QuanONet: Quantum Neural Operator with Application to Differential Equation
Authors: Ruocheng Wang
Zhuo Xia
Ge Yan
Junchi Yan
Venue: ICML 2025 poster (2025)
Keywords: Neural Operator
Quantum Neural Networks
Differential Equations
Primary area: applications->chemistry_physics_and_earth_sciences

Abstract: Differential equations are essential and popular in science and engineering. Learning-based methods including neural operators, have emerged as a promising paradigm. We explore its quantum counterpart, and propose QuanONet -- a quantum neural operator which has not been well studied in literature compared with their counterparts in other machine learning areas. We design a novel architecture as a hardware-efficient ansatz, in the era of noisy intermediate-scale quantum (NISQ). Its circuit is pure quantum. By lying its ground on the operator approximation theorem for its quantum counterpart, QuanONet in theory can fit various differential equation operators. We also propose its modified version TF-QuanONet with ability to adaptively fit the dominant frequency of the problem. The real-device empirical results on problems including anti-derivative operators, Diffusion-reaction Systems demonstrate that QuanONet outperforms peer quantum methods when their model sizes are set akin to QuanONet.

QuanONet: Quantum Neural Operator with Application to Differential Equation
Ruocheng Wang'
Zhuo Xia
Ge Yan Junchi Yan
Abstract
tum algorithms for solving differential equations map equa- tions into higher infinite dimensional linear spaces (Carle- man, 1932; Forets & Pouly, 2017; Kowalski & Steeb, 1991; Jin et al., 2024) or use gradient computation of differentiable quantum circuits (Kyriienko et al., 2021; Xiao et al., 2024b) to solve differential equations. However, these methods are inherently limited to solving specific differential equations and lack the ability learn nonlinear operators or generalize across varying conditions. This restricts their applicability in solving more complex real-world problems.
Differential equations are essential and popular in science and engineering. Learning-baat based meth- ods including neural operators, have emerged as a promising paradigm. We explore its quantum counterpart, and propose QuanONet - a quantum neural operator which has not been well studied in literature compared with their counterparts in other machine learning areas. We design a novel architecture as a hardware-efficient ansatz, in the era of noisy intermediate- quantum (NISQ). Its circuit pure quantum. By lying its ground on the operator approximation theorem for its quantum counterpart, QuanONet in theory can fit various differential equation operators. We also propose its modified version TF- OF-QtunONet with ability to adaptively fit the dominant frequency of the problem. The real-device empirical re- sults on problems including anti-derivative opera- tors, Diffusion-reaction Systems demonstrate that QuanONet outperforms peer quantum methods when their model sizes are akin QuanONet.
Quantum neural networks (QNNs) leverage the exponen- tially large Hilbert space of quantum systems to achieve quadratic speedup online perceptron (Kapoor et al., 2016) and reinforcement learning (Dunjko al., 2016). So there have been many studies using QNN to implement popu- lar classical neural network architectures. However, ex- isting QNN implementations of neural operators, such as QFNO (Jain et al., 2024) and quantum DeepONet (Xiao et al., 2024a), are limited to accelerating the evaluation phase using quantum Fourier transforms (QFT) and quan- tum orthogonal neural networks. The former is not observed to outperform classical operators on differential equation problems, and the latter uses hybrid network structure that brings the communication overhead between quantum and classical devices.
1. Introduction
Nonlincar differential equations are essential tools in mod- eling complex phenomena across various scientific and en- gineering fields. Traditional numerical methods, including finite element methods (FEM) and finite difference methods (FDM), often rely on discretizing the problem domain and solving it iteratively. Recently, artificial intelligence (AI) methods, such as neural operators (Lu et al., 2021: Li et al., 2020), have emerged as promising alternatives by learning nonlinear operators directly and achieving some degree of generalization across different conditions.
We propose QuanONet, a general framework for learning (nonlinear) operators in differential equations using QNN. This means that it can not only be well-suited for the era of noisy intermediate-scale quantum (NISQ) by using any QNN architecture that features low qubit requirements and does not rely on high-coneettivity quantum hardware, but also has potential in the future fault-tolerant quantum era.
Our QuanONet is grounded in the Universal Approxima- tion Theorem for operators, and we show that its QNN im- plementation retains the powerful operator approximation capabilities of classical methods and have good general- ization performance in different cases. Furthermore, we propose TF-QuanONet, a version of QuanONet that uses the trainable- frr.uueey method (Jaderberg et al., 2024). By using the input encoded coefficients as additional trainable parameters, it can efficiently learn the spectrum of opera- tors without the need for extremely deep quantum circuit repetition. These features position QuanONet as a versatile
Quantum computing offers fundamentally different ap- proach to addressing these challenges. Some existing quan-
School of Artificial Intelligence & School of Computer Sci- ence & Zhiyuan College, SJTU. Correspondence Junchi Yan <yanjunchi@ @ sjtu.edu.cn..
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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QuanONet: Quantum Neural Operator with Application to Differential Equation
and scalable tool for solving differential equations, with significant potential applications in fields such as fluid dy- namics, material science, and physical simulation. The contributions of this paper are:
the solution function obtained by the classical numerical method, to obtain the evolution operator of the equation in a short time. Some works (Kyriienko et al., 2021; Xiao et al., 2024b; Joo & Moon, 2021) utilized methods similar to Physics-Informed Neural Networks (PINNs) (Raissi et al., 2019). In this paradigm, quantum circuits are employed for variational optimization of differential equations, where quantum systems compute gradients through circuit eval- uations. These gradients are then used to optimize neural network parameters, approximating solutions to the equa- tions. While effective for specific problems, such quantum approaches have not yet been extended to the learning of nonlinear operators directly from data, an area that remains largely unexplored. More comprehensive discussion and comparison with related works are given in Appendix B.
1) We extend the classic universal approximation theorem for operators (Chen & Chen, 1995) to the quantum state ver- sion, and give the universal approximation theorem for quan- tum computing for operators see Theorem 3.1 and 3.2.
2) Based on our theoretical results, we propose gen- eral trainable quantum neural network framework with pure quantum circuits without hybrid classic-quantum ar- chitecture as a nontrivial limitation suffered from existing work (Xiao et al., 2024a) namely QuanONet for learning (nonlinear) operators.
3) We further propose trainable-frequency version, TF- QuanONet, which is tailored to dominant frequency compo- nent learning in operators and is different from all previous neural operators. It can capture the high frequency part of the operator without depending on the well setting of coefficients, and is thus robust and efficient.
Beyond differential equations, quantum computing has demonstrated potential in other complex problem domains. In combinatorial optimization, quantum algorithms like QAOA aim to find near-optimal solutions to problems such as graph partitioning (Herrman et al., 2021). scheduling and logistics (Azad et al., 2022). Similarly, in quantum chemistry, quantum algorithms are used to simulate molecu- lar structures (Yan et al., 2023) and energy states (Cerezo et al., 2021) with greater precision than classical methods, potentially revolutionizing fields like drug discovery and materials science. These advancements highlight the versa- tility of quantum computing in addressing a wide range of computationally demanding problems.
4) We study a number of differential equation problems. Experimental results show that QuanONet with appropri- ate coefficient initialization can achieve better performance than other quantum solvers, while TF-QuanONet overcomes the difficulty of QuanONet in selecting appropriate hyper- parameter coefficients, and performs well across various coefficient initial conditions.
2.2. Universal Approximation Theorems of QNNs
2. Related Work and Preliminaries
The foundation for the approximation capabilities of QNNs has been recently studied. Yu et al. (2022) showed that a QNN, even with a single qubit, can approximate any contin- uous single-variable function with arbitrary accuracy. This result parallels the Universal Approximation Theorem (Cy- benko, 1989; Hornik, 1991) in classical neural nets. More- over, Pérez-Salinas et al. (2020); Kapoor et al. (2016) proved a universal approximation theorem for QNNs by construct- ing a one-qubit quantum circuit able to arbitrarily approxi- mate any continuous complex-vail valued function. Schuld et al. (2021) showed a similar approach that data encoding can be approximated by infinitely (akin to infinite-widtt classical neural networks) repeating simple encoding schemes as: Theorem 2.1. (Schuld et al., 2021) Denote Hm a univer- sal Hamiltonian family, J' the associated quantum model family:
2.1. Quantum Computing for Complex Problems
Quantum computing has shown great promise in solving complex problems, e.g. combinatorial optimization and quantum chemistry, whereby quantum variational algo- rithms like Quantum Approximate Optimization Algorithm (QAOA) (Farhi et al., 2014) have been well developed.
For differential equations solving, emerging quantum meth- ods have been devised, which can be broadly classified into two categories. One is quantum algorithms based on am- plitude methods and phase estimation, For example, Jin et al. (2024) converted linear differential equations with non-unitary dynamics into Schrödinger-type equations and solved using quantum phase estimation and amplitude am- plification. Oz et al. (2023) used the quantum amplitude estimation by utilizing Chebyshev points for numerical inte- gration to design robust quantum PDE solvers.
(1) For all functions E L2( ) and for all e > 0, there exists some m' E N, some state (T,)) E Cam", and some Hamiltonian M such that
( (x)
Another category is based on variable weight algorithm (VQA). Sarma et al. (2024); Lubasch et al. (2020) repre- sented solution function with the state amplitudes and op- timized the cost function, which is its difference between
(x) - -0(()) Se
(2)
2

