Where can quantum kernel methods make a big difference?
Authors: Muhao Guo
Yang Weng
Venue: ICLR 2022 Submitted (2022)
Keywords: Kernel
Quantum
Classification

Abstract: The classification problem is a core problem of supervised learning, which is widely present in our life. As a class of algorithms for pattern analysis, Kernel methods have been widely and effectively applied to classification problems. However, when very complex patterns are encountered, the existing kernel methods are powerless. Recent studies have shown that quantum kernel methods can effectively handle some classification problems of complex patterns that classical kernel methods cannot handle. However, this does not mean that quantum kernel methods are better than classical kernel methods in all cases. It is still unclear under what circumstances quantum kernel methods can realize their great potential. In this paper, by exploring and summarizing the essential differences between quantum kernel functions and classical kernel functions, we propose a criterion based on inter-class and intra-class distance and geometric properties to determine under what circumstances quantum kernel methods will be superior. We validate our method with toy examples and multiple real datasets from Qiskit and Kaggle. The experiments show that our method can be used as a valid determination method.

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WHERE CAN QUANTUM KERNEL BASED CLASSIFICA- TION METHODS MAKE A BIG DIFFERENCE?
Anonymous authors Paper under double-blind review
ABSTRACT
The classification problem is problem of supervised learning, which is widely present in our life. As a class of algorithms for pattern analysis, Ker- nel methods have been widely and effectively applied to classification problems. However, when very complex patterns are encountered, the existing kernel meth- ods are powerless. Recent studies have shown that quantum kernel methods can effectively handle some classification problems of complex patterns that classi- cal kernel methods cannot handle. However, this does not mean that quantum kernel methods are better than classical kernel methods in all cases. It is still un- clear under what circumstances quantum kernel methods can realize their great potential. In this paper, by exploring and summarizing the essential differences between quantum kernel functions and classical kernel functions, we propose a criterion based inter-class and intra-class distance and geometric properties determine under what circumstances quantum kernel methods will be superior. We validate our method with toy examples and multiple real datasets from Qiskit and Kaggle. The experiments show that our method can be used as a valid deter- mination method.
INTRODUCTION
Since the birth of quantum computing, researchers have been looking for the best place to apply quantum algorithms. The first two quantum algorithms devised were Groverr (1996) and [994). They proved the advantages of quantum algorithms in some specific problems such as search and factorization theoretically. With the rise of artificial intelligence, more and more quantum algorithms combine quantum computing with machine learning. For example, quantum neural network (QNN)
was first proposed by [Ezhov & Ventura (2000) but was only defined in general terms at the physical level. Ricks & Ventura (2003) defined approach to train QNNs, but the complexity of its method exponential. Subsequently. [loyd et al. (2013). Blacoe et al.) (2013), and Reeennrost et al. (2014) tried to introduce quantum computing into clustering, distributed semanties, and SVMs, respectively, but their approaches were too limited to theory. As researchers introduce quantum into various machine learning Serrritnms(SSchuube et al. (2055). Biamonte et al. (2017). Kopezyk (2018). Ciliberto et al. (0188) have started to summarize and sort out the concept of quantum machine learning.
The physical implementation of quantum computers has made great strides in recent years. In 2019, |rute et al. (2019) announced the achievement of quantum hegemony, a milestone event in quantum computing. Also, thanks the emergence of several quantum computing platforms, such as those of IBM and Google, it has become possible for ordinary researchers to translate their research on quantum machine learning algorithms from theory to practice. For example, Farhi & Neven (2018), Quek et al.] (2021). Verdon et al. (2019), Garg et al.) (2019), (Srivastava et al. (2020). and Meichanetzidis et al. (2020) have demonstrated the value of quantum machine learning machine learning tasks by practice, respectively.
On the other hand, one of the most famous machine learning algorithms is the kernel method. A detailed description of the kernel method has been given by Burges] (998a). Muller et al.) (200T). Scholkopf (2001) and Hofmann et al. (2008). Inspired by the classical kernel approach, Rebentrost al. (2014) proposed quantum kernel approach based on SVM, but only theoretically feasible.
was not until Schuld & Killoran (001). and Havličck et al. 2019) systematically proposed two feasible implementations of quantum kernel methods that made quantum kernel methods became

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one of the most mature and practically valuable quantum machine learning methods. Later, Blank jet al. (2020), Wang et al. (202I), Kusumoto et al.] (202I). and Peters et al. (202I) experimentally demonstrated the superiority of quantum kernel methods on some datasets. However, none of them have systematically explored the conditions when quantum kernel methods exist to their advantage. Schuld (202T) summarizes the connection between quantum kernel methods and classical kernel methods. However, it is still unclear when quantum kernel methods will have advantages over classical kernel methods. In this paper, we conclude under what circumstances the quantum kernel method is better or worse than the classical kernel method. Specifically.
