Paper: Power Characterization of Noisy Quantum Kernels (ICLR-2024)

Reviewer comment (Novelty):
This paper does not clearly state its differences from prior works, particularly in terms of technical detail.

Author response:
Firstly, following your constructive advice, we have thoroughly revised Sec. 1.2 Related Work to present our contributions and the differences from prior works more clearly.

Reviewer comment (Novelty):
The result itself is not surprising, as depolarizing channels can be equivalent to a single depolarizing channel with noise strength that grows exponentially close to 1 as the number of layers increases, leading the output state of the encoding circuit to converge to the maximally mixed state.

Author response:
Secondly, although the result itself may not be very surprising, we do explicitly depict the speed of the prediction concentration of noisy quantum kernels in terms of the strength of depolarization noise, the size of training samples, the number of qubits, the number of layers affected by quantum noise, and the number of measurement shots. This helps us quantitatively understand the limits of noisy quantum kernel methods. As an illustration, for a training sample set, whose scale is exponential in the number of qubits $N$, like $q^N$ for some $q>1$, noisy quantum kernel methods fail once the number of noisy layers exceeds $N\log\_{(1−\tilde{p} )^{−2}} q$.

Reviewer comment (Experiments):
This paper could be strengthened by considering more complicated and practical noise models, as well as the effect of error mitigation.

Author response:
We agree with you that much work needs to be done to further investigate the limitations of quantum kernel methods. For example, consider more realistic quantum noise including the effect of mitigation, and conduct experiments on real quantum devices. We leave them for future research. Thank you for pointing out these potential research directions.

Reviewer comment (Writing):
Line -5 before Sec. 1.3: “...required for probably successful training”, what does “probably successful” mean?

Author response:
We mean that for those cases, to have a good training, a necessary condition is that exponentially many resources are required. However, this is not a sufficient condition, namely, exponential number of resources do not necessarily guarantee a good training under those scenarios. In the revised paper, we have removed  “probably successful” to avoid possible confusion.

Reviewer comment (Writing):
Page 3: “indicate that good generalization alone does not necessarily guarantee good prediction for new data”. Does “good generalization” mean the generalization of the noiseless kernel?

Author response:
Here, “good generalization” refers to the generalization for general quantum machine learning methods. Recall that generalization error is the difference between the prediction error and the training error. When both the training and prediction are bad, its generalization error may still be small. Thus, for any quantum machine learning methods, only having good generalization cannot necessarily guarantee good prediction. To have good prediction, both the training and generalization errors should be small. We have revised the paper to make it clearer.

Reviewer comment (Writing):
Eq. (3): it is unclear whether the square is inside or outside of the expectation.

Author response:
We have revised the expression to make it clearer. Thank you for pointing out the potential confusion.

Reviewer comment (Writing):
Sec. 2.2: the introduction to the quantum kernel method is not self-contained. A complete workflow should be given for the ease of readers who are unfamiliar with this field, and it should mention which part is quantum and which part is classical.

Author response:
We have revised Sec. 2.2 to make it more self-contained. Specifically, we have clearly pointed out which part is quantum and which part is classical. Thank you for your suggestion.

Reviewer comment (Theory):
Page 7, line 5: “The depolarization noise model in Theorem 3.1 is weaker than the noise model…” Weaker in what sense? Could you state it more explicitly?

Author response:
Thank you for your constructive comment. We have added more explanations on “weaker” in what sense in the revised manuscript. In one word, our noise model is weaker in the sense that with the same depolarizing rate $\tilde{p}$, under our noise model the quantum state is left untouched with an exponentially larger probability $(1-\tilde{p})$ as compared to that $(1-\tilde{p})^N$ under the noise model considered in previous works. Here, $N$ denotes the number of qubits. 

Specifically, in our work, the noise model, $\mathcal{N}\_{\tilde{p}}=(1-\tilde{p})\rho+\tilde{p}\frac{1}{D}I$ with rate $\tilde{p}$, is referred to as “global” as it applies onto all qubits after each layer of ideal unitary gates. While in Stilck Franca & Garcia-Patron (2021) and De Palma et al. (2023), the considered noise model is the local depolarizing noise. However, it applies to each qubit. Thus, their noise channel applied after each layer of unitary gates is $\mathcal{D}\_{\tilde{p}}^{\otimes N}$, with $\mathcal{D}\_{\tilde{p}}=(1-\tilde{p})\rho+\frac{1}{2}I$ and $N$ being the number of qubits. We have rewritten it using our notation for convenience of comparison. With the same noise rate $\tilde{p}$, we can compare our global depolarizing noise $\mathcal{N}\_{\tilde{p}}$ with its counterpart $\mathcal{D}\_{\tilde{p}}^{\otimes N}$. It can be seen that under our global noise, the quantum state is left untouched with probability $1-\tilde{p}$, while under the so-called local noise, the probability of the state remaining unchanged is only $(1-\tilde{p})^N$, which is exponentially small in $N$. Thus, in this sense our global depolarizing model is weaker than the so-called local depolarizing model. As for the Pauli noise model considered in Thanasilp et al. (2022), following a similar analysis, we can see that the Pauli noise is also stronger than ours. We have made this clearer in the revised manuscript after Eq. (12).