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QuanONet: Quantum Neural Operator with Application to Differential Equation
Gonon & Jacquier (2023) moved one step further and proved precise error bounds for these approximations. Yu et al. (2024) showed that quantum models outperform clas- sical ReLU neural networks by model size, circuit depth, and the number of trainable parameters when approximating high-dimensional functions with specific smoothness condi- tions. In addition, several works (Goto et al., 2021; Pérez- Salinas et al., 2021) have proposed that hybrid quantum classical architectures could provide promising solution. By integrating trainable weights into the QNN, these mod- els enrich the frequency spectrum of the network, enabling effective multivariable function approximation.
(8)
Inper Function =x()
Differontal Equasan Opentor
Ustpont Function G(u](y)
(b)
Draa Me
Nat
m(xx) (u(*)))
C[w)(y)
(b Ap)
Brast Not
(c)
y
Trunk Not
== (01... (p)
hm
The frequency principle (Xu, 2018; Xu et al., 2019) suggested that DNNs train low-frequency components quickly but have poor generalization performance for high- frequency data. Its quantum version Xu & Zhang (2024) shows that QNNs preferentially learn the dominant fre- quency within the spectrum. In this paper, we study the training behavior in the specific operator learning context, especially for the high-frequency terms of operator.
mxxx)
(m(x))m²a
Net
(M Apd
G(u)()) bit
Trank Net
(17))))))))
Figure I. (a) For differential equation, the solution operator G tries to learn a mapping from function w(x) to G(w)(y). (b) Stacked DeepONet consists of number of Branch Nets and Trunk Net, following the form of Theorem 2.2. (c) Unstacked
2.3. The Deep Operator Network (DeepONet)
DeepONet is a deep learning framework designed to approx- imate nonlinear operators that map inputs (e.g.. initial or boundary conditions) to the solutions of differential equa- tions. Unlike traditional machine learning models that pre- dict outputs for fixed onput-output pairs, DeepONet learns the operator itself, mapping input function to its corre- sponding output function. Its network structure is inspired by the universal approximation theorem for operator.
DeepONet is proposed by Lu al. (2021) to reduce the storage and computation cost of Stacked DeepONet.
It has been shown that DeepONet can generalize to new, previously unseen conditions, enabling the rapid solution of differential equations with different initial or boundary conditions without retraining. At present, there have been many improvements for the original version, such as Lu et al. (2021) merging large number of Branch Nets into one network to reduce memory and computational cost, called UnStacked DeepONet. Goswami et al. (2022) proposed a physical-informed variational formulation of DeepONet (V- DeepONet) to alleviate the computational burden. Hc et al. (2024) proposed the Sequential DeepONet (S- (e-DepOONet by introducing sequential learning models in the Branch Net to allow for accurate predictions of the solution contour plots under parametric and time-dependent loading histories.
Theorem 2.2. (Chen& & Chen, 1995) Suppose that g a Tauber- Wiener function, X a Banach Space, Ki S X, K2 S Rd are two compact sets in X and IR" respectively, V is a compact set in C(K1), G is a nonlinear continu- ous operator, which maps V into C(K2). Then for any e > there are positive integers M, N, m. constants ct. Sije 0% Ss E R. Wk R", Xj E K1. i 1.... M,
b= ==1 =1, N. = 1,..,, m, such that
N M
[G(n)(y)
ΣΣ
s)x,)) + 0²) g(wn y + [+)l<< <e
Trank
Branch
(3)
3. Methodology
As shown in Fig. 1, DeepONet consists of two primary components: Branch Net and Trunk Net. The Branch Net encodes the input function (e.g. initial conditions or bound- ary conditions) at fixed number of sensors and the Trunk Net encodes the locations for the output functions. The output of the network is obtained by combining the outputs of the Branch and Trunk Nets, through dot product, to gen- erate the final solution of the operator. The model is trained on pairs of ouput data, generated from numerical solutions of differential equations.
We will first derive the approximation theorem for quantum state functions. provide the quantum state substitution for the vector inner product in the classic operator approxima- tion theorem and give the quantum universal approximation theorem for operators. The theory results inspire us to then propose a operator learning QNN architecture: QuanONet. We also discuss how the coding coefficients affect the ap- proximation error of QuanONet through frequency spectral and how to overcome this effect.
3

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QuanONet: Quantum Neural Operator with Application to Differential Equation
3.1. Theoretical Results
A Hamiltonian family whose eigenspectrum asymptoti- cally contains any integer frequency via the following no- tion (Schuld et al., 2021): Hamiltonian family {Hm) is a universal Hamiltonian family if it has the property that for all E N there exists some m E N such that
3.1.1. QUANTUM UNIVERSAL APPROXIMATION FOR QUANTUM STATE FUNCTIONS
Although works about QNNs have studied universal approx- imation for functions, there still lack the approximation the- orem for quantum state functions. Specifically, for normal- ized quantum state that depends on the inputs, parametric quantum circuits can output its approximation with arbitrary aceuracy, where the inputs are encoded as model parameters. Based on feasible assumptions about QNN (Schuld al., 2021), we give an approximation scheme for an arbitrary quantum state function with the form of:
ZK = f- ... .,, ...... K] C Amm.
(10)
As proposed in Schuld al. (2021), the Hamiltomian family defined the linear combinations of Pauli operators
172
Hm
Σ
o . (,,
(11)
is indeed universal Hamiltonian family with K. This gives the theorem:
(o(x)) = (x)())
(4)
Theorem 3.1. [Quantum Universal Approximation Theo- rem for State Functions] Let Hm be a universal Hamilto- nian family, and 4m the associated quantum state family:
satisfies the normalization conditions Σ, (x,) = {(i)} is the basis of finite dimensional quantum system.
[x)( -WSH (..H,(xx))T)
(12)
Then define a quantum state as the output of parametric quantum circuit as follows:
For all normalized quantum state functions lØ(x))
Ej (x))) with functions 9j E L22(0, Rx)"), ), and for all
[w)(x)) (0)
(5)
<> O, there exists some m' E M, some state (T) E Cam
where 9 and the optimizable parameters and the coding parameters.
and some W such that
Illyme (0(x)))1 Se
(13)
The quantum circuit consists of a data encoding circuit S(x)and a trainable circuit (0) controlled by parameters both alternating L layers. The data encoding block con- sists of gates of the form e iz.f We focus on the role of the data encoding, and view the trainable circuit blocks ar- bitrary unitary operations W W(0)= W. Thus the overall quantum circuit has the form:
Compared with Theorem 2.1, the proof of Theorem 3.1 requires the spectrum A AH,, be able to contain the highest integer frequency K2 = max K, among all amplitudes in the state function. See proof in Appendix D.
Theorem 2.1 has heuristic significance for the wider appli- cation of QNN: it shows that QNN cannot only be used for approximating multivariable functions, but also for learning quantum states that depend on the input, e.g. predicting ground state of quantum systems under different conditions. Based on this theorem, we will provide the construction of the quantum neural operator in the next section.
U(x) (L+11) (()) ... (()) (6)
For notational simplicity and without loss of generality, we consider simple single-layer quantum circuits with L = 1, obtain [1') by acting (1) on the initial state (0) and rewrite (2) as W. Thus we have the equivalent form:
3.1.2. QUANTUM UNIVERSAL APPROXIMATION FOR OPERATORS
(x)x)) = T))
(7)
Based on the above theorem, we give the quantum universal approximation theorem for operators as follows.
Specifically, Schuld et al. (2021) introduced the concept of a Hamiltonian family { (H. Im E N} where Hm acts on m subsystems of dimension d. Such a Hamiltonian family defines a family of quantum states via
Theorem 3.2. [Quantum Universal Approximation The- orem for Operators] Suppose that X is a Banach Space, K2 C X, K₂ C R are two compact sets in X and R. respectively. V is compact set in C(K1), G is nonlinear continuous operator, which maps V into C(K2), then for any e > 0, there exist a positive integer N. a real constant c, a N-dim state function (t) and a N- state functional [b). such that
[Mm(x)) WSH.. (x)
(8)
Given Hamiltonian Hm with eigenvalues (Ai, Aam
the eigenspectrum of Hm is
A A... = (Ayl E (1,... "d""., dd"
(9)
[G(w)(y) - ((()))))((
(14)
4

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QuanONet: Quantum Neural Operator with Application to Differential Equation
A full proof is given in Appendix E. This theorem states that the operator can be approximated by an inner product of quantum states depending on the input function and the output location respectively. implying the existence of QNN structure which can learn the differential operator.
(8)
Encoder Layer Entangle Layer
Ansatz Layer
Pylhi
(b)
Branch Layers
Before introducing the architecture QuanONet inspired by the theorem, we briefly introduce and extend the enror analysis of DeepONet with Lanthaler al. (2022). Specifi- cally, the error sources of DeepONet mainly include three parts: encoding, approximation and re-construction errors, where encoding error depends on the settings of sensors, However, both approximation and re-construction errors depend on the approximation ability of the network and the dimension of the vector (quantum state). Similarly, the error sources of QuanONet also come from these three parts, which inspires that QuanONet may be applicable to the operator version of the QNN dominance problem, namely smooth and spectrally simple operators according to the results of Yu et al. (2024) and Leong et al. (2025).
Input Function Encoder Layer
Ansatzz Layer
Entangle Layer
Repeated he
Repeated la times
Trunk Layers
Output
Entangle Layer
Ansatz Layer
Lecations Encoder Luyer
Repented h, times
Repeated times
(c)
[0)
3.2. The Architecture of QuanONet
(0)
Branch Layers
Trunk Layers
x
Meusure with
If A
(0)
x
As shown in Fig. 2. QuanONet consists of two primary com- ponents: Branch (BNC) Layers and Trunk (TNK) Layers, which encode the input function u = {u(xx)]=, and the output locations y respectively. The output of the circuit is:
Figure 2. QuanONet's detailed architecture. (a) A hardware- efficient class of examples of Encoder, Entangle, and Ansatz layers. (b) Structure of Branch and Trunk Layers with hyper-
(voly, wol u)) =(0)Uggn (0", u. HO)))() y)) Bac u)
parameters ha, hu, lo. 4 representing the number of repetitions Ensatz-EEtannge and encoder-train alternating layers in them. To satisfy the requirement of inner product form, the quantum Layers of Branch and Trunk Layers arranged in reverse order. (c) The overall quantum circuit structure, and the circuit parameters are trained by the gradient of the loss function of the output value.
(15)
Write Hamiltonian H in spectral decomposition form H- Σ, (i). then the output can be written as
(yoly, u) Hoo, u))
= Σ Xel (T Bnc (0", O0*, u)
(16)
range less than 2n. A simple method is multiplying inputs by hyper-parameter coefficients = [A,... so that
If Hamiltonian has only one eigenstate [0) with eigenvalue 1 and all other eigenvalues are zero, then the output corre- sponds to the squared version of the Eq. 3.2.
Allly, ui]li S
(19)
(Voly, u)|l wyly, u)) [0 (0', U) Bnc (0', u) [(t((',))) by)), u)
Ideally, we could set À very small ensure that the model fits wider range of inputs. However as proposed in Xu & Zhang (2024); Jaderberg et al. (2024), the extremely small X means that a higher number of repeated layers is required to achieve Fourier series completeness. How to choose the most appropriate À under the condition that the size of the quantum circuit is limited requires careful consideration.
(17) Since the Hamiltonian is free to choose, we only need to scale it so that the bound of its spectrum includes the range of the output function. An optional Hamiltonian is:
H H
( +
(18)
The order of applying Branch and Trunk Layers does not affect the approximation formula. We adopt the sequence of encoding the input with the Branch I Layers first, followed by the encoding of variable y with the Trunk Layers.
with the upper bound b + na and the lower bound b - na. n is the total number of qubits.
Since parameters of the rotation quantum gates are periodic in the range [0, 2r). inputs u and y should be encoded
As a high-level framework, QuanONet allows for different implementations. To accommodate the current hardware
5

