Paper: Quantum AdaBoost with Supervised Learning Guarantee (ICLR-2024)

Reviewer comment (Experiments):
In section 2.1, using multi-qubit quantum gates for the parametric quantum circuits is not optimal, as real multi-qubit quantum gates have to suffer from more serious quantum noise and are harder to deal with the Barren Plateau problem. Why not use single quantum parametric gates?

Author response:
We utilize multi-qubit quantum gates for the following reasons. 1) Parametric quantum circuits constructed by using only single quantum parametric gates are classical in essence and cannot provide sufficient expressibility for variational quantum algorithms to ensure a good approximation to the solution. 2) We agree with you that multi-qubit quantum gates may suffer from more noise and lead to Barren Plateau issues. There have been many schemes to mitigate the training issues. 3) In our experiments, we only employ 2-qubit quantum gates. 4) 2-qubit quantum gates can be realized in a native way to minimize the hardware noise. 5) 2-qubit quantum gates are necessary components in most of the existing parametric quantum circuits.

Reviewer comment (Theory):
The Eq. (5) associated with the analysis of empirical risk minimizer is incorrect for the quantum neural network (QNN). The output of QNN relies on the measurement, resulting in an additional optimization bias that is related to the expectation of observables. The authors can refer to the Reference as:

Ref. Jun Qi, Chao-Han Huck Yang, Pin-Yu Chen*, Min-Hsiu Hsieh*, "Theoretical Error Performance Analysis for Variational Quantum Circuit Based Functional Regression," Nature Publishing Group, npj Quantum Information, Vol. 9, no. 4, 2023

Author response:
Thank you for pointing out the reference. We have read it in detail. First of all, Eq. (5) in our paper is a widely-adopted figure of merit to depict the generalization error, see e.g., 

  1)	Matthias C. Caro, et.al., Generalization in quantum machine learning from few training data, Nature Communications, 13:4919, 2022. 
  2)	Matthias C. Caro, et.al., Out-of-distribution generalization for learning quantum dynamics, Nature Communications, 14:3751, 2023.
  3)	Hsin-Yuan Huang, et.al., Power of data in quantum machine learning, Nature Communications, 12:2631, 2021.

In quantum machine learning, the prediction accuracy is what we really concern, but it cannot be directly accessed as both the label of unseen data and the distribution is unknown. Nevertheless, we do have access to the empirical or training error. The difference between the prediction error and the training error is referred to as the generalization error. It is unknown but we can bound it as we have done in our work. 

In your mentioned reference, they provided another way for error decomposition, but the so-called training error (whose meaning is different from ours) that results from the optimization bias of gradient-based algorithms is hard to bound. In fact, they did not give such a bound. In addition, to obtain their main results, they assumed a crucial PL condition which assumes that the norm of the cost function gradient is bounded below by the cost function multiplied by a parameter $\mu$, namely, $\frac{1}{2}\\|\nabla\mathcal{L}_S(\boldsymbol{\theta})\\|^2_2\geq\mu\mathcal{L}_S(\boldsymbol{\theta})$. This is a very strong assumption, as it is well-known that training quantum parameterized circuits often suffers from the Barren Plateau issue, where the cost gradient vanishes exponentially fast as the scale of the problem increases. Thus, the strong PL constraint may limit the applicability of their main result. Our work provides the first theoretical proof that for binary classification the prediction error of variational quantum AdaBoost can converge to 0 as the increase of the number of boosting rounds and sample size.

Reviewer comment (Evaluation):
The authors do not provide a deeper discussion for the simulations as shown in Figure 4 and 5, why more Rounds are beneficial to the performance boost for the quantum Adaboost system? and why does the quantum Adboost method even attains worse results at the very beginning?

Author response:
The fact that more rounds are beneficial to the performance of quantum AdaBoost is owing to the characteristic of AdaBoost. Take Algorithm 2 for an illustration. When $t=1$, it starts from a selected base classifier whose error is less than 1/2. Then at each round ($t\in[T]$), a new base classifier, whose error is less than 1/2, is selected and the distribution is updated to have the effect of focusing more on the points incorrectly classified at the next round. After $T$ rounds, the classifier returned by AdaBoost is a non-negative linear combination of the base classifiers selected at each round. In Figure 4 and Figure 5, when $T=1$, namely, at the beginning stage of AdaBoost, the classifier is only a base classifier, which is weak. Thus, its performance is relatively bad. We have made this clearer in the current manuscript.