We propose that the quantum kernel function is probabilistic and classify the existing kernel functions.
We propose a distance-based criterion ó to determine whether the quantum kernel method has the quantum advantage for a given dataset and demonstrate this experimentally.
We explore and find the relationship between the superiority of quantum kernel methods in two dimensions and (1) the complexity of the data pattern (2) the data based on Mersenne Twister random distribution.
2 BACKGROUND AND RELATED WORK
Classical Kernel. Kernel methods, summarised by Muller et al. (2001) and [Hofmann et al. (2008). are an important class of machine learning methods that carry out machine learning by defining which data points are similar to each other and which are not. Mathematically, the similarity is distance in the data space, i.e., the distance between digital representations of data points. Specifi- cally, the kernel method uses a feature mapping function fe to map data from a point the original input space O to a dighee-dimensional Hilbert feature space Fo. i.e., fo O + Fe, making sepa- rability between data classes more explicit. One of the most famous methods is the support vector machine (SVM) proposed by Burges (1998b).
One important factor that makes the kernel method successful is the kernel track. Scholkopf (2001) pointed out that instead of explicitly calculating the distance in high-dimensional Hilbert space, this distance can be calculated implicitly in low-dimensional input space by the kernel function K, but with the same effect. It can reduce the computational effort significantly and avoid large number of calculations. A nonlinear classification problem is one of the classical machine learning problems, and kernel methods effectively handle such problems. Recall that in classical kernel methods, such as support vector machine, a data point 2; E Rn is mapped into a potentially much higher dimensional feature space Fe via nonlinear mapping function Se- where 2n, represented by d(x;), i.e., fe= I, + o(z;). In space F.,, the nonlinear classification problem becomes linear classification problem and simplifies the problem. The inner product of (xx)))((rr) )) is often seen as distances between and Iÿ in the new space Fe.
kernel method designed to run on quantum computers based on quantum computing properties. Its principle is almost identical to the classical kernel method except that maps the data point from the original input space O to the quantum Hilbert space Fg. i.e., Lq O - Fg- The key to the quantum kernel methods is the quantum mapping function fg. We can view the feature mapping function fr as the key to define the quantum kernel methods. Thus, if the quantum kernel approach is superior, the superiority lies in the quantum mapping function. The mechanism of quantum kernel methods is basically the same as that of classical kernel methods. The data point is mapped from the original input space O to the quantum state space F,, i.e., fn Iz + ((x;)), )) where the I-) denotes a vector and physically it represents a state of some quantum system. The (-1 is the Hermitian Conjugate of the vector I-). In practice, the feature map is realized by acting the circuit U (z;) on the initial quantum state (0*), i.e.,
Quantum Kernel. The quantum kernel method
(o(x))) U (x:) 10").
(I)
The quantum kernel can be obtained by running the circuit U (Tx))(a))) the initial quantum state (0"), where Uf is the Hermitian conjugate of U. Then estimate | (+x) J0*) l2 by counting the frequency of the 0" output as a value k(x,, 2,). Fi shows the process flow of the quantum kernel method and classical kernel method.
2

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Quantum Kernel Method Based On Pauli Feature Map. Following the IBM quantum computing platform, we take two qubits as an example. The general expression of a 2-qubit quantum kernel
k 23) (((z)))((x;))
- (z)) (0 12
(2)
By the definition of Havlíčck et al. (2019). the quantum circuit U is realized by U(3) = Us(ay / U H H/ where the X is the Kronecker Product of two matrices. For the Second- order Pauli-Z. evolution circuit, UdCa) exp(i(<0 + I1 Z1 + (I 20)( - (1))) where Zo. Z1 quantum Z-Gates, and H/ the quantum Hadamard-Gare, We denote the corresponding quantum kernel method as the Z-ZZ guantum kernel method, and the corresponding feature map is showed in Fig B3)). The feature maps of the Z quantum kernel method and the ZZ quantum kernel method are shown in Fig (1) and Fig (2),, respectively. In this paper, all references to quantum kernel methods refer the Z-ZZ quantum kernel method unless otherwise stated. In section 4.3, we compare these three quantum kernel methods.