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QuanONet: Quantum Neural Operator with Application to Differential Equation
limitations of quantum computers, e-g. restricted connectiv- ity and depth constraints, we employ a QNN design with Hardware-Efficien Ansatz (HEA) shown in Fig. 2 as our vanilla version. By leveraging this harfware-efficient circuit structure, QuanONet achieves a practical balance be- tween expressiveness and resource requirements, especially remaining deployable on existing quantum hardware.
(8)
True Function
(b)
FF-QNN
(c)
TF-QNN
Heh
3.3. The Frainable-Frequunccy Technique
The frequency principle provides a perspective for under- standing the training behavior and generalization ability of deep neural networks. Since the data input is introduced into the quantum state through trigonometric functions, So Frequency plays a more critical role in training and opti- mization. Specifically, (Xu & Zhang, 2024) developed an evolution equation for gradients and residue dynamics in the frequency domain by introducing quantum neural tangent kernels (QNTK) and empirically showed that the dominant frequency of the data set is learned first by QNNs.
Figure 3. Schematic representation of the function and spectrum of the (a) true function, (b) FF-QNN and (c) TF-QNN, where the QNNs have heen trained the optimum. The FF-QNN model will have a large approximation error when it is not close to the com- plete Fourier series, while TF- ONN can preferentially fit the main frequency part improve the accuracy. This effect is particularly significant when the high-frequency components the function dominate when the quantum circuit is shallow.
Though asymptotically large fixeefrryuencc (FF) QNN are universal function approximators (Schuld et al., 2021), in reality, finite-sized quantum computers will permit models with only a finite range of frequencies. So the approxi- mation effect of quantum variational circuits is strongly related to whether the frequency spectrum can match the spectrum problems. In Jaderberg et al. (2024), trainable- frequency (TF) models are introduced, which extend the conventional fixed-frees greedffrquency parameterizaation of quantum circuits. In typical QNN models, the generator Hamiltoni- ans used for data encoding determine a fixed set of equally spaced frequencies. TF models add trainable parameters to the generator, allowing the circuit to dynamically adjust the frequency spectrum represents. It improves the ability to adapt the spectral characteristics of the target function.
4.1. Protocols
We evaluate QuanONet and TF- TF-QuanONet on a range of benchmark problems: three ordinary differential equation (ODE) and one partial differential equation (PDE) problem:
Anti-derivative operator.
G: u(() - u(x)= [ L. u(T)dr, E [0, (20)
Homogeneous operator.
G: + v(x), du(x)
dr v(x) u(x), E [0. (21)
We propose TF- TF-QuanONet. tailored and improved version of QuanONet. TF-QuanONe employs trinable-frequency model for its Branch and Trunk Layers. By introducing trainable weights and biases in the quantum encoding that A A [y, u] + b, it dynamically adjusts frequency repre- sentation during training. This allows the model to adapt to high-frequency irregularly spaced spectral features of operators that may not be captured by FF-QNN.
Nonlinear ODE.
u(x), G G: - v(x), du(x) -()))(+(())))))))))))))))(())()) (0.
dr
(22)
Diffusion-reaction (D-R) System.
dr
D 02v D dt dr
G: G: u(z)))
v(z,t),
+
kvz + u,x,te [0, 1]
(23)
4. Experiments
all with zero initial/boundary conditions.
Baseline. We comprehensively compare the proposed QuanONet and TF-QuanONet with the following methods:
All the numerical simulations are performed on a machine with 190GB memory, one physical CPU with 32 cores AMD Ryzen Threadripper 3970X CPU, and 5 GPUs (NV GeForce RTX 3090). We implement a Python quantum simulator without noise to simulate QNNs Based on MindSpore (Xu et al., 2024) framework, and designing classical neural net- works using DeepXde (Lu et al., 2020) framework.
Quantum methods: QNN with hardware efficient ansatz (HEA) (Kandala et al., 2017), trainable-frequency HEA (TF-HEA) (Jaderberg et al., 2024).
Classic methods: fully-connected neural net (FNN) (Raissi et al., 2018)), DeepONet (Lu et al., 2021).
6

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QuanONet: Quantum Neural Operator with Application to Differential Equation
Susbreer oce
345E The
180
100
Figure 4. The A distribution of trained TF-QuauQ OF-QQunONeet with initial- ized 0.001 of all runs each problem. The distribution in nonlinear ODE case is significantly more dispersed.
Tea
Data Generation. We generate the dataset by using the mean-zero Gaussian random field with the form: G (2), where the covariance kernel (2), e [1 -1111~2211/22 is the radial-basis function (RBF) kernel with length-scale parameter 1 > 0 to sample input func- tion u. We solve the ODE systems by Munge-Kutta (4, 5) and PDEs by a second-order finite difference method.
301 1900
200
Figure 5. Errors of all methods trained to learn the homogeneous operator. The solid and dash lines are the training error and test error in training, respectively. The shaded regions are the one- -andard-derrivttoo from 5 runs with different training/test data and network initialization. All methods are plotted by uniform ordinate scale.
Settings. We scale the various QNNs to ensure that they have close number of parameters (1200 for ODE and 2400 for PDE) and use the Adam optimizer (Kingma, 2014) with learning rate 0.0001. The details are shown in the Ap- pendix A. The number of qubits of all QNNs is 5, and the Hamiltonian is = 2 o² given in Table 2, Table and Table 4.
02-
Detailed configuration
0008
42-
4.2. Results
92
We use 5 runs with different training/test data sampling and network initialization to compute the mean error and the standard deviation (SD), as presented in Table 1.
Method
Branch
Trunk
Params.
TF-QuanONet
Depth
Ansatz Depth
Depth
Ansatz Depth
1200
For ODE e.g. homogeneous operator (Fig, 5), QuanONet performs best under X= = 0.1, but the error increases signif- icantly as the hyperparameter X decreases. The trainable frequency technique improves TF-QuanONet's robustness to X, and makes it perform optimally in all cases. Note that all quantum methods have less generalization error than FNN. In the case of nonlinear ODE, QNNs generally per- form worse than classical methods, because this case has more spread out frequency domain distribution compared to other problems (Fig. 4), so deeper QNNs are needed to ensure frequency completeness. But our method still en- hances the best performance among quantum methods. Four representative cases are shown in Fig. 6 and Fig. 7.
20
Depth
Width
Depth
Width
DeepONet
1261
2
10
10
Figure 6. Anti-derivative operator results for the input function u = sin TT, with hyperparameters of networks in below.
real-devices. We train a 2-qubits TF- TT-QuanONee for the anti- derivative operator and test it on IBM brisbane Q57/Q58*. By leveraging Qiskit's compilation optimization techniques and using standard noise mitigation, the circuit depth is reduced to merely 20 layers. Using u(x)) = x as input func- tion and standard noise mitigation like zero-noise extrap- olation (Giurgica- Mlurgica-Tiron et al., 2020), we achieved MSE- 1.57e-3 (vs. 1.5e-5 simulation). The gap is mainly attributed to non-ideal gate operations and residual decoherence ef- fects.
The hardware noise characteristics in the NISQ are pri- marily influenced by qubit count, gate cost and circuit depth. We conduct extensive benchmarking across various qubit count, layer depths and Hamiltonian, with complete experi- mental results and analysis presented in Appendix F.
We acknowledge Shikun Wei who helped the IBM Quantum platform access and experiments during his stay in Paris, France.
Fig. further supports the potential of TF-QuanONet on

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QuanONet: Quantum Neural Operator with Application to Differential Equation
Table Test error comparison with different initial values A. The best in gray and second best in lightgray.
ODE
PDE
Method
A
Antideriv operator
Homogeneous operator
Nonlinear ODE
D-R System
0.001
0.095871 + 0.002140
0.267378 + 0.006169
0.900696 0 0.034579
0.000601 + 0.000022
HEA
0.01
0.053324 = 0.001074
0.127268 = 0.001941
0.705380 + 0.030721
0.000530 = 0.000039
0.1
0.030112 0.000759
0.084610 0.002567
0.369845 0.017249
0.006988 = 0.001866
0.001
0.002484 + 0.000028
0.00260S = 0.000048
0.045553 0. 0.007749
0.000113 + 0.000021
TF-HEA
0.01
0.002518 + 0.000040
0.002604 = 0.000039
0.044619 +: 0.008089
0.000102 : O 0.000008
0.1
0.002519 0.000035
0.002613 = 0.000032
0.044673 +: 0. 0.007877
0.000807 : 0.000131
0.001
0.131876
0.003809
0.330244 = 0.005671
0.869680 0.0315-16
0.000601 + 0.000033
QuanONet
0.01
0.043198
0.001002
0.11 = 0.003816
0.630765 + 0.031 184
0.0003417 = 0.000015
0.1
0.001341
0.000190
0.002005 = 0.000253
0.059582 + 0.008323
0.001802 + 0.001053
0.001
0.000113
0.000010
0.000192 = 0.000028
0.089072 0.007277
0.000055 1 0.000014
TF-QuanONet
0.01
0.000121 + 0.000010
0.000198 + 0.000013
0.039-169 + 0.008081
0.000051 + 0.000010
0.1
0.000112 2 + 0.000008
0.000181 = 0.000021
0.039074 + 0. 0.0074422
0.0001455 + 0.000028
DeepONet
0.000596 0.000215
0.000665 + 0.000117 17
0.027237 + 0.003972
0.000088 + 0.0U0036
PNN
0.000934 + 0.000594
0.001 187 + 0.000732
0.028487 + 0.006250
0.000089 + 0.000066
11/0
613
Desodhed E/W
V-ousdhet
Yut
Sup
TT-Cardive
Method
Qubits
Params.
Gates
Trerations
TF-QuanONet
2
960
1080
100000
Figure 8. Inference results of a 2-qubits TF- QuanONet and Deep-
ONet anti-derivative operator. The input function is u(x)
Desplies
ser
sea
niques help TF-QuanONet learn the dominant frequency of the operator, thus providing robustness.
Broad Applicability. TF-QuanONet effectively handles various operator learning tasks, showcasing its versatility and potential for solving complex differential equations.
Method
Branch
Trunk
Params.
TF-QuanONet
Depth
Ansutz
Depth
Amsatz Depth
20
Depch
Width
Depth
Widhh
DeepONer
2521
5. Conclusion and Outlook
Figure 7. D-R system results for input functions u (ahove) and sin न. (helow). The hyperparameters of networks are shown below. The predictions and the truth share the same colorbar, and errors of two methods plotted on another same colorbar.
We have derived the universal approximation theorem of QNNs for quantum state functions, giving a quantum ver- sion the operator approximation theorem and construct- ing quantum neural operator QuanONet based entirely on quantum circuits. To address the frequency spectral locality issue that may arise due to the coding coefficient setting, introduce the trainable frequency method and propose an improved version: TF-QuanONet.
The experiments collectively show that:
Expressiveness. TF-QuanONet achieves lower approxi- mation errors over diverse tasks, outperforming existing quantum methods at similar scale and model size.
A Acknowledgment
Optimization Efficiency. Coefficient setting significantly influences approximation error. Trainable frequency tech-
Work was partly supported by NSFC (92370201, 62222607).
8

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QuanONet: Quantum Neural Operator with Application to Differential Equation
Impact Statement
Farhi, E., Goldstone, J., and Gutmann, S. quantum
approximate optimization algorithm. arXiv preprint arXiv: arXiv:1411.4022 1411,4421, 2014.
This paper concerns the between of machine learning and quantum computing for the fundamental problem of dif- ferential equation solving. It has wide technical impact to Al and other disciplines and need to be careful the fast development of the potential technology.
Forets, M. and Pouly, A. Explicit error bounds for carleman linearization, arXiv preprint
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---

