Physics-Informed Bayesian Optimization of Variational Quantum Circuits

Summary

The paper introduces a new approach for BO of VQEs. BO can be a good match for this problem since it models the noise (measurement and circuit level) of quantum circuits. The main idea of the paper is to use a kernel that is adapted to the form of the VQE objective assumed when variational parameters are associated to single qubit gates. The paper also introduces EMICoRE and NFT-with-EMICoRE, acquisition function and parameter update strategies. Overall, the idea of using an adapted kernel for this task is well motivated, the paper is well written and presents numerical evaluation of their method. I think however that in the current state, the experimental evaluation is too limited to be convincing of the promise of this method with respect to other optimisation algorithms for VQEs.

Strengths

The paper addresses an important problem in quantum computing. It is well written and accessible for both quantum computing and ML communities I think. It introduces the following technical innovations: - Kernel adapted to VQEs: the kernel decomposition based on the form of the cost function of VQEs is a sound contribution. - NFT-with-EMICoRE: a new BO algorithm that builds on NFT and a new acquisition function.

Weaknesses

1. Specific ansatz: the paper deals only with parametrised single qubit gates. While I believe this is not a theoretical limitation - as one can always decompose any unitary as a product of single qubit gates and CNOTs - it is not clear to me how practical is this parametrisation. The authors could add an explanation of whether this is a limitation for practitioner and what steps one needs to do to apply their algorithm to a generic parametrised quantum circuit 2. Limited experimental evaluation: the paper only discusses Q=3 and 5 qubit systems, while classically one could simulate easily bigger systems. Also the paper does not compare against other methods in the literature, it only compare against NFT. More benchmarking for larger systems and against other techniques would be required to assess the promise of the method. 3. Unclear benefit of the kernel: looking at figure 2, it is unclear that the red curve is better than blue curve - in fact on the left figure it looks to me that RBF is doing better. 4. Missing ablation for the EMICoRE acquisition function. One contribution of the paper is to introduce this new acquisition function, but ablations showing its importance are missing. Minor: - $\mu_X$ depends on y as well in section 2.1 - "the the" in line 308 - unclear what conjugacy means in line 97

Questions

1. Could you benchmark your method against other methods for VQE optimisation than NFT? And for larger systems? 2. Why is the ground state energy not reached in figure 3 say? I would expect that it is possible to find a VQE ansatz that can solve this easy problem. 3. Can you add ablations for the EMICoRE acquisition function? Have you tried different heuristics for updating $\kappa$? 4. Can you explain how your method can be used to optimise general parametrised circuits beyond those that have single qubit gates?

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

1 poor

Presentation

3 good

Contribution

2 fair

Limitations

- limited experiments and benchmarking - limited explanation of their choice of parametrised circuits

Summary

In this work, the authors integrate a quantum kernel method with the EMICoRe architecture to further improve the NFT framework of Bayesian Optimization. The simulation results show the advantages of the proposed approach.

Strengths

(1) The method of leveraging the quantum method for Bayesian Optimization is interesting. (2) The investigation of how to incorporate the quantum kernel method is significant.

Weaknesses

(1) Since no parametric circuits are implemented in the quantum kernel model shown in Eq. (9), the proposed VQE-kernel is nothing but a quantum kernel learning method, which has been comprehensively studied in previous work in Refs. [1-3]. In particular, Ref. [4] has exhibited the use of quantum kernel learning for improving the performance of Bayesian optimization. --- resolved (2) The circuit diagram for the quantum kernel learning is not provided such that the experiments cannot be easily reproduced. --- resolved (3) There are two main contributions to this work: the use of quantum kernel learning and an introduction to EMICoRe. Although the simulation results demonstrate the performance improvement, it is still unknown where the performance gains come from, and the quantum advantages of the quantum kernel are not analyzed at all. --- resolved [1] Havlíček, Vojtěch, et al. "Supervised learning with quantum-enhanced feature spaces." Nature 567.7747 (2019): 209-212 [2] Wang, Xinbiao, et al. "Towards understanding the power of quantum kernels in the NISQ era." Quantum 5 (2021): 531 [3] Blank, Carsten, et al. "Quantum classifier with tailored quantum kernel." npj Quantum Information 6.1 (2020): 41 [4] Rath, Yannic, Aldo Glielmo, and George H. Booth. "A Bayesian inference framework for compression and prediction of quantum states." The Journal of Chemical Physics 153.12 (2020)

Questions

(1) What are the quantum advantages of the quantum kernel method? Since the quantum kernel method is combined with another proposed method `EMICoRe', we do not know which part contributes to the improvement of the Bayesian Optimization method. --- resolved

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

2 fair

Presentation

2 fair

Contribution

3 good

Limitations

It would be better to reformulate the quantum approach in the authors' proposed Bayesian Optimization framework.