Reviewer comment (Novelty):
There are several previous works on noisy quantum kernels, such as Wang et al. (2021), Stilck Franca & Garcia-Patron (2021); De Palma et al. (2023), Thanasilp et al. (2022). It is not clear how this work compares to existing research and what new limitations are discovered in this paper. The advantages or significance of this work is not clear enough to me. It would be better if the authors could explain this angle in a more organized way.

Author response:
Thank you very much for your helpful comments and suggestions. 

First of all, we would like to point out that, for the problem of power characterization under consideration, our global noise model is actually weaker than the noise models considered in previous works you mentioned, such as, Stilck Franca & Garcia-Patron (2021), De Palma et al. (2023) and Thanasilp et al. (2022).

Specifically, in our work, the noise model, $\mathcal{N}\_{\tilde{p}}=(1-\tilde{p})\rho+\tilde{p}\frac{1}{D}I$ with rate $\tilde{p}$, is referred to as “global” as it applies onto all qubits after each layer of ideal unitary gates. While in Stilck Franca & Garcia-Patron (2021) and De Palma et al. (2023), the considered noise model is the local depolarizing noise. However, it applies to each qubit. Thus, their noise channel applied after each ideal unitary layer is $\mathcal{D}\_{\tilde{p}}^{\otimes N}$, with $\mathcal{D}\_{\tilde{p}}=(1-\tilde{p})\rho+\frac{1}{2}I$ and $N$ being the number of qubits. We have rewritten it using our notation for convenience of comparison. With the same noise rate $\tilde{p}$, we can compare our global depolarizing noise $\mathcal{N}\_{\tilde{p}}$ with its counterpart $\mathcal{D}\_{\tilde{p}}^{\otimes N}$. It can be seen that under our global noise, the quantum state is left untouched with probability $1-\tilde{p}$, while under the so-called local noise, the probability of the state remaining unchanged is only $(1-\tilde{p})^N$, which is exponentially small in $N$. Thus, in this sense our global depolarizing model is weaker than the so-called local depolarizing model. As for the Pauli noise model considered in Thanasilp et al. (2022), following a similar analysis, we can see that the Pauli noise is also stronger than ours.

When presenting negative results concerning noisy quantum kernels, it is better to assume a relatively weaker noise model. Once the kernel methods fail under weaker noises, they fail under stronger ones in general. Note that the noise rate $\tilde{p}>0$ in our work can be arbitrarily small. Thus, it can depict the case where the noise influence is very weak, namely, after the noise the quantum state is left untouched with a very high probability $1-\tilde{p}$. Thus, in this sense, our global depolarizing model is an appropriate starting point to study the limitations of noisy quantum kernel methods. We have made this clearer in the revised manuscript after Eq. (12). As you suggested, much work needs to be done to further investigate the limitations of quantum kernel methods, e.g., under more realistic quantum noise. We leave this for future research. 

To address your constructive question "What are the key new findings on the noisy quantum kernel model compared with previous works", we have thoroughly revised Sec. 1.2 Related Work to compare our work with previous works. Your question definitely helps us to present our key new findings and the differences from the existing works more clearly. For more details, please refer to revised Sec. 1.2 Related Work as well as the above response to your question regarding global depolarizing noise.

Reviewer comment (Novelty):
What are the key new findings on the noisy quantum kernel model compared with previous works?

Author response:
Thank you very much for your helpful comments and suggestions. 

First of all, we would like to point out that, for the problem of power characterization under consideration, our global noise model is actually weaker than the noise models considered in previous works you mentioned, such as, Stilck Franca & Garcia-Patron (2021), De Palma et al. (2023) and Thanasilp et al. (2022).

