Original: W HERE C
Revised: W HERE C

Original: AN Q UANTUM K ERNEL  M ETHODS M AKE
Revised: AN Q UANTUM K ERNEL B ASED C LASSIFICATION M ETHODS M AKE

Original: A 
Revised: A B

Original: B IG D IFFERENCE ?
Revised: IG D IFFERENCE ?

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: The classiﬁcation problem is a core problem of supervised learning, which is widely present in our life.
Revised: The classification problem is a core problem of supervised learning, which is widely present in our life.

Original: As a class of algorithms for pattern analysis, Kernel methods have been widely and effectively applied to classiﬁcation problems.
Revised: As a class of algorithms for pattern analysis, Kernel methods have been widely and effectively applied to classification problems.

Original: However, when very complex patterns are encountered, the existing kernel methods are powerless.
Revised: However, when very complex patterns are encountered, the existing kernel methods are powerless.

Original: Recent studies have shown that quantum kernel methods can effectively handle some classiﬁcation problems of complex patterns that classical kernel methods cannot handle.
Revised: Recent studies have shown that quantum kernel methods can effectively handle some classification problems of complex patterns that classical kernel methods cannot handle.

Original: However, this does not mean that quantum kernel methods are better than classical kernel methods in all cases.
Revised: However, this does not mean that quantum kernel methods are better than classical kernel methods in all cases.

Original: It is still unclear under what circumstances quantum kernel methods can realize their great potential.
Revised: It is still unclear under what circumstances quantum kernel methods can realize their great potential.

Original: 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.
Revised: 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.

Original: We validate our method with toy examples and multiple real datasets from Qiskit and Kaggle.
Revised: We validate our method with toy examples and multiple real datasets from Qiskit and Kaggle.

Original: The experiments show that our method can be used as a valid determination method.
Revised: The experiments show that our method can be used as a valid determination method.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Since the birth of quantum computing, researchers have been looking for the best place to apply quantum algorithms.
Revised: Since the birth of quantum computing, researchers have been looking for the best place to apply quantum algorithms.

Original: The ﬁrst two quantum algorithms devised were Grover (1996) and Shor (1994).
Revised: The first two quantum algorithms devised were Grover (1996) and Shor (1994).

Original: They proved the advantages of quantum algorithms in some speciﬁc problems such as search and factorization theoretically.
Revised: They proved the advantages of quantum algorithms in some specific problems such as search and factorization theoretically.

Original: With the rise of artiﬁcial intelligence, more and more quantum algorithms combine quantum computing with machine learning.
Revised: With the rise of artificial intelligence, more and more quantum algorithms combine quantum computing with machine learning.

Original: For example, quantum neural network (QNN) was ﬁrst proposed by Ezhov & Ventura (2000) but was only deﬁned in general terms at the physical level.
Revised: For example, quantum neural network (QNN) was first proposed by Ezhov & Ventura (2000) but was only defined in general terms at the physical level.

Original: Ricks & Ventura (2003) deﬁned an approach to train QNNs, but the complexity of its method is exponential.
Revised: Ricks & Ventura (2003) defined an approach to train QNNs, but the complexity of its method is exponential.

Original: Subsequently, Lloyd et al.
Revised: Subsequently, Lloyd et al.

Original: (2013), Blacoe et al.
Revised: (2013), Blacoe et al.

Original: (2013), and Rebentrost et al.
Revised: (2013), and Rebentrost et al.

Original: (2014) tried to introduce quantum computing into clustering, distributed semantics, and SVMs, respectively, but their approaches were too limited to theory.
Revised: (2014) tried to introduce quantum computing into clustering, distributed semantics, and SVMs, respectively, but their approaches were too limited to theory.

Original: As researchers introduce quantum into various machine learning algorithms, Schuld et al.
Revised: As researchers introduce quantum into various machine learning algorithms, Schuld et al.

Original: , Biamonte et al.
Revised: , Biamonte et al.