Support Vector Machine. Support Vector Machine is a maximal margin classifier. It is seen one of the most successful cases of the kernel approach. SVMs are dedicated to finding hyper-plane that separates different classes and makes the margin as large as possible. In general cases, i.e., nonlinear cases, the data is mapped non-linearly to high dimensional Hilbert space by mapping function. Then the distance between two data points can be calculated using the kernel function. Suppose have of data points D {z1 y1), (Inz Vn)]. }. where z; Rd and y/ (-1,+1) According to Burges (1998b), the nonlinear SVM can be modified and expressed by an optimization problem maximize:
2
LD
(3)
1,j=0
0 < ai C and
Ex 0gli 0, where =1, 1,..* n. The decision function is
N₂
N,
f(x) = sign( Σ
asyid(s;)
(x)) + b) = sign Σ
a.yik(si, x) + b).
(4)
where 8; are the support vectors and Ng is the number of support vectors. The SVM-based quantum kernel method is very similar in principle the traditional SVM, except that the computation of the kernel performed on a quantum computer. Havliček et al. (2019) refer to as quantum kernel estimation. We show the specific estimation method in Section 4.2.
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(2.0r210))
Classical D
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caloulation of results
(2)
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02(2000x10))
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Figure 1: A. Working process of classical kernel method and quantum kernel method. B. Based on the IBM quantum computing platform, (1), (2), and (3) denote the feature maps of Z, ZZ, and Z-ZZ quantum kernel methods, respectively. Note that only show one repetition here, and in the experiments, the number repetitions each method is set to two.
METHODS
QUANTUM KERNEL IS A PROBABILISTIC KERNEL
The kernel function is equation for measuring similarity. In vector space, estimate the simi- larity vectors utilizing vector kernel functions. Similarly, graph kernels describe the similarity of two graphs, and tree kernels compare the similarity of trees, which often used in natural language

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processing. A question arises to how to define kemells what kind of kernel functions be effectively used or not. There is no answer to this question. Mercer (1909) argues that a valid kernel function needs satisfy symmetry and positive definiteness. However, some kernel functions that do not obey Mercer's condition still achieve good results in some specific tasks, such as the widely used sigmoid kernel function proposed by Lin & Lin (2003). Mix kernel function proposed by Smits) & Jordaan (2002) tries to achieve better properties when combining different kernel functions.
The mechanism of the quantum kernel function is similar some traditional kernel functions. follows the Mercer theorem and is practical kernel function that expands the family of kernel functions. However, its implementation is based on quantum superposition states and entanglement. Since the values obtained are based on probabilities in statistical sense, we call it probabilistic kernel function. For correspondence, we call the classical kernel function deterministic kernel function. We try to clearly show the relationship and difference between different kernel functions by diagram (Fig 2. It is worth noting that there are no guarantees for one kernel work better than the other in all cases, according to the No Free Lunch Theorem (Wolpert & Macready (197)) Choosing different kernel functions in various subjects will achieve better results. The primary purpose of this paper is to investigate under what circumstances the quantum kernel method is better or worse than the classical kernel method.
Kernel
Vector Kernel
Satisfy Mercer's Condition
Don't Satisfy Mercer's Condition
Deterministic
Linear Kernel
Palynomial
Tree Kernel
Sigmold Kemel
Laplacian Kemel
RBF Kernel
Cosine Similarity
Kernel
Customer Defined Kernel
Graph Kernel
Today's Quantum Kernel
Probabilistic
Pauli Feature Map
DLP Feature Map
Markov Kernel
Kernel
Based Kernel
Customer Defined
Basic Inner
Kernel
Customer Kemel
Product Based
Figure 2: Kernels Category
3.2
THE PROPOSED PATTERNS AND CRITERIA FOR JUDGING QKM
First, we would like to demonstrate the advantages of quantum kernel methods over classical kernel methods when dealing with data patterns based on the Mersenne Twister random distribution. The Mersenne Twister is pseudo random number generator which was first proposed by Matsumoto & Nishimura (1998). The Mersenne Twister is used as default pseudo random number generator by many software, such Python, R, and PHP. The Mersenne Twister random distribution is distribution that be generated by the Mersenne Twister method.
Theorem 3.1 (Advantages of QKM for random distributions) In two dimensions, the Z-ZZ quantum kernel method has better learning ability than classical kernel methods for randomly dis- tributed data patterns based on Mersenne Twister.
Proof. We assume that the Z-ZZ feature map can effectively simulate the efficacy of the feature map proposed by Liu et al. (2021). By Matsumoto & Nishimura (1998), for a k-bits binary number, the Mersenne Twister algorithm generates discrete uniformly distributed random numbers in the range [0, 2$ 1]. Solving this problem is discrete logarithm problem (DLP). For DLP, Liu et al. (2021) say efficient classical algorithm achieve inverse-poll porress-polnmmillll better accuracy than random guessing. Therefore, the Z-ZZ quantum kernel method can demonstrate quantum superiority over the classical kemel methods for Mersenne Twister-based randomly distributed data patterns.