QuanONet: Quantum Neural Operator with Application to Differential Equation
A. Details of the Experiments
Table 2. Default parameters for each problem
Case
u space
Sensors #
Training
Testing #
Batch Size
Anti-deriv operator
GRF(/ 0.2)
100
10000
100000
100
ODE
Homogeneous operator
GRF(I = 0.2)
100
10000
100000
100
Nonlinear ODE
GRF(/ 0.2)
100
10000
100000
100
PDE
D-R system
GRF(/ -0.2)
100
100000
1000000
100
Table 3. Hyperparameters for each network in ODE case
Network Type
Branch
Trink
Gates #
Param.#
Iterations
Depth
Ansaez Depth
Depth
Ansozz Depth
QuanONet
20
20
1800
1200
100000
TF-QuanONet
20
2
10
2
1350
1200
100000
Depth
Ansatz Depth
HEA
40
1800
1200
100000
TF-HEA
32
2
1440
1280
100000
Depth
Width
Dqpth
Width
DeepONet
2
10
2
10
1261
100000
Depth
Width
FNN
10
1251
100000
Table Hyperparameterss for each network in PDE case
Network Type
Trink
Gates #
Param.#
Iterations #
Depth
Ansatz Depth
Depth
Ansatz
QuanONet
40
2
40
2
3600
2400
100000
TF-QuanONet
40
2
20
2
2700
2400
100000
Depth
Ansatz Depth
HEA TF-HEA
2880
2400
100000
3600
2560
100000
Banch
Trink
Depth
Width
Depth
Width
DeepONet
15
15
2521
100000
Depth
Width
FNN
16
2481
100000
B. Further Related Works
B.1. Neural operators accelerated by quantum computing
DeepONet entails quadratic complexity concerning input dimensions during evaluation. Given the progress in quantum algorithms and hardware, Xiao al. (2024a) proposed to utilize quantum computing to accelerate DeepONet evaluations, yielding complexity that is linear in input dimensions. The quantum DeepONet they proposed integrates unary encoding and orthogonal quantum layers. However, since the quantum DeepONet only utilizes quantum circuits to speed up DeepONet's
12

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QuanONet: Quantum Neural Operator with Application to Differential Equation
disidcal
Pyramd33
Pycomid
Brunch
Quastum Laper
Querium Layer
Output layer
Figure (Xiao et al., 2024a) Architecture of quantum DeepONet. The Branch Net and Trunk are replaced these with QOrthoNN, which is composed of several quantum layers arranged sequentially. The nonlinear operations performed on classical computers.
evaluation, this means that we cannot expect it to achieve better results than DeepONet (or even worse if noise is taken into account), provide results of DeepONet alternative.
B.2. Fourier neural operator with quantum Fourier layers
Jain al. (2024) proposed a new quantum circuit for performing a quantum fourier transform (QFT) on an input encoded superposition of unary states which can be sequentially combined to form QFNO, quantum version of trainable Fourier neural network for solving Parametric PDFs.
(a)
(b)
W
No
[A]
IFT
fix, 0)
1OFT
Figure 10. (Jain et al., 2024) (a) The proposed Sequential Quantum Circuit, which replicates the classical FNO operation from (b) if we measure at the end.
Compared with classical FNO, QFNO uses QFT to construct Fourier transform layer, but the advantage is not obvious. In numerical experiments. QFNO does not observe better performance than classical FNO. Jain et al. (2024) proposed that due to the efficiency of the QFT, QFNO has logarithmic time complexity compared to N, input dimension, while its classical counterpart has linear time complexity as shown in Table 5. However, the number of qubits and circuit depth of QFNO will still bring huge noise to current quantum devices, making the QFT impossible.
provide the results of FNO with 100 initial functions as training instances, as shown Tab.
13

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QuanONet: Quantum Neural Operator with Application to Differential Equation
Table (Jain al., 2024) Comparison the order of time/depth complexities of the quantum Fourier Layer (QFL) in QFNO eircuits with the classical Fourier Layer (FL). Here N, denotes the sampling dimension, N, denotes the channel dimension and K denotes the maximum number modes allowed. If QENO is used solve benchmark problem in this paper, means that multiple low-noise quantum circuits with up to 100 qubits are needed, which cannot be realized using the existing quantum devices.
Method
Qubits
Circuits
Gate Complexity
Depth Complexity
Classical FL
Ne + N₂ log No
Parallel QFL
Ne + N₂
K
K KN log No + KN,,,, log N, KN log N: N,N, log N₂ (Ne + log((N + K) + N.N, No N.
No + 1og N,
Sequential QFL
Ne + N₂
1
KN. + Ne log N,
Composite QFL
No + N.
1
(og(N. + K) + N₂ log N.
Table Test error comparison of TF-QQanONet, DeepONet and FNO
Method
Data type
Iteration
Antideriv operator
Homogeneous operator
Nonlinear ODE
Aligned
10000
0.002763 + 0.000107
0.003424 + 0.000270
0.114749 = 0.021397
TF-QuanONet
Semi-aligned
10000
0.002524 + 0.000153
0.003020 + 0.000151
0.077709 + 0.011518
Aligned
10000
0.004308 + 0.000332
0.006661 + 0.002313
0.856181 + 0.127804
DeepONet
Semi-aligned
10000
0.003557 + 0.000342
0.006016 + 0.003654
0.279827 0.071935
Method
Batch Size
Iteration
Antideriv operator
Homogeneous operator
Nonlinear ODE
100000
0.031340 + 0.008230
0.086708 = 0.029126
0.330201 = 0.149577
FNO
10
10000
0.004615 + 0.001619
0.009267 + 0.002576
0.077072 + 0.015786
50
2000
0.006853 + 0.001724
0.015811 + 0.003861
0.096177 + 0.026694
B.3. Physics-informed quantum neural network For the output of a QNN as follows:
fo(x) = (vo(x)
(24)
Define another two sets of inputs 0 and x()) as:
(+)
[20, 21, ..... 2y 2: x()) = [20, ".. 24 -''.)
(25)
Thus the partial derivative of f(x) can easily calculated by
vox) -10(((00)))) = -
(26)
Based on the differentiability of QNNs like this, Xiao et al. (2024b) introduced network (PI-QNN) by employing the QNN as the function approximator for solving forward and inverse problems of PDEs. Numerical results indicate that PI-QNN demonstrates superior convergence over PINN when solving PDEs with exact solutions that are strongly correlated with trigonometrie functions. Kyriienko et al. (2021) proposed similar approach earlier and called it differentiable quantum circuits (DQC), and designed Chebyshev quantum feature map that offers powerful basis set of fitting polynomials and furnishings rich expressivity.
quantum neural
They are all quantum methods designed for solving differential equations with specific input functions, thus are fundamentally different from the quantum neural operators proposed in this paper.
B.4. Fault-tolerant quantum algorithms for differential equations
Berry (2014) proposed a fault-tolerant quantum algorithm which solves sparse systems of linear ODEs by discretising the system of ODEs and subsequently employing the HHL algorithm (Harrow et al., 2009) to solve the resulting system of linear equations. Childs & Liu (2020) also proposed a quantum algorithm for solving linear ODEs, which, however, relies
14

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QuanONet: Quantum Neural Operator with Application to Differential Equation
Equation residuals
QPU
f(x.t) Uy + 4,50
Layer I
Layer
Loss Lyoz Aep Aauas
Laane Spensal Y
Done
Figure 11. Schematic diagram of PI-QNN (Xiao et al., 2024b).
on spectral methods. Spectral methods use linear combinations of basis functions (e.g., Fourier basis) to approximate the solution. This approach also ends with solving a linear system of equations on quantum computer. A quantum algorithm to solve quadratically nonlinear ODEs under certain conditions is described in (Liu et al., 2021), using the Carleman linearisation (Carleman, 1932; Forets & Pouly, 2017; Kowalski & Steeh, 1991) to approximate the nonlinear part. The Carleman linearisation represents a finite-dimensional polynomially nonlinear system by an deinite-dimensional linear system. To make use of the Carleman linearisation, the dimeniiodi linear system is truncated a certain point. Subsequently, Liu et al. (2021) again discretised the resulting system and solve the linear system with HHI. (Harrow et al., 2009). The algorithm presented in (Liu et al., 2021) may also be applied to solving certain PDEs (with restricted kind of nonlinearity), as the discretisation of PDE in all but one dimension generally results in system of ODEs. Liu et al. (2023) presented an algorithm for solving (nonlinear) reaction-diffusion equations. Using Euler's 's method, as well as the Carleman linearisation for the nonlinearity, they discretise the PDE and solve the resulting system with HHL. Childs et al. (2021) presented quantum algorithms for solving linear PDEs by making use the finite difference method (FDM) and spectral methods separately. In the FDM approach, they discretise the PDE on a grid. Both cases result in a linear system of equations which needs to be solved. The authors of (Arrazola et al., 2019) also present a quantum algorithm for solving linear PDEs, which relies on Hamiltonian simulation of cleverly chosen Hamiltonian, which encodes certain properties of the PDE. Lloyd et al. (2020) outlined a quantum algorithm to solve nonlinear ODEs by mapping the ODE to the nonlinear Schrödinger equation, which is then solved using Trotterisation. Another numerical scheme, the finite element method (FEM, which approximates the solution by using interpolating functions within each discretised element) and HHL are used to solve elliptic PDEs in (Montanaro & Pallister, 2016). Jin & Liu (2022); Jin et al. (2023; 2024) derived quantum algorithms for solving nonlinear ODEs as well as the nonlinear Hamilton-Jacohh (HJ) equation (which a special case of the nonlinear Hamilon--Jcobi-Bellln
PDE). They do so, by mapping the nonlinear ODEs and the nonlinear HJ equation to linear ODEs and HJ equations using linear representation methods, such the level set method, and then use HHL to solve the linear system.
C. Partial Fourier Series Representation of State Functions
We show how a certain class of L = 1 quantum models naturally realise states with amplitudes of multivariate Fourier series. We consider a output quantum state of the form:
(x(()) = (X)))
(27)
with
S(x) = e iiz 3 ... @ ixx
(28)
where
2
r
T sx dxlix @ 1 ijn)
(29)
fi =x=1
is some arbitrary state, and W is some arbitrary unitary operator. To simplify the index handling, we introduce the
15

---

QuanONet: Quantum Neural Operator with Application to Differential Equation
multi-indices (2 with which rewrite
IT) = Σ mili)
(30)
Additionally, as argued before we can without loss of generality assume that all Hamiltonians are diagonal, i.e.:
HA di (k)
(K)
(31)
With this assumption, we note that Sxx is diagonal with entries
[S(x)]is = ix
(32)
where we have defined
\ - )
(33)
Given this, we see that
(x)x)
=
J
k
(34)
ΣΣ
73 W A1
which is indeed a partial multivariate Fourier series, with the accessible frequencies fully determined by the spectra of the encoding Hamiltonians {H and the Fourier coefficients determined by the trainable unitaries (or equivalently, the state and Hamiltonian).
D. Proof of the Quantum Universal Approximation Theorem for State Functions
We provide in this section proof of the Theorem 3.1.
Theorem 3.1 Let Hm be universal Hamiltonian family, and v/m the associated quantum state family:
1m = W WS (x)[T).
(35)
For all normalized quantum state functions (x(x)) Ez (x) with functions 9j [2[(0,2m]*) and for all >> 0,
there exists some mn E N, some state IT) E Cam and some W such that
I1l (x)) - [o(x))]1
(36)
Proof. To begin with, we note that we can approximate any given E [2( 2n) ) ), up to an arbitrarily small error in L2 norm, by using a truncated Fourier series (Weisz, 2012). More specifically, for any e 0, there exists some K N and some set of coefficients {c,/n e Z ). such that
K ix.n
o(x)
Σ
(37)
DEZN
satisfies
11x(x) g(x)) 5
(38)
Thus we rewrite (o(x)) with (x) EnEzy, Gpeien
(x(x)) = Σ Σ
Cine = Σ
Σ Cinexxm
(39)
3 nEZK
nEEN
where K, max KK)).
16