Original: (2017), Kopczyk (2018),
Revised: (2017), Kopczyk (2018),

Original: Ciliberto et al.
Revised: Ciliberto et al.

Original: (2018) have started to summarize and sort out the concept of quantum machine learning.
Revised: (2018) have started to summarize and sort out the concept of quantum machine learning.

Original: The physical implementation of quantum computers has made great strides in recent years.
Revised: The physical implementation of quantum computers has made great strides in recent years.

Original: In 2019, Arute et al.
Revised: In 2019, Arute et al.

Original: (2019) announced the achievement of quantum hegemony, a milestone event in quantum computing.
Revised: (2019) announced the achievement of quantum hegemony, a milestone event in quantum computing.

Original: Also, thanks to 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.
Revised: Also, thanks to 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.

Original: For example, Farhi & Neven (2018), Quek et al.
Revised: For example, Farhi & Neven (2018), Quek et al.

Original: (2021), Verdon et al.
Revised: (2021), Verdon et al.

Original: , Garg et al.
Revised: , Garg et al.

Original: Srivastava et al.
Revised: Srivastava et al.

Original: (2020), and Meichanetzidis et al.
Revised: (2020), and Meichanetzidis et al.

Original: (2020) have demonstrated the value of quantum machine learning in machine learning tasks by practice, respectively.
Revised: (2020) have demonstrated the value of quantum machine learning in machine learning tasks by practice, respectively.

Original: On the other hand, one of the most famous machine learning algorithms is the kernel method.
Revised: On the other hand, one of the most famous machine learning algorithms is the kernel method.

Original: A detailed description of the kernel method has been given by Burges (1998a), Muller et al.
Revised: A detailed description of the kernel method has been given by Burges (1998a), Muller et al.

Original: Scholkopf (2001) and Hofmann et al.
Revised: Scholkopf (2001) and Hofmann et al.

Original: Inspired by the classical kernel approach, Rebentrost et al.
Revised: Inspired by the classical kernel approach, Rebentrost et al.

Original: (2014) proposed a quantum kernel approach based on SVM, but only theoretically feasible.
Revised: (2014) proposed a quantum kernel approach based on SVM, but only theoretically feasible.

Original: It was not until Schuld & Killoran (2019), and Havl´ıˇcek et al.
Revised: It was not until Schuld & Killoran (2019), and Havl´ıˇcek et al.

Original: (2019) systematically proposed two feasible implementations of quantum kernel methods that made quantum kernel methods became one of the most mature and practically valuable quantum machine learning methods.
Revised: (2019) systematically proposed two feasible implementations of quantum kernel methods that made quantum kernel methods became one of the most mature and practically valuable quantum machine learning methods.

Original: Later, Blank et al.
Revised: Later, Blank et al.

Original: (2020), Wang et al.
Revised: (2020), Wang et al.

Original: , Kusumoto et al.
Revised: , Kusumoto et al.

Original: (2021), and Peters et al.
Revised: (2021), and Peters et al.

Original: (2021) experimentally demonstrated the superiority of quantum kernel methods on some datasets.
Revised: (2021) experimentally demonstrated the superiority of quantum kernel methods on some datasets.

Original: However, none of them have systematically explored the conditions when quantum kernel methods exist to their advantage.
Revised: However, none of them have systematically explored the conditions when quantum kernel methods exist to their advantage.

Original: Schuld (2021) summarizes the connection between quantum kernel methods and classical kernel methods.
Revised: Schuld (2021) summarizes the connection between quantum kernel methods and classical kernel methods.

Original: However, it is still unclear when quantum kernel methods will have advantages over classical kernel methods.
Revised: However, it is still unclear when quantum kernel methods will have advantages over classical kernel methods.

Original: In this paper, we conclude under what circumstances the quantum kernel method is better or worse than the classical kernel method.
Revised: In this paper, we conclude under what circumstances the quantum kernel method is better or worse than the classical kernel method.