Specifically, in our work, the noise model, $\mathcal{N}\_{\tilde{p}}=(1-\tilde{p})\rho+\tilde{p}\frac{1}{D}I$ with rate $\tilde{p}$, is referred to as “global” as it applies onto all qubits after each layer of ideal unitary gates. While in Stilck Franca & Garcia-Patron (2021) and De Palma et al. (2023), the considered noise model is the local depolarizing noise. However, it applies to each qubit. Thus, their noise channel applied after each ideal unitary layer is $\mathcal{D}\_{\tilde{p}}^{\otimes N}$, with $\mathcal{D}\_{\tilde{p}}=(1-\tilde{p})\rho+\frac{1}{2}I$ and $N$ being the number of qubits. We have rewritten it using our notation for convenience of comparison. With the same noise rate $\tilde{p}$, we can compare our global depolarizing noise $\mathcal{N}\_{\tilde{p}}$ with its counterpart $\mathcal{D}\_{\tilde{p}}^{\otimes N}$. It can be seen that under our global noise, the quantum state is left untouched with probability $1-\tilde{p}$, while under the so-called local noise, the probability of the state remaining unchanged is only $(1-\tilde{p})^N$, which is exponentially small in $N$. Thus, in this sense our global depolarizing model is weaker than the so-called local depolarizing model. As for the Pauli noise model considered in Thanasilp et al. (2022), following a similar analysis, we can see that the Pauli noise is also stronger than ours.

When presenting negative results concerning noisy quantum kernels, it is better to assume a relatively weaker noise model. Once the kernel methods fail under weaker noises, they fail under stronger ones in general. Note that the noise rate $\tilde{p}>0$ in our work can be arbitrarily small. Thus, it can depict the case where the noise influence is very weak, namely, after the noise the quantum state is left untouched with a very high probability $1-\tilde{p}$. Thus, in this sense, our global depolarizing model is an appropriate starting point to study the limitations of noisy quantum kernel methods. We have made this clearer in the revised manuscript after Eq. (12). As you suggested, much work needs to be done to further investigate the limitations of quantum kernel methods, e.g., under more realistic quantum noise. We leave this for future research. 

To address your constructive question "What are the key new findings on the noisy quantum kernel model compared with previous works", we have thoroughly revised Sec. 1.2 Related Work to compare our work with previous works. Your question definitely helps us to present our key new findings and the differences from the existing works more clearly. For more details, please refer to revised Sec. 1.2 Related Work as well as the above response to your question regarding global depolarizing noise.

Reviewer comment (Theory):
The paper focuses on global depolarizing noise, which is not considered realistic for quantum devices (as well as noisy quantum kernel methods). The natural noise model would be local depolarizing noise, where each qubit is subject to a small amount of noise.

Author response:
We agree with you that the local depolarizing noise is a natural noise model, where each qubit is subject to a small amount of noise. As compared with it, our global depolarizing model is a stronger assumption about the noise at first glance. However, on second thought, our global depolarizing model is actually weaker than the so-called local one for our problem in this paper. The reason is as follows. 

In our work, as illustrated in Fig. 2(c), after each layer of ideal unitary gates, a depolarizing channel, $\mathcal{N}\_{\tilde{p}}=(1-\tilde{p})\rho+\tilde{p}\frac{1}{D}I$ with rate $\tilde{p}$ is applied. We refer to it “global” as it applies onto all qubits. In your mentioned paper ( https://arxiv.org/abs/2210.11505), two kinds of noise models are considered. The first one is the so-called local depolarizing noise acting on each qubit, which reads $\mathcal{D}\_{\tilde{p}}=(1-\tilde{p})\rho+\frac{1}{2}I$. We have rewritten it using our notation for convenience of comparison. As demonstrated by Fig. 1 in the mentioned paper, the noise channel applied after each layer of ideal unitary gates should be $\mathcal{D}\_{\tilde{p}}^{\otimes N}$ with $N$ being the number of qubits. Now we can compare our global depolarizing noise $\mathcal{N}\_{\tilde{p}}$ with its counterpart $\mathcal{D}\_{\tilde{p}}^{\otimes N}$ under the same noise rate $\tilde{p}$. It can be seen that under our global depolarizing noise, the quantum state is left untouched with probability $1-\tilde{p}$, while under the so-called local noise, the probability of the state remaining unchanged is only $(1-\tilde{p})^N$, which is exponentially small in $N$. Therefore, in this sense our global depolarizing model is weaker than the so-called local depolarizing model. Another considered noise model in the mentioned paper is non-unital noise acting on each qubit. Following a similar analysis to the above local depolarizing noise, it can be seen that the non-unital noise is also stronger than ours. 

When presenting negative results concerning noisy quantum kernels, it is better to assume a relatively weaker noise model. Once the kernel methods fail under weaker noises, they fail under stronger ones in general. Note that the noise rate $\tilde{p}>0$ in our work can be arbitrarily small. Thus, it can depict the case where the noise influence is very weak, namely, after the noise the quantum state is left untouched with a very high probability $1-\tilde{p}$. Thus, in this sense, our global depolarizing model is an appropriate starting point to study the limitations of noisy quantum kernel methods. We have made this clearer in the revised manuscript after Eq. (12). We completely agree with you that much work needs to be done to further investigate the limitations of quantum kernel methods under more realistic quantum noise. We leave this for our future research.