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Second, demonstrate that whether quantum kernel methods have quantum advantages is related to the of the data set, where will be defined in Equ. (7). Because any classification problem can be transformed into binary situation, If the distance between classes is large enough, i.e., the simple case, the quantum kernel methods not good the classical kernel methods. Based fourteen Adhoc-Modify datasets (Section 4.1 Qiskit package datasets (6)), we measure the 8 of each dataset and draw graph with å as the horizontal coordinate. Fig shows that when increases certain level, the advantage of the quantum kernel method disappears.
all base study the binary classification problem.
To measure the degree of separation of two classes Cj and Cm. we first define the inter-class distance
D Cm) NgN ΣΣ d(a 3 (m)),
(5)
where Nú and Nm are the number of observations belonging to class C and Cm. respectively. The In is the n-th sample in class Cj and d(.) ) is the euclidean distance between two samples, i.c.,
d(æ, 0) V> - y;)²
For the sake of uniformity. we also need to define the intra-class distance for class C, i.e.
D(C) N(N 1) j- L.oي
Σ Σ
d(rk.
(6)
where the N is the of class C, and In is the n-th sample in class C. We propose a criterion to evaluate whether quantum kernel methods will be better than classical kernel methods. In binary classification problem, we define the degree of integration of C2 and C. as 6pm-
DCC, Cm)
dtm ((CI) + D(C..)
Theorem 3.2 illustrates that the larger the 8, the greater the separation of the two classes. When the delta large enough, the quantum kernel function loses its quantum advantage.
Theorem 3.2 (Deficiencies of the QKM) In the case of a balanced rumber of the two classes, there exists & such that the quantum kernel method will not be better than the classical kernel method in handling classification problems when >> d0- In practice, 60 is usually taken as 0.6.
.. MR which have expectation E(M) = μ and variance D(M) = o², where M is the random variable. R is the number of measurement shots. By the Central Limit Theorem, for any m, the distribution function
Proof. Suppose our measurement independent identical distribution M2 Mr....
FR(m) P *R M,-RF, RE < m} satisfies: lim FR(x) lim ( 2i=1
RR M,-RE
VRD S x)
VDR
V2x c 4 dt. This shows that when R is large enough, the random variable YR
M
VRD
obeys normal distribution N(O, 1). So, E M, v + VRRYR R obeys normal distribution N(R,, Ro²). However, even fully noise caused by measurement. Two types of data are linearly separable or almost linearly separable when (CC,, Cm) D(Ci)= + D(Cm), i.c., 8 >> dg. easy for deterministic kernel method to find a boundary line with an infinitely small error. So, when ô > do, the quantum kernel method is better than the classical kernel method in handling classification problems.
corrected, the quantum methods still have the
In fact, as probabilistic kernel method, quantum kernel method has more error than determinis- tic kernel method caused by the measurement process. especially evident in some simple cases because we assume that deterministic kernel methods must find boundary line. However, errors in probabilistic methods are inevitable.
EXPERIMENTS
4.1 DATA PREPARATION
Qiskit package datasets. Five datasets from qiskit.ml. disttt.s were used in the experiments. (1) Digits datasets. This dataset consists of 1797 8x8 images, and each image is handwritten number
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Figure 3: The relationship between ô and the predicting accuracy in Ad hoc dataset. The top graph shows the training accuracy, and the bottom graph shows the testing accuracy.
0~99 We perform binary classification for every two digits. Thus, 45 small datasets were generated. (2) Breast cancer dataset. (3) Iris dataset. (4) Wine dataset. (5) Adhoc dataset. Note that (1)~(4) are copies from UCI ML Hand-writte Digits Dataset, UCI ML Breast Cancer Wisconsin (Diagnostic) Dataset, UCI ML Iris Plants Dataset, and UCI ML Wine Recognition Dataset, respectively. For (5) Adhoc dataset, separate the two classes and translate all the data points in class by the same distance, leaving the other class unchanged. this way, we get 14 datasets, denoted as (6) Adhoc-Modify datasets.
Kaggle datasets. Five datasets from Kaggle were used in the experiments. (7) Email spam dataset (Balaka Biswas), (8) Heart disease dataset (Zeeshan Mullap. (9) Giants and dwarfs dataset (Vinesm- suic (10) Star type dataset (Baris Dincer). (1I) Drug dataset (Pratham Tripathi).
Geometric toy datasets. We designed the two-dimensional geometric toy datasets to illustrate the learning ability of the quantum kernel method for processing classification problems in geometric patterns. (12) Geometric non-random datasets. We designed 2~55 layers of concentric circles and four squares with different mixing degrees. FAg shows the four circular datasets and four square datasets. (13) Geometric random datasets. Based on the Mersenne Twister random distribution, we designed datasets with random distributions in three geometric patterns: Circular, square, and equilateral triangle patterns. The three graphs on the right side of Fig show a sample of these three datasets, respectively.