Reviewer comment (Theory):
Why more rounds T can be beneficial to the proposed Adaboost method?

Author response:
The fact that more rounds are beneficial to the performance of quantum AdaBoost is owing to the characteristic of AdaBoost. Take Algorithm 2 for an illustration. When $t=1$, it starts from a selected base classifier whose error is less than 1/2. Then at each round ($t\in[T]$), a new base classifier, whose error is less than 1/2, is selected and the distribution is updated to have the effect of focusing more on the points incorrectly classified at the next round. After $T$ rounds, the classifier returned by AdaBoost is a non-negative linear combination of the base classifiers selected at each round. In Figure 4 and Figure 5, when $T=1$, namely, at the beginning stage of AdaBoost, the classifier is only a base classifier, which is weak. Thus, its performance is relatively bad. We have made this clearer in the current manuscript.

Reviewer comment (Experiments):
Why not provide the classical neural networks to compare the Adaboost performance?

Author response:
According to your suggestion, we have performed new experiments to compare our quantum AdaBoost with its classical counterpart. For classical AdaBoost, weak classifiers can be boosted to obtain a strong classifier which has a high level of training accuracy. However, it is generally hard to theoretically guarantee that the generalization error of the strong classifier is small. In contrast, for quantum machine learning, the generalization of quantum classifiers can be guaranteed, while the training is often difficult. In our paper we combine the advantages of quantum machine learning in generalization and classical ensemble methods in training to obtain our main result, namely, variational quantum AdaBoost with supervised learning guarantee for binary classification. In the revised manuscript, as demonstrated in Table 1, we numerically validate that while the training accuracy of quantum AdaBoost is lower than that of classical AdaBoost, the prediction accuracy on new data of quantum AdaBoost is better as compared with classical AdaBoost.

||QCNN+AdaBoost|QCNN+Bagging|CNN+AdaBoost|CNN+Bagging|
|--|--|--|--|--|
|Training Acc. |$0.975 \pm 0.002$| $0.898 \pm 0.006$|$0.980 \pm 0.004$| $0.982\pm0.004$|
|Prediction Acc. |$\boldsymbol{0.973\pm0.001}$|$0.888\pm0.005$|$0.967\pm0.003$|$0.965\pm0.002$|
|Base Classifier |$0.861\pm0.019$|$0.851\pm0.020$|$0.876\pm0.051$ |$0.872\pm0.045$|

Reviewer comment (Novelty):
If using the same Adaboost algorithm, what are the quantum advantages of using quantum neural networks?

Author response:
According to your suggestion, we have performed new experiments to compare our quantum AdaBoost with its classical counterpart. For classical AdaBoost, weak classifiers can be boosted to obtain a strong classifier which has a high level of training accuracy. However, it is generally hard to theoretically guarantee that the generalization error of the strong classifier is small. In contrast, for quantum machine learning, the generalization of quantum classifiers can be guaranteed, while the training is often difficult. In our paper we combine the advantages of quantum machine learning in generalization and classical ensemble methods in training to obtain our main result, namely, variational quantum AdaBoost with supervised learning guarantee for binary classification. In the revised manuscript, as demonstrated in Table 1, we numerically validate that while the training accuracy of quantum AdaBoost is lower than that of classical AdaBoost, the prediction accuracy on new data of quantum AdaBoost is better as compared with classical AdaBoost.

||QCNN+AdaBoost|QCNN+Bagging|CNN+AdaBoost|CNN+Bagging|
|--|--|--|--|--|
|Training Acc. |$0.975 \pm 0.002$| $0.898 \pm 0.006$|$0.980 \pm 0.004$| $0.982\pm0.004$|
|Prediction Acc. |$\boldsymbol{0.973\pm0.001}$|$0.888\pm0.005$|$0.967\pm0.003$|$0.965\pm0.002$|
|Base Classifier |$0.861\pm0.019$|$0.851\pm0.020$|$0.876\pm0.051$ |$0.872\pm0.045$|

Reviewer comment (Novelty):
With the limitation of the system size and circuit depth, the study aims to combine a few weaker quantum classifiers to improve the performance. The manuscript does not show the quantum advantages of the proposed model compared to classical ones from neither theoretical nor numerical. For instance, giving some tasks that are challenging for classical algorithms but can be surpassed by quantum learning models.

