Here, we utilize the number of measurements incurred during the optimization as a quantum resource measure to explore the performance of PALQO and the other benchmark models when applied to 20 qubits TFIM with HEA, 20 qubits Heisenberg model with HVA, and 14 qubits BeH, with UCCSD ansatz, The number of parameters for each case 120, 180, 90, respectively. As shown in Tab. M PALQO achieves significant quantum resource efficiency in aforementioned tasks, with around 90% average reduction in measurement overhead while preserving AE around 10-3
7

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These substantial savings stem from two key factors: 1) compared to the vanilla VQE, PALQO leverages PINN to predict parameter updates, thereby reducing reliance on frequent quantum measurements; 2) the rapid convergence on the classical side enables further reduction in quantum resource expenditure.
Table number of quantum measurement shots (x10*) required for TFIM, Heisenberg model, and BeHfa.
SYSTEM SIZE
HIMM = 20
Hиo = 20
Нвaня = 14
VANILLA VQE
10.97
21.66
464.3
LSTM
3.126
14.49
312.4
QUACK
5.217
14.15
461.8
PALQO
1.535
5.749
28.01
QUACK
LSTM
PALQO
QUACK
LSTM
PALOO
QUACK
LSTM
(a.1)
(b.1)
(c.1)
20
40
Speedup Ratio
Spleduu Ratio
7.5
30
Speeddup
10
20
5.0
10
2.5
12
12
16
(a.2)
(b.2)
Speedup Ratio
20
20
Speedup
Speedup
ME
10
20
20
(0.3)
Speedup Rebio
40
50
Speedup
20
12
16
20
8
12
16
20
8
12
16
20
Iteration
Iteration
Iteration
TFIM
Heisenberg
BeHz
Figure 2: Performance comparison between PALQO and the reference models 20 qubits TFIM HEA, 20 qubits Heisenberg model with HVA, and 14 qubits Bel12 with UCCSD ansatz. Each subplot comprises a AE E curve over iterations performed on quantum device, along with a bar chart depicting the speedup ratios achieved PALQO and competing models. The left column illustrates results for TFIM with J/h {2, 1, 0.5). The central column shows results for Heisenberg model with Je Jg column displays the model performance on BeH2 with the bond length b = {1.3, 1.5).
1, (0.5, 1, 2}. The right
PALQO outperforms benchmark models in accuracy and efficiency. The performance compar- isons of PALQO on 20 qubits TFIM with HEA, 20 qubits Heisenberg model with HVA, and qubits BeH2 with UCCSD ansatz for varying structural parameters observed that PALQO consistently outperformed, up 30x speedup and lower around 10-3 like the case of TFIM with J/h = 2 and HEA ansatz, compared to the other evaluated approaches. Furthermore, the PALOQ predicts the future optimization steps classical hardware, it exhibited faster rate of convergence, achieving a substantial reduction in AE' within fewer iterations performed on a quantum device, compared to the baseline methods. Although the speedup ratio of PALQO has a relatively large variance, its minimum value remains comparable to the average performance of the other approaches. Refer Appendix
presented in Fig. 2 In particular,
E
F for the results of XXZ model and LiH.
(a]50
J/ 0.5
Jih-1
-
(b)50)
j/h 0.5 + - + 1 2
40
30
10
24
28
36
40
2
Number of Qubits
Number of Layer
Figure 3: Scalability analysis of PALQO on TFIMs. (a) The speedup ratio achieved by PALQO in modeling VQE training dynamics with fixed HEA, ranging from 4 to 40 qubits. fixed system size of 12 qubits, assessed under increasing HEA ansatz layers from 2 to 8.
(b) The speedup ratio of PALQO with

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Scalability of PALQO. We next investigate the scalability of PALQO on the TFIM with HEA, examining its performance in increasing system sizes (from n
40) and the number of ansatz layers (from 2 to 8). In Fig. 3 the results reveal that the speedup of PAI.QO is contingent upon the specific system configuration. Nevertheless, as shown in Fig. + (a), while the speedup ratio
fluctuates with the number of qubits varying, it still achieves up to 30x speedup when J/h 0.5.