4.2 PROCESS OF TRAINING AND TESTING OF QUANTUM KERNEL METHOOSS
We briefly introduce the training process of the quantum kernel method here, following the idea of [iu et al. (2021). Suppose training dataset Drrain (()1,)1),= (In, Bn)}. Since the only difference of the quantum kernel method an SVM from the classical SVM kernel calculation, here we only show the kernel calculation process in quantum computer.
Training: For each pair of data points v, and xy (i * j) in Dirains we apply (x,))(())) on input |0@2). After R repetitive runs, we record the number times the output results in |00 Ra denoted as Ro and k(z;, .r2) 2R where + and k(r,, 2x:) 1. In the end, we apply Equ. 3) directly. Testing: When there comes a new sample Knems each data point z; in D we appTy
U (z,))((zeeu)) on the input |00] 2). After R repetitive runs, we record the number of times the output results in (0@2), denoted as R, and k( xz) R. 22R In the end, we apply Equ.44) directly.

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4.3 EXPERIMENTS RESULTS
The quantum kernel method is good at complex data patterns, In this part, we explore the ability of quantum kemel methods to solve classification problems under challenging patterns. Firstly, increase the learning difficulty by increasing the complexity of the geometric patterns and get four patterns, i.e., P1~P4 in 4A. Experiments show that from P1 to P4, the classical kernel approach, which initially performs better, gradually loses its superiority. In contrast, the superiority of the quantum kernel approach begins to emerge.
To further increase the learning difficulty, we hypothesized the existence of pattern in which the data are randomly distributed over the geometry according to the Mersenne Twister. The right three graphs in 4B show this kind of data pattern. To explore the learning ability of quantum ker- nel methods for Mersenne Twister randomly distributed data patterns, we plotted the relationship between the size of the dataset and the prediction accuracy of each method. Considering the ran- domness of the random distribution, we take the average result of 50 times as the result of each experiment. The experiments demonstrate that the quantum kernel method has a more robust learn- ing capability than the classical kernel method in this complex pattern.
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Figure 4: The horizontal coordinates P1~P4 represent the data patterns different difficulties of the geometric non-random dataset, respectively. B. The horizontal coordinates indicate the dataset size, and the vertical coordinates indicate the prediction accuracy. В illustrates the relationship between the prediction correctness and the size of the dataset under three patterns in the geometric random distribution dataset.
The quantum kernel method fails at large values of In this part, we will show that our cri- teria effectively determine whether quantum kernel methods demonstrate advantages over classical kernel methods or not. First, taking the Digits dataset as an example, we perform binary classification for every digits. Thus, we generate 45 small datasets. We then PCA duce the dimensionality of all data to 2 dimensions, as shown in Fig(5) We recorded each small dataset's training and test accuracy after applying four classical kernel methods and quantum kernel method. The bottom left panel in Fig shows the prediction accuracy of each method. Ac- cording the experimental results, we attach corner mark each data set. The red rounded corner markers indicate that the quantum kernel method has the potential for quantum dominance on the corresponding dataset, while the black rounded corner markers indicate that does not. The blue corner markers indicate that the quantum kemel method is indistinguishable from the classical ker- method. Intuitively, the quantum kernel approach for linearly divisible data sets does not show superiority. However, for complex models with relatively high fusion, the quantum kernel approach has the potential to be superior.
To validate our criteria ò, we prepared 81 datasets: 45 small datasets from Digits, 14 datasets from Addoc-Modify, Breast cancer, Iris, Wine, Email spam, Heart disease, Giants and dwarfs, Star Type, Drug, 4 datasets from Geometric non-random, i.c., the top-left, bottom-left. top-right and bottom- right four datasets in Fig 4A, 10 datasets from Geometric random including five circles size thirty and five squares of size forty. The difference in prediction accuracy between classical kernel methods

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0
1
2
3
4
6
8
9
Figure 5: The top right panel shows the 45 datasets after visualization, and the bottom left panel shows the prediction accuracy of the five methods, which are SVM models based on Gaussian kernel, linear kernel, polynomial kernel, sigmoid kernel, and quantum kernel, respectively. The blue line represents the training set, and the red represents the test set. Red corners indicate the case where the training or test scores of the quantum kernel method outperform all classical kernel methods. Black corners indicate the case where the training test scores of the quantum kernel method are worse than any classical kernel method. Blue corners indicate the case where the training or test scores of the quantum kernel method equal to the best value of all classical kernel methods.
and quantum kernel methods on each dataset is calculated. It is shown by the relevant vertical lines in Fig/6. The magnitude of 5 on each dataset is shown with an asterisk in Fig According the experiments, when the quantum kernel method has an advantage, the asterisks appear below the 0.6 level line all the time. We can get the conclusion that (i) the availability of quantum superiority correlates with 8 and (ii) the quantum kernel method can be better than the classical kernel method when 8 < 0.6, although it a sufficient condition.