Author response:
Finding various practical tasks that can demonstrate quantum superiority remains an open question. Here, we combine the advantages of quantum machine learning in generalization and classical AdaBoost in training to provide the first theoretical proof that for binary classification the prediction error of variational quantum AdaBoost can converge to 0 as the increase of the number of boosting rounds and sample size. In the revised manuscript, we also numerically validate that while the training accuracy of quantum AdaBoost is lower than that of classical AdaBoost, the prediction accuracy on new data of quantum AdaBoost is better as compared with classical AdaBoost. We have demonstrated the new numerical results in Table 1 in the revised manuscript. And we have performed more runs of experiments and all the results support our conclusion.

||QCNN+AdaBoost|QCNN+Bagging|CNN+AdaBoost|CNN+Bagging|
|--|--|--|--|--|
|Training Acc. |$0.975 \pm 0.002$| $0.898 \pm 0.006$|$0.980 \pm 0.004$| $0.982\pm0.004$|
|Prediction Acc. |$\boldsymbol{0.973\pm0.001}$|$0.888\pm0.005$|$0.967\pm0.003$|$0.965\pm0.002$|
|Base Classifier |$0.861\pm0.019$|$0.851\pm0.020$|$0.876\pm0.051$ |$0.872\pm0.045$|

Reviewer comment (Experiments):
In numerics, it only provides a single run which is insufficient.

Author response:
In the revised manuscript, we also numerically validate that while the training accuracy of quantum AdaBoost is lower than that of classical AdaBoost, the prediction accuracy on new data of quantum AdaBoost is better as compared with classical AdaBoost. We have demonstrated the new numerical results in Table 1 in the revised manuscript. And we have performed more runs of experiments and all the results support our conclusion.

||QCNN+AdaBoost|QCNN+Bagging|CNN+AdaBoost|CNN+Bagging|
|--|--|--|--|--|
|Training Acc. |$0.975 \pm 0.002$| $0.898 \pm 0.006$|$0.980 \pm 0.004$| $0.982\pm0.004$|
|Prediction Acc. |$\boldsymbol{0.973\pm0.001}$|$0.888\pm0.005$|$0.967\pm0.003$|$0.965\pm0.002$|
|Base Classifier |$0.861\pm0.019$|$0.851\pm0.020$|$0.876\pm0.051$ |$0.872\pm0.045$|

Reviewer comment (Theory):
The algorithm 2 points out the error of the base classifier should be small. However, under the limitation of circuit depth, how to guarantee the classifier $h_t$ has a small test error meanwhile with shallow circuit depth.

Author response:
Algorithm 2 only requires that quantum weak classifiers are slightly better than random guess, which is easy to be satisfied. We have made this clearer in the revised version.

Reviewer comment (Theory):
In theorem 3.1, it points out that the generalization error is bounded by the number of training samples $n$ and independent trainable gates $K$. Since we can increase the number of $K$ to improve the model and decrease the error, however, why do we increase $K$, the bound is getting worse, which is not reasonable.

Author response:
When increasing $K$, the bound is getting worse. This is owing to the Occam’s Razor principle: *Plurality should not be posited without necessity*. Specifically, there is a trade-off between reducing the training error versus controlling the size of the hypothesis set: a larger hypothesis set ($K$ is larger) could help reduce the training error, but is penalized by the second and third terms in Eq. (9). For a similar training error, it suggests using a smaller hypothesis set.

Reviewer comment (Evaluation):
The quantum advantages are not quite clear, is there any evidence that the proposed method gives quantum advantages?

Author response:
Finding various practical tasks that can demonstrate quantum superiority remains an open question. Here, we utilize the advantage of quantum machine learning in generalization to provide the first theoretical proof that for binary classification the prediction error of variational quantum AdaBoost can converge to 0 as the increase of the number of boosting rounds and sample size.

Reviewer comment (Novelty):
Limited novelty in theoretical analysis. The proof seems a straightforward combination of the standard analysis for ensemble learning and well-established lemmas for quantum models. Thus, the novelty of the theoretical analysis in this paper is limited.

Author response:
To the best of our knowledge, our work provides the first theoretical proof that for binary classification the prediction error of variational quantum AdaBoost can converge to 0 as the increase of the number of boosting rounds and sample size. To be specific, for classical AdaBoost, weak classifiers can be boosted to obtain a strong classifier which has a high level of training accuracy. However, it is generally hard to theoretically guarantee that the generalization error of the strong classifier is small. In contrast, for variational quantum machine learning, the generalization of quantum classifiers can be guaranteed, while the training is often difficult. In this paper we combine the advantages of quantum machine learning in generalization and classical ensemble methods in training, and utilize their analytical tools to obtain our main result, namely, variational quantum AdaBoost with supervised learning guarantee for binary classification.