Original: Speciﬁcally,
Revised: Specifically,

Original: • We propose that the quantum kernel function is probabilistic and classify the existing kernel functions.
Revised: • We propose that the quantum kernel function is probabilistic and classify the existing kernel functions.

Original: • We propose a distance-based criterion δ to determine whether the quantum kernel method has the quantum advantage for a given dataset and demonstrate this experimentally.
Revised: • We propose a distance-based criterion δ to determine whether the quantum kernel method has the quantum advantage for a given dataset and demonstrate this experimentally.

Original: • 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.
Revised: • 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.

Original:  B ACKGROUND A ND R ELATED W ORK
Revised:  B ACKGROUND A ND R ELATED W ORK

Original: Classical Kernel.
Revised: Classical Kernel.

Original: Kernel methods, summarised by Muller et al.
Revised: Kernel methods, summarised by Muller et al.

Original: and Hofmann et al.
Revised: and Hofmann et al.

Original: (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.
Revised: (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.

Original: Mathematically, the similarity is a distance in the data space, i.e., the distance between digital representations of data points.
Revised: Mathematically, the similarity is a distance in the data space, i.e., the distance between digital representations of data points.

Original: Speciﬁcally, the kernel method uses a feature mapping function
Revised: Specifically, the kernel method uses a feature mapping function

Original: f c to map data from a point in the original input space O to a higher-dimensional Hilbert feature space F c , i.e., f c : O → F c , making separability between data classes more explicit.
Revised: f c to map data from a point in the original input space O to a higher-dimensional Hilbert feature space F c , i.e., f c : O → F c , making separability between data classes more explicit.

Original: One of the most famous methods is the support vector machine (SVM).
Revised: One of the most famous methods is the support vector machine (SVM) proposed by Burges (1998b).

Original: One important factor that makes the kernel method successful is the kernel track.
Revised: One important factor that makes the kernel method successful is the kernel track.

Original: 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
Revised: 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

Original: K , but with the same effect.
Revised: K , but with the same effect.

Original: It can reduce the computational effort signiﬁcantly and avoid a large number of calculations.
Revised: It can reduce the computational effort significantly and avoid a large number of calculations.

Original: A nonlinear classiﬁcation problem is one of the classical machine learning problems, and kernel methods can effectively handle such problems.
Revised: A nonlinear classification problem is one of the classical machine learning problems, and kernel methods can effectively handle such problems.

Original: Recall that in classical kernel methods, such as support vector machine, a data point x i ∈ R n is mapped into a potentially much higher dimensional feature space F c via a nonlinear mapping function f c , where x  is represented by φ ( x i ) , i.e., f c : x i → φ ( x i ) .
Revised: Recall that in classical kernel methods, such as support vector machine, a data point x i ∈ R n is mapped into a potentially much higher dimensional feature space F c via a nonlinear mapping function f c , where x i is represented by ϕ ( x i ) , i.e., f c : x i → ϕ ( x i ) .

Original: In space F c , the nonlinear classiﬁcation problem becomes a linear classiﬁcation problem and simpliﬁes the problem.
Revised: In space F c , the nonlinear classification problem becomes a linear classification problem and simplifies the problem.

Original: The inner product of φ ( x i ) T · φ ( x j ) is often seen as distances between x i and x j in the new space F c .
Revised: The inner product of ϕ ( x i ) T ϕ ( x j ) is often seen as distances between x i and x j in the new space F c .

Original: Quantum Kernel.
Revised: Quantum Kernel.

Original: The quantum kernel method is a kernel method designed to run on quantum computers based on quantum computing properties.
Revised: The quantum kernel method is a kernel method designed to run on quantum computers based on quantum computing properties.

Original: Its principle is almost identical to the classical kernel method except that it maps the data point from the original input space O to the quantum
Revised: Its principle is almost identical to the classical kernel method except that it maps the data point from the original input space O to the quantum

Original: Hilbert space F q , i.e., f q : O → F q .
Revised: Hilbert space F q , i.e., f q : O → F q .