Summary

The authors propose a method for Bayesian Optimization for Variational Quantum Eigensolvers, which they call NFT with EMICoRe. This method uses a novel VQE kernel, which constrains the function space of the Gaussian Process underlying the BO to include only valid VQE objective functions (using the representation in Prop. 2, derived from NFT). The authors also propose a novel acquisition function for the EMICoRe method (Eq. 11), which optimizes over the expected maximum improvement over confident regions. In their experimentation, the authors show that their VQE kernel is able to outperform other kernels in a BO setting, and that their NFT-EMICoRe approach is able to outperform other (non-BO) NFT approaches.

Strengths

The paper tackles an important problem, the optimization of noisy VQE circuits, and offers a principled solution using BO combined with physical constraints, using state of the art methods (NFT). The paper is well written, and the experimentation is well chosen to support the method.

Weaknesses

The experimentation could be expanded. Particularly, it would be interesting to see how the model performs on an actual quantum implementation. Also, investigation of a broader ranger of Hamiltonians would be desirable (including ones motivated by practical problems).

Questions

The threshold parameter \kappa is introduced in Sec. 3.2, but doesn't appear to be explored in the experimentation. In particular, did the authors verify that an intermediate value of this parameter is advantageous (hence supporting the use of the EMICoRe acquisition function)? I may have missed this in the experimentation. Further, on lines 133-134, they state that they are not concerned with circuit noise on current NISQ devices, but line 320 states that the experimentation confirms the suitability of their method for such devices - is such noise explicitly included in the simulations?

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

4 excellent

Presentation

4 excellent

Contribution

3 good

Limitations

Yes, limitations are addressed.

Summary

I have reviewed the rebuttal, and I intend to maintain my decision. I appreciate the authors made the efforts to clarify most of my concerns. However, I believe it is essential to address the issue of hardware noise in the present study, a point that previous research seems to have overlooked. I strongly encourage the authors to delve deeper into this matter, as a robust algorithm should be capable of practical implementation on real quantum computers. I believe the issue of hardware noise is actually an inescapable and important challenge here. This manuscript presents an innovative approach known as EMICoRe, which leverages the synergy between Bayesian optimization and prior knowledge of variational quantum eigensolvers (VQE) to enhance the efficiency of the optimization. The authors introduce a novel kernel specifically designed for Bayesian optimization and employ it within the EMICoRe framework. Additionally, the paper conducts numerical experiments to compare the proposed method with the state-of-the-art baselines, demonstrating its superior performance.

Strengths

- The manuscript introduces a novel optimization algorithm based on Bayesian optimization that incorporates the specific property of the VQE objective function into its design. - The paper is well-organized and clearly written. The figures in the paper effectively illustrate the improved performance achieved by the proposed method in comparison to the baselines. - The paper establishes the equivalence between two previously proposed VQE properties, providing a foundational basis for the subsequent analysis on VQE objective functions.

Weaknesses

- The numerical results presented in the paper indicate that the proposed method does not exhibit a clear advantage when the number of observed points is limited, as compared to the baselines. - Although the proposed method exhibits improved performance compared to the baseline as the number of observed points increases, it still falls short of achieving the ideal ground state. - The proposed method just considers the shot noise but ignores possible noise originating from the quantum circuit itself.

Questions

1. Based on the numerical results presented in the paper, the fidelity between the ground state and the state obtained through the VQE remains consistently below 0.8, even after reaching 400 observed points. It is noteworthy that the VQE circuit being optimized in the study comprises only three layers and five qubits. Therefore, I believe it is crucial to investigate the potential impact of further increasing the number of observed points and determine the minimum number of observed points required to achieve a higher fidelity, such as surpassing 0.95. 2. The manuscript just includes two parameter settings, one for the Ising model and the other for the Heisenberg model. However, to verify the robustness of the proposed method, I suggest that the authors incorporate more scenarios for validation.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

4 excellent

Contribution

2 fair

Limitations

I believe the authors have provided thorough discussions on the limitations of their work within the Appendix.

Follow-up for the rebuttal letter

Thank the authors for providing the rebuttal letter, which helps the reviewer to better understand the paper. Since the authors have resolved all of my major concerns, I have increased my suggested score for the paper.