The lower speedup at J/h = is due to a smaller energy gap between the ground and first excited states, making the optimization more challenging. Similar behavior also appears in Fig. (b). This suggests that the performance benefits PALOQ are maintained as the computational demands grow, indicating its potential for large-scall scale quantum optimization.
Sta
No
No Lp
Base
Sta
No
No
Base
Sta
No
No
Base
(x)
Speedup Rabo
(b)
20
40
(c)
Raiio
20
20
19
10
Speedup
20
6
10
20
20
No
Mo
20
Zi No
Figure 4: Ablation study on the loss function configuration in PALQO. The three panels show the performance on TFIM with 12, 20, 28 qubits, evaluating the impact of the components Lp and Lp, in C.
Ablation studies on loss function. We further evaluate the performance of PALOQ against variants where specific loss terms in Eq. 6 are removed in the task of TFIM with HEA ansatz. Specifically, we carried out separate ablation studies on the PDE residual and data residual components of the loss function shown in Fig. 4 We noticed that while both the PDE and data residual positively influence model performance, their contributions are not essential. These findings suggest that adopting low- order approximations during the construction of PALQO enables the retention of satisfactory speedup while simultancously reducing the complexity of downstream model training and preventing the degradation of higher-order derivative information.
(a)
NN-50
NN-2D
NN-5
(b)
Depth-2
Death-4
Dopth-6
40-
Speedup ratio
40
20
Speedup
30
20
20
10
10-
Number of Qubits
Number of Qubits
Figure Performance of PATQO evaluated on the TFIM under different neural network architecture configurations. (a) Results obtained using 2-layer neural networks with varying hidden layer widths of W = (50p, 20p, 5p), where denotes the number of parameters O. (b) Results with a fixed hidden layer width of 20p, varying the number of hidden layers from to 6.
To investigate the influence of varying neural network sizes (width W and depth L) within PALQO on its performance, we conducted tests on the TFIM with HEA ansatz and p = 6n parameters, where n is the system size varying from 36. In Fig. 5 (a), we observed that an increase in the width of the hidden layers leads to a corresponding improvement in speedup ratio. However, an inverse phenomenon occurs in Fig. 5 (b), further increasing the neural network depth does not effectively enhance PALQO's performance, which may be related to the vanishing gradient phenomenon, where higher-order derivative information tends to diminish as neural network becomes deeper [К These observations provide guidance for the neural network design in PALQO, indicating that increasing hidden layer width should be prioritized.
Performance on varying the size of PALQO.
The impact of the number of training samples on model prediction. We per- formed experiments on 12-qubit TFIMs with HEA ansatz using various sam- ple sizes to explore how the number of training samples affects the perfor- mance.In Fig. 6 the results validate that PALQO can achieve satisfactory perfor- mance even with limited number of
12.5
r=2 r=4
4
7.5
2.5
10
14
20
Iteration
Figure 6: Performance comparison of PALQO when trained with 2, 4. and data samples.

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training samples. Such data efficiency arises from the direct integration of the physical constraints imposed by the governing PDE into the loss function of the neural network, core characteristic of PINNs
Conclusion
In this study, we devised PALQO towards optimizing large-scale VQAs given restrictive quantum resources. In contrast to previous studies, we derive PALQO from reformulating the training dynamics as a nonlinear PDE and using PINN to approximate the solution, and also provide a generalization analysis. Extensive numerical experiments up to 40 qubits validate the effectiveness of PALQO. Although it is still uncertain whether PALQO can scale to the regime where quantum hardware decisively outperforms classical methods, its results the currently accessible scale are highly encouraging.
Limitations and future works PALQO reduces the need for repeated quantum gradient evaluations by learning the optimization path classically. While this lowers the number of quantum queries compared to vanilla VQE, it may diminish some quantum advantages. Besides, mitigating the high variance in speedup ratios is crucial for achieving more stable and reliable performance. One future research direction is to incorporate adaptive strategies and variance reduction techniques to achieve this goal and further unlock its potential.