Original: The key to the quantum kernel methods is the quantum mapping function F q .
Revised: The key to the quantum kernel methods is the quantum mapping function f q .

Original: We can view the feature mapping function F q as the key to deﬁne the quantum kernel methods.
Revised: We can view the feature mapping function f q as the key to define the quantum kernel methods.

Original: Thus, if the quantum kernel approach is superior, the superiority lies in the quantum mapping function.
Revised: Thus, if the quantum kernel approach is superior, the superiority lies in the quantum mapping function.

Original: The mechanism of quantum kernel methods is basically the same as that of classical kernel methods.
Revised: The mechanism of quantum kernel methods is basically the same as that of classical kernel methods.

Original: The data point x i is mapped from the original input space O to the quantum state space F q , i.e., f q : x i →(cid:104) φ ( x i ) | φ ( x j ) (cid:105) , where the |·(cid:105) denotes a vector and physically it represents a state of some quantum system.
Revised: The data point x i is mapped from the original input space O to the quantum state space F q , i.e., f q : x i → | ϕ ( x i ) ⟩ , where the |·⟩ denotes a vector and physically it represents a state of some quantum system.

Original: The (cid:104)·| is the Hermitian Conjugate of the vector |·(cid:105) .
Revised: The ⟨·| is the Hermitian Conjugate of the vector |·⟩ .

Original: In practice, the feature map is realized by acting the circuit U ( x  ) on the initial quantum state | 0 n (cid:105) , i.e.,| [Equation] [Equation] (cid:105)
Revised: In practice, the feature map is realized by acting the circuit U ( x i ) on the initial quantum state | 0 n ⟩ , i.e.,

Original: [Equation]
Revised: [Equation]

Original: (cid:105) . [Equation] The quantum kernel can be obtained by running the circuit U † ( x j ) U ( x i ) on the initial quantum state | 0 n (cid:105) , where U † is the Hermitian conjugate of U .
Revised: The quantum kernel can be obtained by running the circuit U † ( x j ) U ( x i ) on the initial quantum state | 0 n ⟩ , where U † is the Hermitian conjugate of U .

Original: Then estimate | [-(cid:104) 
-] n | U † ( x j ) U ( x i ) | 0 n (cid:105) | 2 by counting the frequency of the [-
-] n output as a value of k ( x i , x j ) .
Revised: Then estimate | [-(cid:104) 
-]⟨ 0 n | U † ( x j ) U ( x i ) | 0 n ⟩ | 2 by counting the frequency of the [-
-]0 n output as a value of k ( x i , x j ) .

Original: Fig.1A shows the process ﬂow of the quantum kernel method and classical kernel method.
Revised: Fig.1A shows the process flow of the quantum kernel method and classical kernel method.

Original: Quantum Kernel Method Based On Pauli Feature Map.
Revised: Quantum Kernel Method Based On Pauli Feature Map.

Original: Following the IBM quantum computing platform, we take two qubits as an example.
Revised: Following the IBM quantum computing platform, we take two qubits as an example.

Original: The general expression of a 2-qubit quantum kernel is
Revised: The general expression of a 2-qubit quantum kernel is

Original: [Equation]
Revised: [Equation]

Original: By the deﬁnition of Havl´ıˇcek et al.
Revised: By the definition of Havl´ıˇcek et al.

Original: , the quantum circuit U is realized by U ( x i ) =
Revised: , the quantum circuit U is realized by U ( ⃗x ) =

Original: U φ ( x ) H (cid:78) 2 U φ ( x ) H (cid:78) 2 , where the (cid:78) is the Kronecker Product of two matrices.
Revised: U ϕ ( ⃗x ) H (cid:78) 2 U ϕ ( ⃗x ) H (cid:78) 2 , where the (cid:78) is the Kronecker Product of two matrices.