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QuanONet: Quantum Neural Operator with Application to Differential Equation
To prove the theorem we therefore only need show that there exists m' N, some state (T)) and some unitary operator W that the output quantum state It/m (x)) generates the state function [o(x)) Recall that the output quantum state is defined as
(x)) = W S. (x)
(40)
with
SH... (x) = -in @ ... @ izx Hm
in
(41)
In Appendix C, we have seen that express the output quantum state
lx(x)) = ΣΣ
W
(x))xxii
j
k
(42)
Σ
Σ
73 W j.k (x
where the multi-indices j and k have N entries that iterate over all 2 basis states of the d qubit subsystems. Let A be the eigenspectrum of Hm, as defined in Eq. 9. As the {Hm] form a universal family of Hamiltonians by assumption, we can choose an m E N so that
kx (-
K.} S AHT..
(43)
The accessible frequency vectors >j independently contain all possible combinations of the frequencies in A AH
The vector-valued frequency spectrum for the multivariate case is therefore the Cartesian product of N copies of AHr.
Ω AH x ... X AH
(44)
times
As Σg S AH; we naturally have that N C Q which means that the Fourier series generated by the chosen model contains all terms that are necessary construct the Fourier series state (x(xx). With the coefficients which can be freely chosen, Eq. 42 can be easily chosen be the form of 39. 39.
E. Proof of the Quantum Universal Approximation Theorem for Operator
We will often use the following notations.
K: some compact set in a Banach space.
C(K): Banach space of all continuous functions defined on K, with norm [|f max (x())
C,[-1, 1):: All 2-periodic functions with period 2 with respect to every variable 1,...
Lemma E.1. Suppose that K is compact set in IR, f C(K), then there is continuous function E(f) E C(B"), such that (I) fxx) (x) for all x E K; (2) supzeR= [E()((x) S supZek (x)]; (3) there is a constant e such that
sup Ix' I<5
E(f(x') -(()(x") s
sup
1f(x)) f(x")
(45)
Ix'-X" <x_x'
Proof. The proof of Lemma E.1 can be found Stein (p.175).
Lemma E.2. V is a compact set in C(K) if and only if
1. V is a closed set in C(K),
2. there is a constant M such that ()(())))) M for all f EV.
3. V equicontinuous, i..,, for any >> there exists such that [f(x') f(x")) e for all f V, provided that
xxx" K and I|x' x"Ilx
Proof. The proof of Lemma E.2 can be found Dieudonné (2011) 142).
17

---

QuanONet: Quantum Neural Operator with Application to Differential Equation
Lemma E.3. Suppose that K is a compact in I [0, 1J V is a compact in C(K), then can be extended compact set in Cpl-1,1] 11
Proof. By Lemmas E.1 and E.2, V can be extended to be a compact set V1 in C(O,, 1]". Now, for every f E V1, define an even extension of f as follows:
(21. Zg. ... ,nn) f(11
-k:" ; ;nn)
(46)
Then U E if f E ViJ is the required compact set in Cpl-1,
Lemma E.4. Suppose that U compact in 1p(-1,1}
Bulf:x) = Σ (1
) cm cm(() f e
(47)
R²
Jm SR
the Bochner-Ries means of Fourier series of f. where m = [m1 , m,n. |m|
EE= |m,l?, Cm((f are Fourier
coefficients of J, then for any > there is such that
Bn(six)= - s(x) S
(48)
Proof. The proof of Lemma E.4 can be found in Stein & Weiss (1971).
Lemma E.5. Suppose that (f)) a continuous m- state function defined on the Banach space C(K), (o(x)) is a
continuous m-dim state function defined on the compact set K, there exist a real constant c, (m + 2) 2)-mim normalized state functions N (f)) and (b (x))), such that
(x(f))l(x)) = c(P(())((x))
(49)
Proof. Without loss of generality, we can write (r)(s)) and [x(x)) as
m
w(N) Σ a(M)) ((x)) = Σ (x)))
(50)
Since their amplitudes are both bounded, we define the 1-norm of the state function [[)()) = max (f)l. Because the case of [|))()))) 0 [| is trival, we can write (f)) and (+xx) as:
((O) IN
9:(5)l + 1m+1 (f) + 1))
(51)
[x(x) = ||
(x)li) hm+2(x) + 2))
Since the norms of the first dimensions less than or equal 1, there exist fit and 9:42 to normalize (())) and (x)).. Thus exist a real constant such that
(1(())(())) c(V(F))(())
(52)
Theorem E.6. Suppose that K is a compact set in RR, and U is compact set in C(K), then any > 0, there exist positive integer N. a real constant c, a -im state function it), which is independent of f E C(K), and a N-dim state functional (b) depending on f, such that
1s(x) c(bC)))(()) se
(53)
holds for all x E K and f E U.
18

---

QuanONet: Quantum Neural Operator with Application to Differential Equation
Proof. Without loss of generality, we assume that K S 10. By Lemma E.3, can assume that K [-1, 1J" and
U _ [1, By Lemma E.4, for any O, there exists R > O, such that for any [x1, Fx) E [- 1, 1]" and U,
there holds:
m 2 R2
Σ
Cm1 m,,, (x)))xxx((mmm + ...4 + mn)))) - f (. [x)]< <E
(54)
mm|R
By the definition of the Fourier coefficients and the evenness of fxx), we can rewrite as:
Σ dm (S*) (ros(((mma)) m,).n) (xi, , (xi, xn) <e
(55)
m RR
where mm....... real numbers. It is obvious that for every [-1, the first item is the inner product of two states:
Σ dm, mm (m)
mlsR
(56)
cos( + ... + m,)a))m
m R
By Lemma E.5, exist a positive intege N, real constant c, a N-dim state functional (6({*)) and a N-dim state function [t(x),, such that
(F((x) - ((K)))(( <E
(57)
true for all x [-11 and f* E V. Thus
If(x) - c(b(f()(((() <e
(58)
is true for all x E 10, and f E E.
Lemma E.7. (1:) a sequence such that 1n >> 1/2 ... t/2 0 and {n(*)) [=) is sequence positive integers such
that n(m))= < n2(22) < n(Tn)) + such that the first n(T))) elements ((T)) {01,... :nn(na) is nn-net in K.
We define functions
||x
x;lla
i||x - x111x S n
7k
(59)
otherwise
and
T Mk.
(x)
(60)
mal ) T me.t (x)
for j = -,,.. n(()). n(()).. For each V, we define function
Way (x) =
Σ (m.s)Tmm.s())
(61)
1=1
and sets Vm = {un u V} and V* UUUU Vnn).. ). We then have the following results:
1) For each fixed k. Von is compact set in subspace of dimension r((n)) C(K).
2) For every u E V, there holds
||u Uyy (lO(K) S 6x
(62)
3) V* is compact set in C(K).
Proof. The proof of Lemma E.7 can be found Chen Chen (1995).
19

---

QuanONet: Quantum Neural Operator with Application to Differential Equation
Theorem E.8. Suppose that X is Banach Space, K C X is a compact set, V is compaci in C(K), is a continuous functional defined on V, then for any O, there exist a positive integer N, real constant c, a N-dim state function (t), which is independent of f C(K), and state functionaa (b) depending on f. such that
If(u) - <(((f))((())
(63)
holds for all u E V.
Proof. By Tietze's Extension Theorem, can define continuous functional V* such that
(") f(u) if u e
(64)
Because f is continuous functional defined on the compact set V therefore for any c > 0, we can find 5 such that (v) (v)) e/2 provided that u, V* and ||u (u--vi ulccck) ò. Let & be fixed such that dy 6, then by proposition 2. of Lemma E.7 for every V.
||u -aa-uylll llx < 5R
(65)
which implies
(f - (" < 6/2
(66)
for all u E V. By proposition of Lemma E.7, we see that f (umm) is a continuous functional defined on the compact set Vn, in Ru(m)), By Theorem E.6, we can find N. c, (6), (6), such that
(u -(6()))(())))
(67)
We conclude that
If(u) ((b())(((())((())) <E
(68)
Thus, Theorem E.8 is proved.
Theorem E.9. Suppose that X is Banach Space, K1 X, K2 C R" are two compact sets in X and R" respectively, V is compact set in C(K1). G is nonlinear continuous operator, which maps V into C(K2), then for any O. there exist positive integer N, real constant c, a N-dim state function (t) and a N-dim state functional (b), such that
(G(u)(y) ((b((ee((())))))) <e
(69)
Proof. From the assumption that G is continuous operator which maps compact set V in C(K,) into C(K2), it is straightforward to prove that the range G(V) {G(u) V} is also a compact set in C(K2). By Theorem E.6, for any >> 0, there are positive integer N, real constant c, N-dim state function It) and a -idim state functional b), such that
(G(u)()) c(b(G())))(())) <e
(70)
holds for all K2 and u E V.
Since G is a continuous operator, combining with the last proposition of Theorem E.6, we conclude that for each k 1. N, (b))) is continuous state functional defined on V.
(G(w(y)) - c(((())(((())))))) <e
(71)
F. Extensive benchmarking
20

---

QuanONet: Quantum Neural Operator with Application to Differential Equation
Table 6. Test error comparison of TF- QuanONet with different qubits and params. #. where all Trunk Depth is set to 10 and Ansatz Depth is to Under the condition different qubit numbers, performance TF- Ne-QuannNee generally increases with increase of Branch Depth. Interestingly. our method performs better at low qubits, which may be because the deeper Branch Net improves the expressivity in the frequency domain.
Qubits #
Branch Depth
MSE
Param. #
Epochs #
50
6.5e-5
960
1000
2
100
5.8e-5
1760
1000
150
4.9e-5
2560
1000
25
4.4e-5
1120
1000
50
3.2e-5
1920
1000
75
3.7e-5
2720
1000
20
2.2c-4
1200
1000
40
1.7e-4
2000
1000
60
1.3e-4
2800
1000
10
3.15e-4
1600
1000
10
20
1,45e-4
2400
1000
30
1.05e-4
3200
1000
Table Execution time of TF-QuanONet on quantum simulator and DeepONet with 10000 training instances based on different frameworks. The forword time of TF- TTF-QuanONet about 10- 100 times that of DeepONet the same scale
TF- OF-QuanONeet
DeepONet
Framework
Training
Inference
Training
Inference
DeepXde 82.1543
0.000059858s
0.8119s
0.000000414s
Torch 168.0444s
0.000082172s
0.9917s
0.0000004110 Is
MindSpore
10.4797s
0.000008763s
1.5054s
0.000000467s
Differen: Spec-Radius
10-3
10
10-2
10-1
1
10
10
20000
40000
60000
80000
100000
20000
00000 60000 Iteration
80000
100000
Beration
Figure Error curve of TF- Qu--Quan different Hamiltonian anti-derivative operator, means H f of with spectral radius equals 7. Further increasing the spectral radius has little impact on the results, but too small spectral radius will limit the range of solution function and affect prediction accuracy.
21