Acknowledge
We thank Yusen Wu for the helpful discussions and the anonymous reviewers for their constructive feedback and valuable suggestions. This work was supported by Innovation Program for Quantum Science and Technology (Grant No. 20232D0300200), the National Natural Science Foundation of China NSAF (Grant No. U2330201 and No. 92265208), Sichuan Science and Technology Program (Grant No. 2025YFHZ0336). Y. D. acknowledges the support from the SUG grant of NTU. The numerical experiments in this work were supported by the High-Performan Computing Platform of Peking University.
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Reviews:
[{"review_id": "ZsXfmC3Krd", "reviewer": "Reviewer_KUyt", "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.", "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.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Pros\n1. 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.\n2. The experiments are comprehensive and clearly demonstrate the advantages of PALQO.\n\nCons\n1. From my observations of Figure 2., in many cases PALQO has not converged to zero ΔE. What are the causes of this?\n2. 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?", "was_revised": true, "final_justification": "My three concerns are adequately resolved in the rebuttal.", "rebuttal": "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.\n\n**W1: From my observations of Figure 2, in many cases PALQO has not converged to zero $\\Delta E$. What are the causes of this?**\n\n**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 $\\Delta E$.\n\n*$\\bullet$ $\\underline{\\text{Purpose related to Fig. 2}}$*. 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.\n\nIt 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.\n\n*$\\bullet$ $\\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.\n\nThe 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. \n|               |            | TFIM                 | BeH2                 |\n|---------------|------------|----------------------|----------------------|\n| Vanilla VQE   | $\\Delta E$ | $4.266\\times10^{-2}$ | $1.271\\times10^{-3}$ |\n|               | iterations($I_p$) | 334                  | 676                  |\n| PALQO         | $\\Delta E$ | $4.264\\times10^{-2}$ | $1.269\\times10^{-3}$ |\n|               | iterations($I_v$) | 30                   | 29                   |\n| Speedup ratio |  $I_p/I_v$       | 11.13                | 23.31                |\n\nAs 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*.\n\n**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?**\n\n**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. \n\n| condition    |  in main text | metric            | description                                                                                                    |\n|-------------|--------------|-------------------|-------------------|\n| iteration=20 | Figure 2     | ΔE                | how close the estimated energy $\\hat{E}$ is close to target energy $E$, $\\Delta E=\\|\\hat{E}-E\\|$               |\n| ΔE~1e-3 | Figure 2     | speedup ratio     | The ratio ($I_p/I_v$) presents the number of iterations required by vanilla VQE($I_p$) relative to that of PALQO($I_v$). |\n| ΔE~1e-3 | Table 1      | measurement count | the number of measurements incurred during the optimization  |\n\nWe use the above metrics under specific conditions to evaluate the performance of PALQO and other baseline models separately. The detailed settings are as follows:\n1. 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).\n2. 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).\n3. We calculate the total number of measurements throughout the entire optimization process until $\\Delta E$ of the model approach $10^{-3}$ (Table 1).\n\n**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?**\n\n**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.\n\nTo 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.\n\nAssume 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.\n\nThe 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 $$E(t)=(1-\\eta K)^tE(0),$$ \nwhere 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., $K=\\sum_j(\\partial_E/\\partial_{\\theta_j})^2$. \n\nAs 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\n$$K'=K+\\delta K,$$\nwhere $\\delta K$ depends on $\\epsilon_0$ and the number of trainable parameters. Meanwhile, the training dynamics of PALQO-enabled VQE yields \n$$E'(t)=(1-\\eta K')^tE(0).$$\nIn this regard, we can bound the difference between estimated and ideal dynamics, i.e., $E'(t)$ and $E(t)$. That is, we have  $$\\|E'(t) - E(t)\\|\\leq t\\eta\\|\\delta K\\|(\\max(|1-\\eta K|,|1-\\eta K'|))^{t-1}|E(0)|.