Summary
This paper utilizes Koopman operator theory to accelerate the optimization of quantum circuits. Stability analysis is performed and the benefit of alternating between standard and Koopman optimization is discussed. A number of experiments are performed, across several different areas of interest to the quantum computing community, with Koopman optimization performing significantly faster than standard approaches. The effect of noise is studied, with QuACK again finding a benefit.
Strengths
1. This paper motivates the need for efficient, gradient based methods for optimizing quantum circuits.
2. This paper shows, via a number of experiments, the potential of Koopman operator theory to accelerate training, over a number of different regimes.
3. This paper tests multiple implementations for computing the Koopman spectra, including DMD, sliding DMD, and machine learning implementations.
4. The Appendix contains many details on quantum computing and the architectures used (the availability of this information should be mentioned in the main text).
Weaknesses
1. While Theorem 5.1 is true, in the sense that any finite numerical implementation of Koopman mode decomposition will find eigenvalues that are not exactly equal to 1, that does not mean that the eigenvalues computed by DMD will not be very close to 1 (see Redman et al., 2021 Fig. 4 for example of eigenvalue being 1.000), and that the associated Koopman mode could be a good approximation to a true fixed point.
2. Similarly, Theorem 5.2 and Corollary 5.4, while true, does not imply that QuACK will not lead the quantum circuit to converge to a different local minima than the gradient-based optimizer. This should be noted explicitly.
3. Fig. 3 shows that the Koopman based methods converge very quickly to the asymptotic energy and test accuracy - from just the plots alone it seems like the speed-up should be greater than the reported 4x. Why is this the case?
4. The Koopman learning section should be expanded to include mention of the Koopman mode decomposition (Mezic, 2005). Additionally, the need for data to computed the Koopman mode decomposition should be motivated in Sec. 3, before the QuACK algorithm (which needs data from the gradient based optimizer) is presented. The interested reader should be referred to more detailed discussion in Budisic et al., 2012 and Brunton et al., 2022. Finally, it should be noted that the idea of time-delays is related to Taken's theorem, with the implementation presented in Eq. 1 being Hankel DMD (Arbabi and Mezic, 2017). The sliding window DMD method is similar to streaming DMD (Hermati et al., 2014; Giannakis et al., 2023).
5. While this is, to my knowledge, the first approach of using Koopman operator theory to the study of quantum machine learning, it would be appropriate to cite more of the work on studying machine learning using Koopman operator theory (e.g., Mohr et al., 2020; Naiman and Azencot, 2021; Šimánek et al., 2022; Liang et al. 2022). Additionally, the previous work that has optimized neural networks using Koopman operator theory (Dogra and Redman, 2020; Tano et al., 2020) should be discussed in more detail (especially when presenting the speed-up $s$, which was defined similarly by Dogra and Redman, 2020, and when discussing the alternating strategy of Koopman training, which Tano et al. 2020 also utilized). Work using Koopman to study iterative algorithms (Dietrich et al., 2020; Redman et al., 2022) more generally should also be cited when mentioning that "the optimization trajectory...is a dynamical system" (line 52). Relatedly, the references with respect to Koopman operator theory need to be updated. Extended-DMD (EDMD) refers to a specific algorithm, developed by Williams et al. 2015. This was not referenced when EDMD was mentioned in the paper (line 80), and all other papers cited described extensions to DMD that are not EDMD. The connection between DMD and Koopman mode decomposition should be noted (Rowley et al., 2009).
6. Is it true that classical backpropagation is independent of $n_{params}$? I think you mean that you don't need to do individual perturbations along the dimension of each parameter, but this needs to be clarified.
MINOR COMMENTS:
1. The speedup ratio of Fig. 2b and c seem to increase with the number of parameters. Is this expected?
2. Describing Koopman as "renowned for accelerating/predicting nonlinear dynamical systems" (lines 4 and 54) is a little overgeneral. Perhaps, saying "which has found utility in applications, as it allows for a linear representation of nonlinear dynamical systems," would be more specific and accurate.
3. It is mentioned that the quantum natural gradient is linked to imaginary time in the Introduction, but this is not discussed in overview section of Quantum Natural Gradient in Sec. 3.
4. There are a few minor typos:
i. The question "How can we accelerate...", which is not a yes/no question, is posed (line 48), to which you answer "in the affirmative".
ii. You list 3 speedups, but four conditions (lines 61, 62).
iii. "of the system is called wave function" (line 88)
iv. "the a layerwise partitioning" (line 299). Does this refer to the layer Koopman partitioning of Dogra and Redman, 2020?
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
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.
Limitations
The authors do a sufficient job addressing their limitations in the Limitations subsection of Sec. 4. They also study several different examples, across multiple different subregions of quantum computing, as well as different noise condition, providing strong evidence for their claims that Koopman training accelerates quantum optimization.