Reviewer comment (Reproducibility):
As there is no code for this work, it is unclear whether the empirical results are reproductible.

Author response:
We will release the codes when preparing the camera-ready version. If you are interested in reproducing our results, we are also ready to share these codes with you.

Reviewer comment (Novelty):
What are the main differences between this work and existing works for ensemble learning and quantum machine learning? Please respond precisely.

Author response:
We have revised Sec. 1.2 Related Work to make it clearer. For your convenience, we list the differences between our work and existing works in the following.

   1)	As compared with classical AdaBoost, the training accuracy of quantum AdaBoost is lower, since it is often harder to train a quantum classifier than to train a classical classifier. However, as the generalization is easy to be guaranteed for quantum classifiers, the prediction accuracy on new data of quantum AdaBoost is better as compared with classical AdaBoost. We have performed new experiments to demonstrate this in Table 1 in the revised manuscript.
   2)	Various quantum versions of AdaBoost have been proposed, such as Arunachalam & Maity, 2020; Wang et al., 2021b; Ohno, 2022. In their works, they employed quantum subroutines, e.g., mean estimation and amplitude amplification, to update quantum weak classifiers and estimate the weighted errors. Therefore, the realizations of these quantum versions of AdaBoost are beyond the scope of current noisy intermediate-scale quantum (NISQ) devices. In contrast, in our work we utilize variational quantum classifiers realized on the current NISQ circuits, which are obtained through a quantum-classical hybrid way.
   3)	Although the variational quantum AdaBoost has been investigated, e.g., in Li et al., 2023, its performance is validated only through numerical simulations. In our work, we theoretically guarantee the performance of quantum AdaBoost.
   4)	In the current NISQ era, quantum machine leaning is generally weak owing to limited scale of quantum circuits and inevitable influences of noise. Thus it is better to utilize the ensemble methods, e.g., AdaBoost, to combine weak learners to obtain a strong leaner.

Reviewer comment (Experiments):
What are the main differences in experiments between the quantum weak learners and weak (classical) classifiers?

Author response:
The training accuracy of quantum weak learners is lower than that of classical ones, as it is often harder to train quantum learners than to train classical learners. In our new experiments (illustrated in Table 1), under the same number of optimization iterations, the average training accuracy of quantum AdaBoost (having 120 parameters) is 0.975 with standard deviation 0.002, while the classical AdaBoost (having 787 parameters) has a higher level of average training accuracy which is 0.980 with standard deviation 0.004. In contrast, as the generalization is easy to be guaranteed for quantum classifiers, the average prediction accuracy on new data of quantum AdaBoost is 0.973 with standard deviation 0.001, which is better than that of the classical AdaBoost being 0.967 with standard deviation 0.003. We have demonstrated these results in Table 1 in the revised manuscript.

||QCNN+AdaBoost|QCNN+Bagging|CNN+AdaBoost|CNN+Bagging|
|--|--|--|--|--|
|Training Acc. |$0.975 \pm 0.002$| $0.898 \pm 0.006$|$0.980 \pm 0.004$| $0.982\pm0.004$|
|Prediction Acc. |$\boldsymbol{0.973\pm0.001}$|$0.888\pm0.005$|$0.967\pm0.003$|$0.965\pm0.002$|
|Base Classifier |$0.861\pm0.019$|$0.851\pm0.020$|$0.876\pm0.051$ |$0.872\pm0.045$|

Reviewer comment (Novelty):
The proposed method, named quantum AdaBoost, does not exploit the underlying quantum information. Effectively, it treats the PQCs as arbitrary weak classifiers that can take quantum states as input, and use AdaBoost to ensemble them. The novelty of this work is not thoroughly justified by the evidence presented.

Author response:
We have thoroughly revised the manuscript to more clearly present our contributions and the differences from other previously proposed algorithms. For example, we have changed the title to “Variational Quantum AdaBoost with Supervised Learning Guarantee” and rewritten Sec. 1.2 Related Work and Sec. 1.3 Our Contributions to make them clearer.