$$ \nAccording to Theorem 3.5 in Ref [3] and our assumptions, $\\delta K$ can be arbitrarily small by increasing the PINN width and depth, and $\\max(|1-\\eta K|,|1-\\eta K'|)=1-\\eta \\min(K,K')<1$. \n\nTaken 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.\n\n[1] Di Luo, Jiayu Shen, Rumen Dangovski, and Marin Soljacic. Quack: accelerating gradient-\nbased quantum optimization with koopman operator learning. Advances in Neural Information\nProcessing Systems, 36, 2024.\n\n[2] Guillaume Verdon, Michael Broughton, Jarrod R McClean, Kevin J Sung, Ryan Babbush,\nZhang Jiang, Hartmut Neven, and Masoud Mohseni. Learning to learn with quantum neural\nnetworks via classical neural networks. arXiv preprint arXiv:1907.05415, 2019.\n\n[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.\n\n[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.\n\n[5] Guillaume Verdon, Michael Broughton, Jarrod R McClean, Kevin J Sung, Ryan Babbush,\nZhang Jiang, Hartmut Neven, and Masoud Mohseni. Learning to learn with quantum neural\nnetworks via classical neural networks. arXiv preprint arXiv:1907.05415, 2019."}, {"review_id": "9QssMNMr0y", "reviewer": "Reviewer_KJmn", "summary": "This 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.", "questions": "1. 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.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n**W1: Whether the robustness of PALQO is due to its fixed [0,1] parameter initialization and suggests testing with varied schemes to validate generality.**\n\n**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. \n\n|               | TFIM-12 | BeH2  |\n|---------------|------|-------|\n| Speedup Ratio | 18±9 | 23±12 |\n\nThe results show that PALQO can still achieve good performance under various initialization schemes. \n\n**W2: The suggestion about broader statistical analysis to verify whether measurement reductions hold across various Hamiltonians and circuit depths.**\n\n**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. \n\n|       | TFIM-12qubits, depth=3 | TFIM-8qubits, depth=3 | TFIM-12qubits, depth=5 | TFIM-8qubits,depth=5 |\n|-------|------------------------|-----------------------|------------------------|----------------------|\n| VQE   | $3.882 \\times 10^8$             | $1.688 \\times 10^8$            | $6.649 \\times 10^8$             | $2.813 \\times10^8$           |\n| PAlQO | $5.432 \\times 10^7$             | $2.362 \\times 10^7$            | $9.053 \\times 10^7$             | $8.016 \\times10^7$           |\n\nThis result shows that even in systems of different sizes and depths, PALQO can still effectively reduce measurement cost.\n\n**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?**\n\nThanks 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. \n\n**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.**\n\n**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. \n\n**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.**\n\n**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.\n\n**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.**\n\n**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. \n|               | 10% depolarizing noise | 20% depolarizing noise | Amplitude Damping Channel |\n|---------------|------------------------|------------------------|---------------------------|\n| Speedup Ratio | 10±6                   | 6±3                    | 13±5                      |\n\nIn 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."}, {"review_id": "1Seiq91hQc", "reviewer": "Reviewer_xXH5", "summary": "To 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.", "questions": "To 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.", "limitations": "Yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "The authors have provided very detailed responses, and many of my concerns have been addressed. However, it should be noted that within the quantum computing community, there remains a general skepticism regarding the potential for quantum advantage in variational quantum algorithms. While this manuscript claims to offer a solution by reducing the required number of measurements, I have the following concerns:\n- The inherent difficulty of training variational quantum circuits remains unresolved.\n- The claimed reduction in measurement counts is not supported by strong theoretical guarantees but rather demonstrated empirically.\n- The experiments presented involve training PINNs with only a few dozen samples, leading to high uncertainty in the reported results.