Simple quantum kernel methods do not offer quantum advantages. When the feature space is so large that the kernel function is computationally expensive, quantum kernel methods can effectively estimate their kernel functions, but classical kernel methods cannot. So, quantum kernel methods are preferred over classical kernel methods for classification problems with complex patterns. However, simpler quantum kernel function can be simulated classically. For simple-kerell-brte kernel method, it no more has the superiority. Its ability to handle complex pattern problems is significantly reduced and even inferior classical kernel methods. Interestingly, sometimes this simple quantum kernel function works well with simple pattern problems.
quantum
We illustrate the above opinion experimentally. In this part, prepared 68 datasets: 45 small datasets from Digits, Breast cancer, Iris, Wine, Adhoc, Email spam, Heart disease, Giants and dwarfs, Star Type, Drug, 4 datasets from Geometric non-random, i.e., the top-left, bottom-left, top-
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Figure 6: The horizontal coordinate is the 81 training datasets, the left vertical coordinates indicate the method prediction accuracy, and the right vertical coordinates indicate the value of 6. The green circles represent the best classical kernel methods in RBF, Linear, Polynomial, and Sigmoid kemel- based kernel methods. The purple circles represent the quantum kernel methods. Under the same dataset, if the classical kernel method is worse than the quantum kernel method, we use the green vertical line to indicate how better the classical kernel method is than the quantum kernel method and the black asterisk to indicate the magnitude of 6. Otherwise, we use the purple vertical line to indicate how better the quantum kernel method is than the classical kernel method and use the red asterisk to indicate the magnitude of 6.
right and bottom-right four datasets in Fig 4A 10 datasets from Geometric random including five circles of size thirty and five squares of size forty. By comparing the performance of the best classi- cal kernel method and the three quantum kernel methods on 68 training datasets, find that some kernel methods based on simple quantum kernel functions, such as the Z quantum kernel method, do outperform the Z-ZZ kernel method on some datasets, but do not outperform the best classical kernel methods. However, when quantum superiority exists, it is often achieved by Z-ZZ kernel methods. The experiments show that the quantum kernel method superiority on only one dataset achieved by the Z quantum kernel method.
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Figure 7: The horizontal coordinate the 68 training datasets, the left vertical indicates the method's prediction accuracy, and the right vertical coordinate indicates the value of 5. The hollow black circles, red circles, orange circles, and pink circles represent the best classical kernel methods, Z-ZZ quantum kernel-based, 77 quantum kernel, and Z quantum kernel-based quantum kernel methods, respectively. We use blue asterisks indicate the value of å when the Z-quantum kernel-based method is optimal, which has only case, and orange asterisks to indicate the value of ó when the ZZ-quantum kernel-based method is the best, although this case does not occur in the experiment. Correspondingly. red and black asterisks used to represent the value of 5 when the Z-ZZ quantum kernel method and the classical kernel method are best, respectively.
CONCLUSION
The classification problem is one of the most common problems machine learning, and both clas- sical kernel methods and quantum kernel methods can effectively handle the classification problem. However, is difficult to determine what situation each of them is suitable for. This paper explores when quantum kernel methods take quantum advantage by comparing different kernel functions. Moreover, a judgment criterion is proposed to help one decide when quantum kernel methods can achieve better results than classical kernl methods. Experiments show that method is effective.

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prat brathamttnn Eeaatrippatin drrggdruuurooorr www.ssaassaaaaaaa.aoco.a.a
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and -hnddddwarf.