To the best of our knowledge, our work provides the first theoretical proof that for binary classification the prediction error of variational quantum AdaBoost can converge to 0 as the increase of the number of boosting rounds and sample size. To be specific, for classical AdaBoost, weak classifiers can be boosted to obtain a strong classifier which has a high level of training accuracy. However, it is generally hard to theoretically guarantee that the generalization error of the strong classifier is small. In contrast, for variational quantum machine learning, the generalization of quantum classifiers can be guaranteed, while the training is often difficult. In this paper we combine the advantages of quantum machine learning in generalization and classical ensemble methods in training to obtain our main result, namely, variational quantum AdaBoost with supervised learning guarantee for binary classification.

Reviewer comment (Novelty):
It is confusing to claim that multiple quantum AdaBoost algorithms have already been proposed in the literature while naming the proposed framework as quantum AdaBoost. The comparison between these algorithms and frameworks is also not explained.

Author response:
Various quantum versions of AdaBoost have been proposed, such as Arunachalam & Maity, 2020; Wang et al., 2021b; Ohno, 2022. In their works, they employed quantum subroutines, e.g., mean estimation and amplitude amplification, to update quantum weak classifiers and estimate the weighted errors. Therefore, the realizations of these quantum versions of AdaBoost are beyond the scope of current noisy intermediate-scale quantum (NISQ) devices. In contrast, in our work we utilize variational quantum classifiers realized on the current NISQ circuits, which are obtained through a quantum-classical hybrid way.

Reviewer comment (Novelty):
The main theorem of the paper seems to be a combination of previously known results, with an additional bound on the Rademacher complexity of PQCs. The ideas justifying its novelty are not illustrated.

Author response:
We acknowledge the well-established research on the generalization error in Adaboost. However, we would like to emphasize two significant distinctions between our work and the results presented in Chapter 4 of [1].

Firstly, our results characterize the relationship between the generalization error in terms of the number of independent parameterized gates $K$ and the size of training dataset $n$. This can be easily checked, as opposed to relying on the abstract concept of VC dimension in [1]. It provides a direct guidance enabling us to estimate the order of the required data size for achieving a smaller generalization error.

Secondly, in our results, the upper bound of the generalization error is independent of the number of rounds $T$,  whereas in [1] the upper bound of generalization error is $T$-dependent. As $T$ increases, the upper bound of the generalization error also increases. In contrast, in our results, as $T$ increases, the upper bound of the training error exponentially decreases, but the upper bound of the generalization error remains unchanged. So, our work provides the theoretical proof that for binary classification the prediction error of variational quantum AdaBoost can converge to 0 as the increase of the number of boosting rounds and sample size. 

[1] Schapire R E, Freund Y. Boosting: Foundations and algorithms[J]. Kybernetes, 2013, 42(1): 164-166.

Reviewer comment (Experiments):
The experiments have not considered other ensemble methods and only compare the proposed framework with the base models.

Author response:
Following your suggestion, we have performed new experiments comparing our quantum AdaBoost with Quantum Bagging, classical neural networks+AdaBoost, and classical neural networks+Bagging. We find that our quantum AdaBoost has the best prediction accuracy on new data among all these results. The results are demonstrated in Table 1 in the revised manuscript.

||QCNN+AdaBoost|QCNN+Bagging|CNN+AdaBoost|CNN+Bagging|
|--|--|--|--|--|
|Training Acc. |$0.975 \pm 0.002$| $0.898 \pm 0.006$|$0.980 \pm 0.004$| $0.982\pm0.004$|
|Prediction Acc. |$\boldsymbol{0.973\pm0.001}$|$0.888\pm0.005$|$0.967\pm0.003$|$0.965\pm0.002$|
|Base Classifier |$0.861\pm0.019$|$0.851\pm0.020$|$0.876\pm0.051$ |$0.872\pm0.045$|

Reviewer comment (Experiments):
The experiments where noises are present do not suffice to justify that the proposed method is robust to the noises.

Author response:
In addition to the depolarizing noise, we also perform new experiments under the amplitude damping noise and phase damping noise. All experimental results demonstrate that quantum AdaBoost can help mitigate the influence of these noises. The reasons are twofold. First, in our variational quantum AdaBoost, weak learners can be boosted to obtain a strong learner as long as the weak learners are slightly better than random guess. Noise may degrade the weak leaners, however, as long as they are still better than random guess, they can be boosted to obtain a strong learner. Second, as the quantum parameterized circuit is shallow, quantum learners are weak, but also, the classifiers are less affected by noise due to shallow circuits.