\n\nFor these reasons, I will maintain my current evaluation.", "rebuttal": "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. \n\n**[W1 & Q1] The paper lacks sufficient qualitative discussion of PALQO, i.e., how it addresses key challenges in VQAs, such as barren plateau (BP).** \n\n**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.\n\n$\\bullet~\\underline{\\text{Key challenges of VQAs}}$. 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.\n\nTo 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. \n\nThis 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**.\n\n$\\bullet~\\underline{\\text{Relation between BP and PALQO}}$.  PALQO is **complementary to approaches aimed at mitigating barren plateaus**, as both seek to improve the optimization efficiency of VQAs at scale.\n\nTo 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.\n\n**[W2 & Q2] It remains unclear how PALQO differs from existing methods. How is the reduction in shot count theoretically justified?**\n\n**Response**:  Let us address the reviewer's concern in sequence. \n\nAs 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**. \n\nAs for the concern \"*How is the reduction in shot count theoretically justified?*\", we address this from two perspectives. \n\n$\\bullet$ 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.\n\n$\\bullet$ **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.\n\nIn 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.\n\n**[W2  & Q4] The paper lacks sufficient discussion on the key factors driving PALQO's superior performance.**\n\n**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.\n\nWe further evaluate PALOQ with variants where specific loss terms are removed, and compare its performance with LSTM and QUACK, as follows:\n\n| TFIM-20qubits | PALQO | No $L_{p1},L_{p2}$ | No $L_{D}$ | LSTM | QUACK |\n|---------------|-------|-------------------|------------------|------|-------|\n| Speedup Ratio | 30±12 | 2.2±1             | 24±8             | 6±4  | 6±2   |\n\nAs 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.\n\n**[W3 & Q3] The validity of the proposed approach is unclear. Can PALCO accurately predict learning dynamics?**\n\n**Response**: To address the reviewer's concern, we have conducted the following experiments to illustrate that PALCO can **indeed accurately predict learning dynamics**.\n\nWe 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.\n\n|      | H2        | BeH2    | TFIM-8qubits | TFIM-12qubits |\n|------|-----------|---------|--------------|---------------|\n| Norm | 0.03±0.01 | 1.2±0.6 | 0.14±0.03    | 0.192±0.01    |\n\nThese 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.\n\n**[W3 & Q5]  Is this method applicable only to quantum optimization?**\n\n**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: \n1) 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. \n2) 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.\n\n[1] Park C Y, et al. Hamiltonian variational ansatz without barren plateaus[J]. Quantum, 2024\n\n[2] Zhang K, et al. Escaping from the barren plateau via gaussian initializations in deep variational quantum circuits[J]. NeurIPS, 2022\n\n[3] Di L, et al. Quack: accelerating gradient-based quantum optimization with koopman operator learning. NeurIPS, 36, 2024.\n\n[4] Guillaume V, et al. Learning to learn with quantum neural networks via classical neural networks. arXiv preprint arXiv:1907.05415\n\n[5] Cervera-Lierta A, et al. Meta-variational quantum eigensolver: Learning energy profiles of parameterized hamiltonians for quantum simulation[J]. PRX Quantum, 2021\n\n[6] Karim A, et al. Fast and Noise-aware Machine Learning Variational Quantum Eigensolver Optimiser[J]. arXiv:2503.20210.\n\n[7] Jiang Y, et al. Fantastic generalization measures and where to find them[J]. arXiv:1912.02178, 2019.\n\n[8] Meng F, et al. Conditional Diffusion-based Parameter Generation for Quantum Approximate Optimization Algorithm[J]. arXiv:2407.12242\n\n[9] Khairy S et al. Learning to optimize variational quantum circuits to solve combinatorial problems[C]//AAAI. 2020"}, {"review_id": "zDKW459kSb", "reviewer": "Reviewer_2QJF", "summary": "This 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.", "questions": "My 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?", "limitations": "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.\n\nOn the other hand, the work does not have any direct negative societal impact implications.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "I have revised my score positively after an extensive discussion with the authors.", "rebuttal": "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.