QuanONet: Quantum Neural Operator with Application to Differential Equation

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.

We 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?

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.

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.

The formatting style of Figures 4 and 5 is not good; I suggest to increase the font size of numbers and labels.

Thanks for your suggestions, we have updated the figure in the new link.

What is the computational complexity (perhaps execution time) of QuanONet, TF-QuanONet and other baselines?

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. The 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.

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.

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. For our sentence, we will remove it and give a more specific discussion based on our new results.

How many Qbits we would need to run this model?

Based 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).

What PDEs does quantum methods have an advantage? Maybe high-dimension Schrodinger equation?

Quantum 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). For higher-dimensional PDE, since the operator approximation theorem requires input function discretization, they all face fundamental limitations of exponential input dimension.

It would be helpful to add Quantum DeepONet and Quantum FNO as baseline models.

Since 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, so we provide results for DeepONet as an alternative.

In 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. (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.

We provide the results of FNO with 100 initial functions as training instances, as shown in Tab. 4

Meanwhile, the authors mentioned noisy intermediatescale quantum (NISQ) and fault-tolerance in the introduction. It seems not supported in the experiments.

The 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.

The Fig. 5 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.

QuanONet 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. 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.

if somehow we get exponential speedups using quantum method, how do we extract the solution? Is the observation costly?

QuanONet extracts the solution by measuring the expectation value of a Hamiltonian composed of commuting single-qubit Z-terms and scaling the result.

The observation cost primarily stems from three factors: the number of measurement groups, shots per group, and error mitigation overhead.

1.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.

2.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.

3.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.

The performance gain over classical methods isn't consistently demonstrated across all cases, which weakens the overall impact.

We add more experiments with larger scale with batch size 100, 100K max iterations and for five runs.

1. ODE: 10K train instances and 10K test instances.
2. PDE: 100K train instances and 100K test instances.

The results are given in Tab. 1. TF-QuanONet outperforms other quantum methods on all problems and is superior to classical methods except for nonlinear operators.

How would quantum hardware noise affect QuanONet performance, and what error mitigation strategies would be appropriate for practical implementation?

Compared 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.

Besides, 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.

What mechanisms drive the overfitting phenomenon observed with TF-QuanONet, and what strategies might mitigate this behavior?

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. 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.

Beyond trainable frequencies, what alternative approaches might address the coefcient initialization challenge?

We 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.

How does QuanONet relate to Neural Network Quantum States research, and could insights from that domain inform further development?

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.

What specific properties of the diffusion-reaction system make it particularly amenable to quantum approaches?

D-R system exhibits a lower frequency distribution compared to the other problems as shown in Fig. 1.

The 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.

Given the specific Hamiltonian requirements for diferent problems, how feasible would transfer learning or multi-task learning be with this architecture?

The choice of Hamiltonians are discussed detailly in the first response to reviewer LZaG.

The 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 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.

The choice of 5 qubits and fixed Hamiltonian may limit exploration of quantum advantages.

We 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. The 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.

Experimental 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 and Fig. 3

Quantum vs. classical architectures are fundamentally diferent, making direct comparisons less meaningful.

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.

Could you provide a more detailed construction of $W$ and $|\Gamma\rangle$ in Theorem 3.1 to clarify how arbitrary states are approximated?

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 $SU(2^n)$).

Theorem [Controllability of molecular systems. Physical Review A, 51(2):960, 1995].

1. 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.
2. 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.

Based 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 $SO(2^n)$).

The detailed discussion of Theorem 3.2 can be found in the first reply torReviewer xycX.

Why are training iterations for PDEs vastly diferent between QuanONet (100) and DeepONet (10,000)? Would QuanONet maintain its advantage with equal iterations?

Our 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.
Taking the antiderivative operator as an example, the loss curves of all methods are as Fig. 4. Here the (TF-)QuanONet's performance improves as the number of iterations increases more intuitively.

How does QuanONet address limited qubit connectivity or noise in real NISQ devices?

The 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.