Reviews:
[{"review_id": "m8HKZ2Y6YW", "reviewer": "Reviewer_CFgz", "summary": "The paper introduces QuanONet, a quantum neural network framework for learning nonlinear operators in differential equations. The primary contribution is extending classical universal approximation theorems for operators to quantum state versions. Two variants are proposed: a standard QuanONet with hardware-efficient pure quantum circuits, and TF-QuanONet featuring trainable frequency encoding to improve generalization and overcome coefficient selection challenges. Experimental evaluations demonstrate competitive performance compared to classical approaches and superior results relative to existing quantum methods on various differential equation problems.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> The performance gain over classical methods isn't consistently demonstrated across all cases, which weakens the overall impact.\n\nWe add more experiments with larger scale with batch size 100, 100K max iterations and for five runs. \n\n1) ODE: 10K train instances and 10K test instances.\n2) PDE: 100K train instances and 100K test instances.\n\n The results are given in [Tab. 1](https://anonymous.4open.science/r/ICML-2025-rebuttal-42ED/Table1.png). TF-QuanONet outperforms other quantum methods on all problems and is superior to classical methods except for nonlinear operators.\n\n    \n>How would quantum hardware noise affect QuanONet performance, and what error mitigation strategies would be appropriate for practical implementation?\n\nCompared to other influential quantum algorithms, QuanONet enjoys the advantages of fewer qubits, lower demand for topological connectivity, and a simpler gate set that is easy to implement (e.g., avoiding complex multi-controlled gates). It can be further improved by noise mitigation (e.g. ZNE, MPS pretraining, and noise modeling), similar to [W. Sun, J. Xu, and C. Duan, Noise-Mitigated Variational Quantum Eigensolver with Pre-training and Zero-Noise Extrapolation](https://arxiv.org/abs/2501.01646).\n\nBesides, we implemented physical device testing on IBM brisbane using the y=x anti-derivative problem as benchmark with standard noise mitigation (gate twirling, ZNE, TREX). The hardware implementation achieved MSE = 1.57e-3 (vs simulation MSE = 1.5e-5), with residual error primarily attributable to non-ideal gate operations and residual decoherence effects (T1/T2 = 252/178 μs, ECR error = 7.9e-3, readout error = 1.51e-2). Extended details and nalysis are provided in our fourth response to reviewer Eswa.\n\n\n>  What mechanisms drive the overfitting phenomenon observed with TF-QuanONet, and what strategies might mitigate this behavior?\n\n 2k training instances in our initial submission is too small to fully reflect the frequency characteristics of the problems, thus TF-QuanONet mislearns the dominant frequency. We also find that DeepONet exhibits overfitting on these 2k training instances.\n In added experiments, as the number of instances increases to 100K, the performance of TF-QuanONet is significantly improved and is much better than other quantum methods.\n\n> Beyond trainable frequencies, what alternative approaches might address the coefcient initialization challenge?\n\nWe envision that the dominant frequency of the problem can be learned by a specialized small-scale TF-QNN. We do not care how the small-scale QNN performs, but rather provide a good coefficient setting strategy for the larger scale QuanONet through its trained coefficient distribution. For the time being, we leave it as future work due to the short rebuttal period.\n\n\n> How does QuanONet relate to Neural Network Quantum States research, and could insights from that domain inform further development?\n\n It's an interesting association. QuanONet uses parameterized quantum state to construct Neural Operator (Quan4AI); NN Quantum state uses NN to learn quantum states (AI4Quan). They both embody the research promise of combining AI with scientific problems. We will add discussion in our final version.\n\n> What specific properties of the diffusion-reaction system make it particularly amenable to quantum approaches?\n\n D-R system exhibits a lower frequency distribution compared to the other problems as shown in [Fig. 1](https://anonymous.4open.science/r/ICML-2025-rebuttal-42ED/Fig1.jpg).\n\nThe frequency characteristics are closely related to the performance of QuanONet. For low frequency problems, QuanONet only needs a lower depth to achieve spectrum coverage. With this condition, it prioritises learning the dominant frequency of the problem (unlike NN learning from low frequencies) based on the frequency principle and hence more efficient. [Xu Y H, Zhang D B. Frequency principle for quantum machine learning via Fourier analysis.](https://arxiv.org/pdf/2409.06682)\n\n> Given the specific Hamiltonian requirements for diferent problems, how feasible would transfer learning or multi-task learning be with this architecture?\n\n The choice of Hamiltonians are discussed detailly in the first response to reviewer LZaG. \n    \nThe input of neural operator can be extended to a tensor product of multiple functions (initial functions, driving terms, boundary conditions, etc.) to construct a multi-task learning method for differential equations. [Jin P, Meng S, Lu L. MIONet: Learning multiple-input operators via tensor product](https://arxiv.org/pdf/2202.06137) uses low rank approximation to avoid exponential dimension. For QuanONet, if a number of branch nets act independently on states $|b_i(u_i)\\rangle$ and input functions $f_i$, the entire quantum system $|b_1(u_1)\\rangle\\otimes\\cdots\\otimes |b_n(u_n)\\rangle$ is naturally in tensor product form, as shown in [Fig. 2](https://anonymous.4open.science/r/ICML-2025-rebuttal-42ED/Fig2.jpg).", "claims_and_evidence": "The theoretical foundations presented in the paper are robust, with formal proofs (Theorems 3.1 and 3.2) providing sound theoretical grounding. Empirical results across several differential equation problems validate that QuanONet outperforms alternative quantum methods when properly configured, while TF-QuanONet demonstrates enhanced robustness across different initialization settings.\n\nClaims regarding NISQ device applicability require additional substantiation, particularly concerning error propagation and mitigation in authentic quantum hardware environments. The comparison methodology with quantum approaches is methodologically sound, though comparisons with classical methods present limitations due to implementation differences.", "methods_and_evaluation_criteria": "The experimental framework uses appropriate differential equation test cases with varying complexity levels. \n\nHowever, the analysis primarily focuses on approximation error metrics without exploring training efficiency, hardware requirements, or scalability characteristics. While the selected test problems serve as effective benchmarks, they may not fully represent the complexities encountered in advanced scientific applications.\n\nThe convergence analysis of TF-QuanONet would benefit from expanded investigation, particularly regarding the overfitting phenomena observed in error evolution plots. The simulations are conducted in idealized quantum environments rather than actual quantum hardware, leaving implementation challenges unaddressed.", "theoretical_claims": "The connection between theoretical principles and architectural implementation is clearly established, particularly in demonstrating how the frequency spectrum analysis informs the advantages of TF-QuanONet over the standard implementation. The theoretical formulation represents a significant strength of the paper. The extension from classical to quantum approximation theorems follows logical progression with well-structured proofs.", "experimental_designs_or_analyses": "The experimental design is generally appropriate but has some limitations. The experiments cover a range of differential equation problems with increasing complexity, and the authors test under different coefficient settings to demonstrate the robustness of their TF-QuanONet approach. The parameter counts are controlled across models for fair comparison, which strengthens the validity of the results.\n\nHowever, the authors use simulated quantum environments rather than actual quantum hardware, which leaves questions about practical implementation challenges. The analysis of convergence issues with TF-QuanONet (mentioned on page 7) is somewhat superficial and would benefit from deeper investigation. The error evolution plots (Fig. 4) show interesting patterns that warrant more detailed analysis, especially the overfitting phenomenon mentioned for TF-QuanONet in Fig. 4(c). Furthermore, the performance gain over classical methods isn't consistently demonstrated across all cases, which weakens the overall impact.", "relation_to_broader_scientific_literature": "The paper properly positions itself at the intersection of quantum computing and neural operators for differential equations. The authors provide a comprehensive background on both quantum neural networks and neural operators, especially DeepONet. The comparison with related work is thorough, covering quantum algorithms for solving differential equations, universal approximation theorems for QNNs, and neural operator approaches. The distinction between existing hybrid quantum-classical approaches (like QFNO and quantum DeepONet) and their pure quantum approach is clearly articulated.\n\nHowever, the paper could benefit from more discussion of how this work relates to broader quantum advantage questions, particularly given recent work on quantum neural network expressivity. Clearer explanations of how this approach compares to other quantum PDE solvers in terms of computational complexity, not just empirical performance, would strengthen the work.", "essential_references_not_discussed": "The paper covers most relevant literature, although it could benefit from some more discussion of nnqs/vmc with symmetries, some recent pinn related works, and  barren plateau.", "other_strengths_and_weaknesses": "The paper introduces a novel, theoretically-grounded approach to quantum neural operators, which represents a significant advance in the field. The trainable frequency technique is an innovative solution to the coefficient initialization problem, and the theoretical analysis connecting quantum circuits to Fourier series representations provides valuable insights. The pure quantum circuit design (without hybrid classical-quantum components) represents an advance over existing approaches.\n\nDespite these strengths, several weaknesses should be addressed. The practical implementation on near-term quantum devices is not addressed sufficiently, leaving questions about the actual feasibility in NISQ-era hardware. The performance advantages over classical methods are inconsistent and depend heavily on hyperparameter settings. The paper lacks analysis of computational complexity or potential quantum advantages in terms of asymptotic scaling. The convergence challenges with TF-QuanONet mentioned briefly deserve more investigation. The D-R system case seems to be treated differently from the other examples with less detailed analysis.", "other_comments_or_suggestions": "The paper would benefit from a more explicit discussion of limitations, particularly regarding current quantum hardware constraints. A clearer roadmap for how this work might lead to practical quantum advantage would strengthen the impact. More ablation studies on the effects of quantum circuit depth, Hamiltonian choice, and encoding strategies would provide valuable insights.", "questions_for_authors": "1. How would quantum hardware noise affect QuanONet performance, and what error mitigation strategies would be appropriate for practical implementation?\n2. What mechanisms drive the overfitting phenomenon observed with TF-QuanONet, and what strategies might mitigate this behavior?\n3. Beyond trainable frequencies, what alternative approaches might address the coefficient initialization challenge?\n4. How does QuanONet relate to Neural Network Quantum States research, and could insights from that domain inform further development?\n5. What specific properties of the diffusion-reaction system make it particularly amenable to quantum approaches?\n6. Given the specific Hamiltonian requirements for different problems, how feasible would transfer learning or multi-task learning be with this architecture?