Reviewer comment (Evaluation):
The key behaviors of noisy quantum kernel methods illustrated by the analytical results are expected. As the total noise increases, the generalization error decreases while the training error increases. Hence, the generalization error can be close to zero while the prediction performance is bad.

Author response:
We agree with you that the key behaviors of noisy quantum kernel methods are expected. However, one of our key contributions is that we explicitly depict the speed of the prediction concentration of noisy quantum kernels in terms of the strength of depolarization noise, the size of training samples, the number of qubits, the number of layers affected by quantum noise, and the number of measurement shots. This helps us quantitatively understand the limits of noisy quantum kernel methods. As an illustration, for a training sample set, whose scale is exponential in the number of qubits $N$, like $q^N$ for some $q>1$, noisy quantum kernel methods fail once the number of noisy layers exceeds $N\log\_{(1−\tilde{p} )^{−2}} q$.

Reviewer comment (Theory):
My major issue with the paper is the correctness of the main result (Theorem 3.1). I might be missing something, as I am not convinced this theorem is correct. Equation (20) in the theorem bounds the distance between the quantum kernel's choice $\tilde{h}$ and the worst possible predictor $\bar{h}$. The bound is in terms of the noise bias $p$, the number of samples (n), and the dimension (D). Based on this bound $|\bar{h} -\tilde{h}|$ converges to zero as $n$ and $D$ grow large regardless of the value of p<1.

Author response:
Thank you very much for your feedback. 

In fact, our main result holds only for the noisy case. Note that the function $f(z)$ in the RHS of Eq. (20) has the form of $f\left( z\right) =\frac{z+8\sqrt{\frac{z}{\lambda}}}{1-z}$. To make it meaningful, its argument must be less than 1. Thus, in Eq. (20), $\frac{n}{\lambda}(1-p)(1+\frac{1}{D}) $ should be less than 1. This corresponds to noisy channels rather than the ideal case. This is because $1-p=(1-\tilde{p})^{2L}$. In the idea case $\tilde{p}=0$, then $1-p=1$. This leads to $\frac{n}{\lambda}(1-p)(1+\frac{1}{D}) > 1$.

To sum up, in this paper, we focus on the effect of quantum noise and provide a crucial warning to utilize noisy quantum kernels. Our main results are indicative for noisy deep quantum circuits.

Reviewer comment (Theory):
If we set $p=0$, implying no noise, the bound in Theorem 3.1 still converges to zero. This does not make any sense as we expect when $p=0$, the kernel method should work better than the worst predictor. This raises concerns about the result’s validity.

Author response:
Thank you very much for your feedback. 

In fact, our main result holds only for the noisy case. Note that the function $f(z)$ in the RHS of Eq. (20) has the form of $f\left( z\right) =\frac{z+8\sqrt{\frac{z}{\lambda}}}{1-z}$. To make it meaningful, its argument must be less than 1. Thus, in Eq. (20), $\frac{n}{\lambda}(1-p)(1+\frac{1}{D}) $ should be less than 1. This corresponds to noisy channels rather than the ideal case. This is because $1-p=(1-\tilde{p})^{2L}$. In the idea case $\tilde{p}=0$, then $1-p=1$. This leads to $\frac{n}{\lambda}(1-p)(1+\frac{1}{D}) > 1$.

To sum up, in this paper, we focus on the effect of quantum noise and provide a crucial warning to utilize noisy quantum kernels. Our main results are indicative for noisy deep quantum circuits.

Reviewer comment (Theory):
Why does the bound in Theorem 3.1 still converge to zero as $p=0$?

Author response:
-  Why does the bound in Theorem 3.1 still converge to zero as $p=0$? 

As you mentioned, Equation (20) in the theorem bounds the distance between the quantum kernel's choice $\tilde{h}$ and the worst possible predictor $\bar{h}$. It consists of three terms. The third term converges to 0 as $n$ increases, and the second term approaches to 0 as $n$ and $D$ increase (as long as $n$ is not in the order of $D$). While the first term $f(\cdot)$ goes to 0 if and only if its argument goes to 0. If we set $p=0$, then the argument of the first term $f(\cdot)$ is a constant of $\frac{n}{\lambda}(1+\frac{1}{D})$, and in turn the bound does not converge to zero. Thank you for pointing out this potential ambiguity. Following your insightful comments, we have made it clearer in the revised manuscript.

Reviewer comment (Experiments):
The samples are classical in this work, but the kernel is quantum. What can be done with your work for the quantum samples?

Author response:
- The samples are classical in this work, but the kernel is quantum. What can be done with your work for the quantum samples?

In our work, the samples are classical and encoded into quantum states to compute their kernels. If the samples are quantum, our technique can still be applied to analyze the impact of noise on quantum kernel methods if the depolarizing noise occurs during the generation of quantum samples or during the computation of kernels. We leave this as future work for a more detailed analysis. Thank you for your insightful comments.