Problem-dependent Quantum Circuit Design Based on Entropy Matching

• It seems that the authors uses the linear entropy in a more heuristic way with few theoretical justicfications. Could the authors explain more about why to use the linear entropy, not other measure like Renyi entropy? For instance, could the authors provide a a comparative analysis between linear entropy and Renyi entropy to show the advantages for variational quantum circuit design?
• In the experimental results section, more details could be provided. For example, could the authors explain more about how they compute the average linear entropy mentioned in line 367? A step-by-step description of how the average linear entropy was calculated, including any preprocessing of the MNIST data will be helpful for better understanding. Also, could the author provide more details about the experiments in section 4.2 about the training process? For example, it would be helpful if the authors could explain the optimization algorithm used, number of iterations, and how the error rates were calculated in these experiments.
• The writing and presentations in the submission could be improved. For example, in section 3.3 "DETERMINING THE OPTIMAL CIRCUIT DEPTH BASED ON ENTROPY MATCHING", it would be helpful to provide more diagrams to show the overall methodology and strategy for the entropy matching. Also, there are some typos, see the question part for more details.

1. The manuscript provides limited information, as its 9 pages focus primarily on a single concept: linking the trace of the quantum state to the expressivity of the ansatz. This approach lacks depth and leaves much to be desired in terms of exploration of new ideas or techniques.

2. As far as I understand, the trace of a quantum state remains constant under unitary transformations unless the system interacts with an external environment. For any valid quantum state, the trace should remain 1, regardless of its distribution. Generally, the expressivity of a quantum ansatz relates to its ability to span the Hilbert space and requires the ansatz to remain unitary. It is unclear how the authors justify using entropy to assess expressivity, as entropy typically measures non-unitary characteristics. In particular, considering the quantum ansatz illustrated in Figure 2, it’s confusing where non-unitary transformations might be involved. Could the authors clarify this conceptual basis?

3. The experimental section lacks essential details. It is unclear whether the experiments were conducted on a quantum device or simulated on a classical computer. If simulations were performed, was circuit noise considered? Furthermore, the dotted line in Figure 3 is ambiguous; why does it indicate that layers 9 and 10 yield the best results? Additionally, the MNIST dataset includes images of digits 0–9, not +1 and -1. How was amplitude encoding performed as described, and what was the fidelity of state preparation?

4. The quantum circuit $V(\theta)$ should be a unitary matrix for a 10-qubit system. Therefore, it is unclear how $V(\theta)^\dagger$ could be applied directly to single-qubit states $|0\rangle$ and $|1\rangle$. This application requires further explanation to ensure consistency with the properties of unitary operators.

5. The manuscript’s technical writing lacks precision. Equations, numbers, and variables should be formatted in LaTeX math mode (e.g., "28x28" on line 362, $\psi$in and $\psi$out on line 371, and instances of L in section 4.2). Additionally, line 344 mentions "The entropy curves for states labeled +1 and -1 ..." prematurely, as it relates to content in the following section. The figures lack proper captions and do not clearly label the x- and y-axes, making it difficult to interpret the results effectively.

Linear entropy only considers one-qubit gates. To assess the power/complexity of a VQC, entanglement capabilities are essential as well. More metrics are needed to give a full picture of VQC's expressivity and computational power. For example, the Meyer-Wallach (MW) entanglement Q-measure [1] can be used to quantify the global entanglement measure of pure-state qubits entanglement.

Only one evaluation of a small subset of MNIST images are performed. More evaluations with a variety of dataset on real quantum hardware are needed to prove the robustness. MNIST images only consider binary image classification, which is a small subset of quantum machine learning applications. VQCs have other applications such as finding the ground state energy of molecules in quantum chemistry and portfolio optimization in finance. Testing the framework’s scalability and generalization to common quantum applications will further prove its feasibility. Including randomized benchmarking will also verify the validity.

Regarding the Haar measure introduction, more discussion of its usage/explanation in quantum computing field might be better than pure mathematical definitions. For example, explaining how single-qubit gates affect the Haar measure and including graphs could facilitate understanding.

[1] Gavin K. Brennen. 2003. An observable measure of entanglement for pure states ofmulti-qubitsystems. arXivpreprint,arXiv:quant-ph/0305094.

1. I had a very hard time to understand the experimental results. For instance, the text explains that single-qubit entropies are quantified (Fig. 3), but it is not specified for which qubit. If it is an average, there is no explicit mention to it, and no error bars are provided nor is the overall behaviour explained.

2. It is not explained if such entropy is given for a particular configuration of the parameters or as a statistical measure - averaged over some quantities. This point is crucial in particular for this paper, because all barren plateau phenomena that the authors claim to study are precisely about the variances of the loss landscapes. The given presentation made it very difficult for me to properly understand the findings.

3. In general, the results could be made much more clear with proper explanations of the figures. For example, Figure 1 does not seem to correspond to the actual experiments. In Figure 3 there is a blue dashed line without any explanation. Figure 4 does not have axes labels, it is not clear what this plot represents.

4. Even on multiple reads, I did not fully understand the claims. It is mentioned that the average linear entropies of an amplitude-encoding data embedding is given by these numbers ∼ 0.45 (again with the same problems as before). Then, the entropies from data and circuits are compared, and it is stated that those circuits with comparable entropies give optimal performances. Should it be clear why this is the case? Or is this what the experiments check? I do not see sufficient evidence to support these claims, a priori, but also when all the presented results are taken into account. Additionally, the text claims for theoretical optimization of number of layers, but I have not seen any such theory. In particular, in the section "EXPERIMENTAL RESULTS" the authors have the subsection "THEORETICAL DETERMINATION OF OPTIMAL LAYER DEPTH", which is a seeming contradiction (is it expermental or theoretical), and indeed it is not theoretical.

5. This paper assumes that expressibility is a guarantee for the existence, a priori, of a configuration of parameters with good performance. While this is trivially true if expressivity = capacity to represent all states; but here expressivity is used as proxy for "emergence of barren plateaus" and this requires much less - just a 2 design, which do not cover a dense subset of unitaries... Thus claim is not supported by any known result in the literature, including the barren plateau saga. On the other hand, random states are known to exhibit high entropies and entanglements, so the results here stated are not surprising.

6. Topics related to this paper have been already explored, see for instance [1]. In this paper, the authors explore how entanglement is built in a quantum neural network as the number of layers increases. This key reference is missing.

7. From a broader perspective, having to circuits with the same entropies for a given bipartition, does not guarantee trainability or expressivity, since entropies are global properties of states. Much more information is required to recover properties of the system.

8. From a presentation point of view, I find that there is too much dedication to concepts that are already well established, such as entropy, quantum computing, parameter shift rule (whose presence I cannot connect to the content of the paper), Haar-measure and so on. Also, some sections of the paper do not yield valuable information. For instance, section 3 is called "theoretical ". Is there a word missing?

[1] Marco Ballarin, Stefano Mangini, Simone Montangero, Chiara Macchiavello, and Riccardo Mengoni. Entanglement entropy production in Quantum Neural Networks