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "daa9vQ0U5S", "reviewer": "Reviewer_LZaG", "summary": "The paper introduces QuanONet, a quantum neural operator framework designed to solve differential equations using pure quantum circuits. The authors extend classical universal approximation theorems to quantum settings, proving that quantum neural networks (QNNs) can approximate operators for differential equations. They propose two architectures: QuanONet, a hardware-efficient quantum neural operator, and TF-QuanONet, which incorporates trainable frequencies to improve robustness. Experiments on antiderivative operators, homogeneous/nonlinear ODEs, and a diffusion-reaction PDE demonstrate that QuanONet outperforms existing quantum methods and competes with classical baselines like DeepONet in certain cases. The theoretical foundation, empirical results, and focus on NISQ-compatibility position QuanONet as a novel contribution to quantum machine learning for operator learning.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> The choice of 5 qubits and fixed Hamiltonian may limit exploration of quantum advantages. \n\nWe choose the number of qubits to be 5 mainly based on the experience of DeepONet that operator problems can be solved well by taking vector dimensions as 10-1000, so the quantum state dimension $2^5=32$ can balance accuracy and efficiency. This is also an acceptable number of bits for existing quantum devices.\nThe choice of the Hamiltonian does make sense. In our attempts, further increasing the spectral radius has little impact on the results, but too small spectral radius will limit the range of the solution function and thus affect the prediction accuracy.\n\n \n\nExperimental results of TF-QuanONet across varying qubit counts and branch depths for the antiderivative operator (note: not all models reached full convergence due to time constraints) as shown in [Tab. 2](https://anonymous.4open.science/r/ICML-2025-rebuttal-42ED/Table2.jpg) and [Fig. 3](https://anonymous.4open.science/r/ICML-2025-rebuttal-42ED/Fig3.jpg)\n\n    \n>  Quantum vs. classical architectures are fundamentally diferent, making direct comparisons less meaningful.\n\n The training efficiency of QNN is indeed currently not comparable to NN, so we focus on highlighting the advantages of QuanONet over other QNN frameworks. Although QuanONet experimentally performs better than classical methods when aligning parameters and number of iterations in most cases, it is currently still a common bottleneck to the community beyond our paper, for maintaining the performance on real quantum device due to the limited quantum hardware and noise. We will clarify this point in our revision.\n\n\n> Could you provide a more detailed construction of $W$ and $|\\Gamma\\rangle$ in Theorem 3.1 to clarify how arbitrary states are approximated?\n\n We apologize for the confusion caused by the expression brevity to the theory. $W$ is realized by a quantum repeated parameter layer that the whole Hilbert space is completely controllable, that is, the generators of the used parameter quantum gates $g=\\set{H_p}_{p=1}^P$ can span a dynamic Lie algebra(DLA) $\\mathfrak g$ of dimension $4^n$ ($n$ is the number of qubits, the corresponding algebra is $SU(2^n)$).\n\nTheorem [Controllability of molecular systems. Physical Review A, 51(2):960, 1995]. \n1. All coherent superpositions of states can be achieved if $S$ equals $U(N)$. This is equivalent to requiring that $\\mathfrak g$ be the Lie algebra of all $N\\times N$ skew-Hermitian matrices, which in turn is equivalent to requiring that the dimension of $\\mathfrak g$ as a vector space over the real numbers is precisely $N^2$. The latter two of the above equivalent conditions are also necessary for controllability.\n2. All probability amplitudes can be achieved if $S$ is compact and contains $SU(N)$ which is equivalent to demanding that $\\mathfrak g$ is the Lie algebra of all $N\\times N$ skew-Hermitian matrices. In particular, if all probability amplitudes can be achieved then one can obtain all coherent superpositions of states.\n\nBased on the above theorem, $W$ satisfying this property can approximate any $n$ bit quantum operator with ideal depth and  $|\\Gamma\\rangle=W|0\\rangle$ which is an initial state acted by $W$ can approximate any $n$ bit quantum state. We adopt HEA as the structure of Branch and Trunk layers with {Rz, Ry, CNOT} that act on any qubit (or near-connected qubits) to span $SU(2^n)$. But purely one type of Pauli rotation gate or RBS gate does not conform (generators of RBS gates only span a subalgebra of $SO(2^n)$).\n    \nThe detailed discussion of Theorem 3.2 can be found in the first reply torReviewer xycX.\n\n> Why are training iterations for PDEs vastly diferent between QuanONet (100) and DeepONet (10,000)? Would QuanONet maintain its advantage with equal iterations?\n\nOur added experiments in the first of our response to reviewer CFgz, provide the results of five runs on a much larger dataset, aligning all methods for 100K iterations, as a more convincing setting.  Our approach achieves robust generalization.  \nTaking the antiderivative operator as an example, the loss curves of all methods are as [Fig. 4](https://anonymous.4open.science/r/ICML-2025-rebuttal-42ED/Fig4.jpg). Here the (TF-)QuanONet's performance improves as the number of iterations increases more intuitively.\n\n>  How does QuanONet address limited qubit connectivity or noise in real NISQ devices?\n\nThe two-bit quantum gates in QuanONet are all nearest neighbor connected. Due to its hardware efficient design, it can benefit from a very small number of high-quality qubits, which fits well with the current trend from NISQ to FTQC era. We provide complementary experiments on IBM brisbane to further precisely this point, and detailed results can be found in the fourth reply to reviewer Eswa.", "claims_and_evidence": "- **Claim:** QuanONet is the first pure quantum neural operator.  \n  **Evidence:** The architecture uses only quantum circuits without classical components, differentiating it from hybrid approaches like QFNO. This is supported by the circuit design in Fig. 2.  \n- **Claim:** TF-QuanONet improves robustness by dynamically adjusting frequency spectra.  \n  **Evidence:** Table 1 and Fig. 4 show TF-QuanONet performs consistently across coefficient settings, unlike QuanONet.  \n- **Claim:** QuanONet outperforms classical methods like FNN and DeepONet.  \n  **Evidence:** Results in Table 1 show lower errors for some tasks, but comparisons are limited by differing parameter scales and training iterations (e.g., 100 vs. 10,000 iterations for PDEs). This claim requires further validation under matched computational budgets.", "methods_and_evaluation_criteria": "- **Strengths:** The use of Gaussian random fields for data generation aligns with prior work (Lu et al., 2021). The focus on NISQ-compatible hardware-efficient ansatzes is pragmatic.  \n- **Weaknesses:**  \n  - Parameter counts are matched across methods, but quantum vs. classical architectures are fundamentally different, making direct comparisons less meaningful.  \n  - The choice of 5 qubits and fixed Hamiltonian $\\(H = \\sum \\sigma_z\\)$ may limit exploration of quantum advantages.  \n  - Training iterations for PDEs (100 for QuanONet vs. 10,000 for DeepONet) skew performance comparisons.", "theoretical_claims": "- **Theorem 3.1 (Quantum Universal Approximation for State Functions):** The proof in Appendix D relies on Fourier series approximations, building on Schuld et al. (2021). While plausible, the proof lacks detailed steps for critical transitions (e.g., constructing $\\(W\\)$ and \\$(|\\Gamma\\rangle\\))$.  \n- **Theorem 3.2 (Quantum Universal Approximation for Operators):** The proof in Appendix E maps classical DeepONet structures to quantum states but does not rigorously address continuity or compactness in Banach spaces. A more formal treatment is needed.", "experimental_designs_or_analyses": "- **Training Data Size:** Only 2,000 samples for ODEs may be insufficient for robust generalization.  \n- **Coefficient Sensitivity:** QuanONet’s performance heavily depends on $\\(\\lambda\\)$ (Fig. 4), raising concerns about practicality.  \n- **PDE Experiment:** The 100-iteration limit for QuanONet vs. 10,000 for DeepONet undermines claims of efficiency. A fair comparison requires equal computational effort.", "relation_to_broader_scientific_literature": "The paper situates QuanONet within quantum methods for differential equations (e.g., QFNO, quantum DeepONet) and classical neural operators (DeepONet). However, it does not engage with recent hybrid quantum-classical operator approaches (e.g., Smith et al., 2023) that combine classical neural networks with QNNs, which could provide valuable context.", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "None", "other_comments_or_suggestions": "None", "questions_for_authors": "1. **Theoretical Proofs:** Could you provide a more detailed construction of $\\(W\\)$ and $\\(|\\Gamma\\rangle\\)$ in Theorem 3.1 to clarify how arbitrary states are approximated?  \n   *Impact:* A clearer proof would strengthen theoretical claims.  \n2. **Training Iterations:** Why are training iterations for PDEs vastly different between QuanONet (100) and DeepONet (10,000)? Would QuanONet maintain its advantage with equal iterations?  \n   *Impact:* If QuanONet’s performance degrades with more iterations, claims about efficiency would weaken.  \n3. **Hardware Constraints:** How does QuanONet address limited qubit connectivity or noise in real NISQ devices?  \n   *Impact:* Practical applicability hinges on robustness to hardware limitations.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "11wpdfYKvQ", "reviewer": "Reviewer_Eswa", "summary": "This paper proposes a new model namely QuanONet (and TF-QuanONet) which is the first model purely based on quantum circuit. The paper further generalizes the approximation theorem of DeepONet to quantum setting. It has experiments on ODEs and PDEs and it shows advantages of QuanONet compared to previous quantum neural networks.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> How many Qbits we would need to run this model?\n\nBased on the experience of DeepONet, operator problems can be solved well by taking vector dimensions as 10-1000, so only less than 10 qubits are needed to build QuanONet. More discussion is visible in our first response to reviewer LZaG. Notably, our method performs well even with only two qubits (MSE<1e-4).\n    \n> What PDEs does quantum methods have an advantage? Maybe high-dimension Schrodinger equation?\n\nQuantum methods exhibit particular advantages for operator problems at low frequencies, as the spectral lines can be effectively captured through finite-depth circuit implementations (see our detailed analysis in the first and seventh responses to reviewer CFgz). \nFor higher-dimensional PDE, since the operator approximation theorem requires input function discretization, they all face fundamental limitations of exponential input dimension.\n\n> It would be helpful to add Quantum DeepONet and Quantum FNO as baseline models.\n\n\nSince Quan-DeepONet only utilizes quantum circuits to speed up the linear layer of DeepONet inference, the same classical network setup is used for the nonlinear layer as well as the training part, so it's unable to achieve better results than DeepONet (or even worse) [Xiao P, et al. Quantum DeepONet: Neural operators accelerated by quantum computing](https://arxiv.org/pdf/2409.15683), so we provide results for DeepONet as an alternative.\n    \nIn the existing research on Quan-FNO, no advantage over FNO on operator problems is observed, and more than 100 qubits are needed, which exceeds the limitations of all Quantum simulators and quantum real machines, as shown in [Tab. 3](https://anonymous.4open.science/r/ICML-2025-rebuttal-42ED/Table3.jpg). (Quan-)FNO is only applicable to aligned data (i.e., sampling all sensors for each initial function) while QuanONet and DeepONet do not have such restrictions.\n    \nWe provide the results of FNO with 100 initial functions as training instances, as shown in [Tab. 4](https://anonymous.4open.science/r/ICML-2025-rebuttal-42ED/Table4.jpg)\n\n\n\n> Meanwhile, the authors mentioned noisy intermediatescale quantum (NISQ) and fault-tolerance in the introduction. It seems not supported in the experiments.