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12


Reviews:
[{"review_id": "lEf8d02wlKp", "reviewer": "Reviewer_DX2D", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The introduction of the article is well-written and reflects a good knowledge of the literature. However, the authors restrict the comparison between quantum kernel methods and classical kernel methods to the case of binary classification. Moreover, the theoretical results are presented informally. ", "summary_of_the_paper": "This article studies the superiority of quantum kernel methods over classical kernel methods. The authors relate the quantum advantages for classification tasks to the separation between the classes.  Several numerical simulations were conducted to support the theoretical findings.\n", "summary_of_the_review": "Unfortunately, I have to recommend that the paper would be rejected for the following reasons:\n\n1.  the analysis concerns binary classification, which is one among many learning tasks to study. It is not clear how to deal with other learning tasks. \n2. The presentation of the theoretical results (Theorem 3.1 and Theorem 3.2) is poor and informal, and the proofs lack rigor. For example, the threshold 0.6 in Theorem 3.2 Is mysterious and does not seem to stem from any grounded argument. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "U3o_-embdAQ", "reviewer": "Reviewer_kXVf", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Although the question raised by this paper is interesting, I am not convinced enough by the proposed answer to warrant acceptance. In particular,\n\n**There are many writing flaws**:\n- last sentence of 2nd paragraph in unclear\n- 1st sentence of 3rd paragraph in unclear: *kernel methods* (and not *the kernel method*) are a family of algorithms\n- kernel methods bec*o*me (last word of p.1)\n- We *conjecture* (1st contribution bullet point)\n- please cite the SVM paper when introducing them (1st par. of sec. 2)\n- kernel *trick* (2nd par. sec. 2)\n- SVM paragraph: remove *kind of*\n- Eq. (4), a $\\top$ is missing between the $\\phi$ (also support vectors are never defined)\n- top of page 4: there is a very clear theoretical answer about what kernel is a valid or not. The good performances of the sigmoid kernel (reference needed here) does not change anything\n- I cannot understand what authors mean by *that expands the family of kernel functions*\n- Fig.2 seems to imply that tree and graph kernels do not satisfy Mercer's conditions: why such a separation? Also, the ... boxes are not relevant, and \"Don't satisfy...\" would be a better legend\n- the Mersenne Twister distribution could be introduced and commented before the theorem statement\n- Eq. (5): I assume $N_l$, $N_m$ instead of $N_1$, $N_2$\n- the \"number of observations belonging to class X\" would be more clear than the *size*\n- the $\\rightarrow$ vector notation is introduced once and is not consistent with the rest of the paper\n- expliciting the formulas in eq. (7) is not necessary\n\n**The exposition of classical and quantum kernels is particularly messy**, which is very harmful as it is a key point in the paper:\n- weird $x$ notation\n- $f_c$ is mapping $x$ to $\\phi(x)$ ?! \n- the dot product denoted with both $^T$ and $\\cdot$\n- note also that to compute a distance, one would use $|| \\phi(x) - \\phi(y)||$ rather that $<\\phi(x), \\phi(y)>$ as claimed by the authors\n- in the quantum paragraph, I assume $F_q$ means $f_q$\n- it seems to me that the definition of $f_q$ from $\\mathcal{O}$ to $\\mathcal{F}_q$ is repeated. Furthermore, I cannot see any difference from the standard kernel definition\n- $f_q$ is mapping $x_i$ to a quantity that depends on $x_j$ ?!\n- the $|\\cdot >$ notation is very confusing for people used to kernel dot products. Even if it is the standard notation in the quantum literature, why don't we have two $|$ in the definition of $f_q$?\n- the circuit $U$ is never defined\n\nIn the same vein, **the maths seem not rigorous to me**:\n- how can the proof of Theorem 3.1 begin by \"we assume\" ?! If you have an extra assumption, just put it in the theorem statement\n- the same goes for the proof of Theorem 3.2\n- the exact statement of Theorem 3.2 should be \"there exists $\\delta_0$ such that...\"\n- the CLT does not imply the existence of a $R$ such that the distribution is the standard Gaussian (what if the empirical mean is $1/n$ for instance?)\n- what are the \"measurements\" referring to?\n- where do $\\sigma$ and $\\mu$ play a role?\n- please prove \"it is easy for a deterministic kernel [...] with small error\"\n\nAnd **the contribution is at best incomplete**, if not erroneous (see comments about the proofs above):\n- Theorem 3.1 only states that quantum kernels are superior than standard kernels on this example because the latter cannot learn anything. Can we prove that quantum **can** learn something in this case, i.e., are strictly better. Otherwise the statement is pretty vacuous.\n- the definition of $\\delta$ is based on empirical quantities, is it suitable? Shouldn't a quantity in expectation make more sense?\n- what happens for kernels defined on non-vectorial inputs? Do they have a quantum analog? It should be discussed\n- can we use another distance than the Euclidean one?\n- if we add an offset to the distance, it seems that the 0.6 threshold exhibited changes, making me dubious about such an absolute value\n- Theorem 3.2 actually only shows that quantum kernels are worse than standard ones in a precise regime, not that they are better in the opposite scenario\n\nMore generally, the general idea I retain from this paper is that computing quantum kernels requires some randomness, and that if this randomness is greater than the \"task randomness/difficulty\", it is vain to use quantum kernels. If this message is the right one, I would suggest the authors to be more clear on this, and to drop the $\\delta$ criterion, which seems to be a proxy for the task difficulty.", "summary_of_the_paper": "This paper tries to identify a condition that makes quantum-based kernel methods superior to standard ones.", "summary_of_the_review": "Although the question raised by this paper is interesting, the proposed answer is not clearly exposed, not rigorous enough on the math side, and not enough significant to warrant acceptance.