\n\n**[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.**\n\n**Response:** We thank the reviewer for raising the important point regarding the potential impact of our method on quantum advantage. \n\nThe 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].\n\n\n*$\\bullet$ $\\underline{\\text{Preserves quantum advantage.}}$* \nWe 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.\n\nThe 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].\n\n*$\\bullet$ $\\underline{\\text{Need for a quantum VQE.}}$* 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. \n\n**[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?**\n\n**Response:** We thank the reviewer for the insightful comment. \n\n*$\\bullet$ $\\underline{\\text{Retain inherent quantum advantage.}}$* 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. \n\n*$\\bullet$ $\\underline{\\text{The scale of PALQO.}}$* \nWe 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.\n\nTo 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. $p\\sim O(n)$.\nIn 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).\nIn 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.\nThese scaling results suggest that PINN achieves strong performance with only linear increases in width and depth, avoiding the need for exponential scaling.\n\n|      |       | Width(l=2)|       |       | Depth(w=20p) |       |\n|------|-------|-------|-------|-------|-------|-------|\n|      | 5p    | 20p   | 50p   | 2     | 4     | 6     |\n| n=4  | 17±8  | 18±11 | 22±9  | 18±12 | 20±7  | 9±8   |\n| n=12 | 9±8   | 22±11 | 26±12 | 21±10 | 19±11 | 24±16 |\n| n=20 | 14±11 | 21±7  | 30±12 | 22±7  | 21±15 | 8±4   |\n| n=28 | 8±6   | 14±9  | 23±7  | 13±10 | 26±15 | 9±6   |\n| n=36 | 11±9  | 26±13 | 25±16 | 25±15 | 26±18 | 6±3   |\n\n\n**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.**\n\n**Response:** The observed high variance in the speedup ratio may be attributed to the following factors: \n1. 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.\n\n2. 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.\n\n3. 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.\nBelow, 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.\n\n|                                  | J/h=0.5,1    | J/h=2       |\n|-----------------------|--------------|-------------|\n| Vanilla VQE steps(Ip) | 120,115,130  | 150,140,135 |\n| PALQO steps (Iv)      | 4,3,4        | 26,25,20    |\n| speedup ratio (Ip/Iv) | 40,38.3,32.5 | 5.7,5.6,6.7 |\n| variance  of speedup ratio            | 13.4         | 0.37        |\n\n**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.**\n\n**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.\n\n[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.\n\n[2] Cerezo M, Arrasmith A, Babbush R, et al. Variational quantum algorithms[J]. Nature Reviews Physics, 2021, 3(9): 625-644.\n\n[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.\n\n[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.\n\n[5] Leimkuhler O, Whaley K B. Exponential quantum speedups in quantum chemistry with linear depth[J]. arXiv preprint arXiv:2503.21041, 2025.\n\n[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.\n\n[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."}]

Author rebuttal:

Meta-review: The authors present a method for accelerating VQE that dramatically reduces the number of expensive evaluations needed on quantum hardware to achieve a good estimate of the ground state of a given Hamiltonian. Due to the no-cloning theorem, analytic gradients cannot be computed on quantum hardware, and instead a method similar to finite differences is used, which becomes expensive as the number of parameters grows. By augmenting the quantum updates with the results from the PINNs, the number of parameter updates using real quantum hardware is reduced by nearly an order of magnitude. This is somewhat reminiscent of synthetic gradients for neural networks, using a learned function to replace gradient calculations. I should note that for the systems investigated (e.g. LiH, BeH2), accurate calculations can be performed with _zero_ expensive calls to quantum hardware. However, I recognize that comparison against classical computation is out of the scope of this paper, and it is assumed that at some point, quantum hardware will be able to scale beyond what can be done with classical hardware. While it remains to be seen if PALQO can scale to the point where quantum hardware has a real advantage over classical hardware, the results on the scale achievable today are promising and the reviewers mostly liked the paper. I recommend acceptance.
Decision: Accept (poster)