\n\nThe hardware noise characteristics in the NISQ era are primarily influenced by qubit count, gate cost and circuit depth. We conducted extensive benchmarking across various qubit count and layer depths, with complete experimental results and analysis presented in the first response to reviewer LZaG2. Remarkably, TF-QuanONet demonstrates exceptional precision (MSE <1e-4) even with 2 qubits. Moreover, QuanONet's hardware-friendly circuit design philosophy and ultra-low qubit requirements enable full exploitation of modern quantum compilation optimizations, which result in a significant reduction in circuit depths. \n\nThe [Fig. 5](https://anonymous.4open.science/r/ICML-2025-rebuttal-42ED/Fig5.jpg) further supports the performance of our method on real-devices. We trained a 2-qubits TF-QuanONet for antiderivative tasks and tested it on IBM brisbane Q57/Q58 (T1/T2 = 252/178 μs, ECR error = 7.9e-3, readout error = 1.51e-2). By leveraging Qiskit's compilation optimization techniques, the circuit depth was reduced to merely 20 layers. Using y=x with 100 points in [0,1] and standard noise mitigation (gate twirling, ZNE, TREX), we achieved MSE = 1.57e-3 (vs. 1.5e-5 simulation). The gap is mainly attributed to non-ideal gate operations and residual decoherence effects.\n\nQuanONet shows the high cost-efficiency in utilizing the limited and currently available qubit resource. This feature is especially welcomed along the FTQC trend as recently discussed in the Nature paper: [R. Acharya et al., Quantum error correction below the surface code threshold](https://doi.org/10.1038/s41586-024-08449-y). Unlike the other QML areas like vision, which still requires large number of qubits for real-world data, We believe our technique is more promising for showing the value of QML.\n\n> if somehow we get exponential speedups using quantum method, how do we extract the solution? Is the observation costly?\n\nQuanONet extracts the solution by measuring the expectation value of a Hamiltonian composed of commuting single-qubit Z-terms and scaling the result. \n\n The observation cost primarily stems from three factors: the number of measurement groups, shots per group, and error mitigation overhead.\n\n1.Due to the Hamiltonian’s commuting structure, we can measure them simultaneously in a single group, avoiding the grouping overhead required for non-commuting observables. \n\n2.The shots required to achieve precision $\\epsilon$ scale as $O(n _{qubits}/\\epsilon^2)$, where the numerator arises from the independent variances of the $Z$-terms, independent of problem dimension.\n\n3.Experimental results on IBM brisbane demonstrate that $10^4$ shots suffice to achieve < 4e-2 absolute error, validating the practicality of our approach on NISQ hardware.", "claims_and_evidence": "I think the quantum neural operator needs more motivation -- Are quantum methods fundamentally faster for operator learning problems? For example, for which type of PDEs do we expect quantum methods to be faster?\n\nMeanwhile, the authors mentioned noisy intermediatescale quantum (NISQ) and fault-tolerance in the introduction. It seems not supported in the experiments.", "methods_and_evaluation_criteria": "The paper contains experiments on operator learning consisting of antiderivative operator, simple ODE, and 1-dimensional PDE. These problems can be perfectly solved using classical methods such as numerical solver and classical ML models. It would be more interesting to discuss which problem quantum method can potentially show advantage, and add experiments on these. \n\nBesides, while previous works such as Quantum DeepONet and Quantum FNO have classical components, it is still very interesting to add these baselines to see the gap of pure quantum and hybrid methods.", "theoretical_claims": "The paper contains an approximation theorem for quantum neural operator. The theorem looks reasonable to me. Whilemean, the proof seems a translation of linear algebra into quantum setting.", "experimental_designs_or_analyses": "N/A", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "While the motivation can be improved, the paper show potential to design neural operator using pure quantum circuits.", "other_comments_or_suggestions": "Comments:\n- It would be helpful to add Quantum DeepONet and Quantum FNO as baseline models.\n- Figure 4 does not provide much information. A table should be sufficient.", "questions_for_authors": "- What PDEs does quantum methods have an advantage? Maybe high-dimension Schrodinger equation?\n- How many Qbit we would need to run this model?\n- if somehow we get exponential speedups using quantum method, how do we extract the solution? Is the observation costly?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "yGTOWUwnSN", "reviewer": "Reviewer_xycX", "summary": "This paper proposes QuanONet, A quantum analogy of neural operators, which can be executed in quantum computers. The paper extends the classical universal approximation theorem. The proposed architecture QunONet retains powerful generalization of classical neural operators. The work also highlights a version called TF-QuanONet based on the trainable-frequency method, reducing the requirement of deep quntum circut repetition.\n\n## update after rebuttal\n\nAfter carefully reading the rebuttal and other reviews, I decided to keep my original score. The authors are encouraged to incorporate suggested updates in the revision.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> Judging by the brevity of Appendix E, I am not fully convinced whether these theorems show how the architectural advancements presented in this works is justified. \n\nWe apologize for the confusion caused by the over-brevity of the theory and add a detailed proof of a stronger version of quantum universal approximation theorem for operators with the form of $|f(x)-c\\langle b(f)|t(x\\rangle|\\leq \\epsilon$ and rigorously address continuity and compactness in Banach spaces. Due to space limitations in Rebuttal, we would like to ask if it is possible to provide the details of the proof in pictorial form?\n\n> The explanation of TF-QuanONet is not sufficient for me to understand. Furthermore---if I understand the results correctly---TF-QuanONet strictly performs worse than QuanONet(\"error evolution of TF-QuanONet is poor,\" and \"shows obvious overfitting phenomenon.\") Even tough, TF-QuanONet reduces the burden of setting coefficients, I think TF-QuanOnet does not have noticeable benefits. This raises the question of presenting claims TF-QuanOnet as in the introduction section.\n\n In initial experiments, the 2k training instances was too small to generalize. We provide the results of five runs on a much larger dataset with batch size 100 (Training instances: ODE 10k, PDE 100k; Testing instances:  ODE 100k, PDE 1000k), aligning all methods for 100K iterations, as a more convincing setting.  The results can be seen in our first response to reviewer CFgz and fourth response to reviewer LZaG.  TF-QuanONet outperforms other quantum  methods on all problems and is superior to classical methods except for nonlinear operators. Its performance almost  does not depend on coefficient selection.\n\n> The formatting style of Figures 4 and 5 is not good; I suggest to increase the font size of numbers and labels. \n\n Thanks for your suggestions, we have updated the figure in the new link.\n    \n> What is the computational complexity (perhaps execution time) of QuanONet, TF-QuanONet and other baselines?\n\n  For QNNs, params-shift method is widely used for gradient calculation, that is, the gradient of the parameters is obtained by two measurements. Therefore, the training complexity of QNNs mainly depends on the number of parameters and the selection of observations. This is all aligned in our experiments. More discussion of the complexity of the measurements can also be found in our fourth response to reviewer Eswa.\nThe comparison between classical and quantum methods is not easy, especially since current quantum computers require more shots to mitigate noise. We provide a comparison of training and inference time (averaged over 1e6 iterations on 10k training instances and 1e7 inferences) between quantum simulators (implementing TF-QuanONet with classical computation) and DeepONet under different frameworks as shown in [Tab. 5](https://anonymous.4open.science/r/ICML-2025-rebuttal-42ED/Table5.jpg).\n\n\n\n> Regarding the conclusion \"How to compare QuanONet and classical neural operators is a controversial issue.\", I do not agree with this opinion. As the QNNs are capable of modeling various ODEs and PDEs, the authors are encouraged to point out actual performance comparisons and discuss the particular reason QNNs are not as efficient as classical neural operators.\n\n Thanks for your suggestion. We further add new comparison between QNN and classic operators, and give more discussion based on the new results. Please also check our first response to reviewer CFgz and fourth response to reviewer  LZaG for specific new experiments and discussion which we will add to our revisio.\nFor our sentence, we will remove it and give a more specific discussion based on our new results.", "claims_and_evidence": "The main claims presented in this paper are versatility and scalability incorporating advancements of DeepONet, such as Branch Net and Trunk Net.  With unique architectural properties for QNNs such as hardware efficient encoder, entangle, and ansatz layers the proposed architecture was able to outperform other quantum methods. Compared to other QNNs, such as QFNO and Quantum DeepONet, the authors argue that QuanONet truly integrates the spirit of QNNs into the core architecture applicable to quantum computing machines. It reads that this property enables the authors to extend QNN-related approximation theories (Theorems 2.1 & 2.2) to Quantum State Functions (Theorem 3.1). Lastly, this facilitates of TF-QuanONet, which overcomes the difficulty of setting coefficients and improve the robustness.", "methods_and_evaluation_criteria": "Benchmark data such as antiderivative operators and diffusion-reaction systems are synthetically generated, and can be replicated by the main paper and supplementary materials.", "theoretical_claims": "The theoretical claim of the paper is the justification of the proof of Theorem 3.1 seems to rely on the universal Hamiltonian family. Theorem 3.2 deals with Quantum Universal Approximation. Judging by the brevity of Appendix E, I am not fully convinced whether these theorems show how the architectural advancements presented in this works is justified. Therefore, I would say some theoretical results are properly placed, but I think the implication of the theory does not fully cover every implementation details of QuanONet architectures.", "experimental_designs_or_analyses": "Experiments are based on four types of ODE and PDE problems. Based on the results QuanONet shows best results depending on the hyperparameter $\\lambda$. Since I am not expert in this field, experiment results for QuanONet seems to be quite good but I am not fully confident on this initial assessment. Also, I have skepticism for TF-QuanONet since they are strictly worse than QuanONet. Therefore, performance of QuanOnet might be mainly due to extensive architecture and hyperparameter searches by the authors before the actual experiments.", "relation_to_broader_scientific_literature": "This work could bridge to gab between machine learning and natural science, especially particle physics. There has been multiple ways of simulating quantum systems, and QuanONet might be beneficial for finding scientific breakthrough if this is combined with broader scientific literature.", "essential_references_not_discussed": "Essential references are well-discussed in this paper.", "other_strengths_and_weaknesses": "In my reading, the explanation of TF-QuanONet is not sufficient for me to understand. Furthermore---if I understand the results correctly---TF-QuanONet strictly performs worse than QuanONet(\"error evolution of TF-QuanONet is poor,\" and \"shows obvious overfitting phenomenon.\") Even tough, TF-QuanONet reduces the burden of setting coefficients, I think TF-QuanOnet does not have noticeable benefits. This raises the question of presenting claims TF-QuanOnet as in the introduction section.", "other_comments_or_suggestions": "The formatting style of Figures 4 and 5 is not good; I suggest to increase the font size of numbers and labels. Regarding the conclusion \"How to compare QuanONet and classical neural operators is a controversial issue.\", I do not agree with this opinion. As the QNNs are capable of modeling various ODEs and PDEs, the authors are encouraged to point out actual performance comparisons and discuss the particular reason QNNs are not as efficient as classical neural operators.", "questions_for_authors": "* What is the computational complexity (perhaps execution time) of QuanONet, TF-QuanONet and other baselines?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The rebuttal addressed the concerns from the reviewers and provided comprehensive feedback. It is useful for assessing the paper's contributions. All reviewers recommend acceptance after discussion, and the ACs concur. The final version should include all reviewer comments, suggestions, and additional discussion from the rebuttal.
Decision: Accept (poster)