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "1: The main claims of the paper are incorrect or not at all supported by theory or empirical results.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "MFZl7iSdse9", "reviewer": "Reviewer_3XVL", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "The background section was clear and thorough.\n\nI have some difficulties with section 3:\n\n- proof of theorem 3.1.  The proof hinges on an assumption - \"the Z-ZZ feature map can effectively simulate the efficacy of the feature map proposed by Liu et al. (2021)\" - but I don't see any proof of this conjecture, at least in the mathematical sense.  If this is axiomatic to the theorem then it should be presented as such in the theorem: otherwise the proof doesn't work afaict.  Have I missed something here?\n\n- theorem 3.2: why is $\\delta_0$ usually taken as 0.6?  Is this based on the experimental results alone, or is there some intuition as to why this particular threshold is important?\n\nHowever the experimental section is very thorough and quite convincing, so I am willing to overlook my misgivings regarding the theorems.\n\nMinor point:\n\n- in the paragraph before equation (1), is the definition of the feature map $f_q : x_i \\to < \\phi (x_i) | \\phi  (x_j) >$ correct?  And if so, how does $x_j$ fit here?\n\n\n-- EDIT AFTER FURTHER CONSIDERATION --\n\nAfter further consideration and reading other comments I am persuaded that the informality of proofs is a more serious problem than I initially thought and adjusted my recommendation accordingly.", "summary_of_the_paper": "The paper investigates the circumstances under which quantum kernel methods will be superior to classical kernel methods.  The criteria is based on a threshold of the ratio of the inter- and intra-class distances of the (binary) training data.  Validation of the proposed criteria is carried out on a range of toy and real datasets.", "summary_of_the_review": "- Overall a well-written paper.\n- I have some problems with the proof of theorem 3.1.\n- I am also curious how the threshold value 0.6 is arrived at, and whether this is purely experimental or if there is some alternative intuition.\n- The experimental section is very thorough and quite convincing.\n\n\n-- EDIT AFTER FURTHER CONSIDERATION --\n\nAfter further consideration and reading other comments I am persuaded that the informality of proofs is a more serious problem than I initially thought and adjusted my recommendation accordingly.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "KrZwM_fesKq", "reviewer": "Reviewer_cR4C", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "1. The writing quality of the paper is not ideal. Sections 2 and 3 are long but not informative. The figures in experiments are hard to understand. The paper has a lot of grammar errors.\n\n2. The paper seems to focus on the classification problems. Quantum kernels can be applied to a much wider range of problems. The terminology ``quantum kernel methods\" in the title may need to be changed to quantum kernel based classification methods.\n\n3. The proof of theorem 3.1 is just a summary of statements in the literature. The proof of theorem 3.2 is just an application of CLT (and with some errors). For example, the first sentence in the proof of Theorem 3.2. ``Suppose our measurement independent identical distribution ... where M is the random variable, R is the\nnumber of measurement shots.\" is not correct in both English and mathematics. Also, I am not sure if the proof really did the job to prove the statements in Theorem 3.2.\n\n4. The experiments are not convincing. Many implementation details are missing. The results, figures, and explanations are hard to understand. No replication codes are provided. My guess is the experiments are done by simulations run on the classical computer rather than real quantum computers? ", "summary_of_the_paper": "This paper studies quantum kernel methods for classification problems. It compares the differences between quantum kernel functions and classical kernel functions. A degree of integration and two theorems are proposed. Some empirical experiments are done on the datasets from Qiskit and Kaggle.", "summary_of_the_review": "Correctness: \nThis paper only considered limited scenarios in classification problems. Therefore, the title and the statements in the paper may be exaggerated by saying quantum kernel methods instead of quantum kernel based classification methods. The proofs of the major theorems are not convincing to me. The experiments are not strong enough to support the claims made in the paper.\n\nTechnical Novelty And Significance:\nThe theoretical contributions of the paper are limited if there are some at all. \n\nEmpirical Novelty And Significance:\nI am not sure if the authors have proposed any new quantum methods for the experiments. The results are hard to understand.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "1: The main claims of the paper are incorrect or not at all supported by theory or empirical results.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: There was consensus that though  the paper introduces an interesting question, but not enough exploration has been made. The reviews point out several mathematical in-accuracies, and points out  several issues including that the delta criterion needs to be examined.
Decision: Reject