Original: For the Secondorder Pauli-Z evolution circuit, U φ ( x ) =
Revised: For the Secondorder Pauli-Z evolution circuit, U ϕ ( ⃗x ) =

Original: exp ( i ( x 0 Z 0 + x 1 Z 1 + ( π − x 0 )( π − x 1 ) Z 0 Z 1 )) , where
Revised: exp ( i ( x 0 Z 0 + x 1 Z 1 + ( π − x 0 )( π − x 1 ) Z 0 Z 1 )) , where

Original: Z 0 , Z 1 are quantum Z-Gate s, and H is the quantum Hadamard-Gate .
Revised: Z 0 , Z 1 are quantum Z-Gate s, and H is the quantum Hadamard-Gate .

Original: We denote the corresponding quantum kernel method as the Z-ZZ quantum kernel method , and the corresponding feature map is showed in Fig.1B(3).
Revised: We denote the corresponding quantum kernel method as the Z-ZZ quantum kernel method , and the corresponding feature map is showed in Fig.1B(3).

Original: The feature maps of the Z quantum kernel method and the ZZ quantum kernel method are shown in Fig.1B(1) and Fig.1B(2), respectively.
Revised: The feature maps of the Z quantum kernel method and the ZZ quantum kernel method are shown in Fig.1B(1) and Fig.1B(2), respectively.

Original: In this paper, all references to quantum kernel methods refer to the Z-ZZ quantum kernel method unless otherwise stated.
Revised: In this paper, all references to quantum kernel methods refer to the Z-ZZ quantum kernel method unless otherwise stated.

Original: In section 4.3, we compare these three quantum kernel methods.
Revised: In section 4.3, we compare these three quantum kernel methods.

Original: Support Vector Machine.
Revised: Support Vector Machine.

Original: Support Vector Machine is akind of maximal margin classiﬁer.
Revised: Support Vector Machine is a maximal margin classifier.

Original: It is seen as one of the most successful cases of the kernel approach.
Revised: It is seen as one of the most successful cases of the kernel approach.

Original: SVMs are dedicated to ﬁnding a hyper-plane that separates different classes and makes the margin as large as possible.
Revised: SVMs are dedicated to finding a hyper-plane that separates different classes and makes the margin as large as possible.

Original: In general cases, i.e., nonlinear cases, the data is mapped non-linearly to high dimensional Hilbert space by a mapping function.
Revised: In general cases, i.e., nonlinear cases, the data is mapped non-linearly to high dimensional Hilbert space by a mapping function.

Original: Then the distance between two data points can be calculated using the kernel function.
Revised: Then the distance between two data points can be calculated using the kernel function.

Original: Suppose we have a set of data points D
Revised: Suppose we have a set of data points D

Original: = { ( x 1 , y 1 ) , · · · , ( x n , y n ) } ,
Revised: = { ( x 1 , y 1 ) , · · · , ( x n , y n ) } ,

Original: where x i ∈ R d and y i ∈ {− 1 , +1 } .
Revised: where x i ∈ R d and y i ∈ {− 1 , +1 } .

Original: According to Burges (1998b), the nonlinear SVM can be modiﬁed and expressed by an optimization problem as maximize:
Revised: According to Burges (1998b), the nonlinear SVM can be modified and expressed by an optimization problem as maximize:

Original: [Equation]
Revised: [Equation]

Original: s.t. 

Revised: s.t.

Original: ≤ α i ≤ C and (cid:80) ni α
Revised: ≤ α i ≤ C and (cid:80) ni α

Original: i y i = 0 , where i = 1 , · · · , n .
Revised: i y i = 0 , where i = 1 , · · · , n .

Original: The decision function is
Revised: The decision function is

Original: [Equation]
Revised: [Equation]

Original: where s i are the support vectors and N s is the number of support vectors.
Revised: where s i are the support vectors and N s is the number of support vectors.

Original: The SVM-based quantum kernel method is very similar in principle to the traditional SVM, except that the computation of the kernel is performed on a quantum computer.
Revised: The SVM-based quantum kernel method is very similar in principle to the traditional SVM, except that the computation of the kernel is performed on a quantum computer.

Original: Havl ´ ı ˇ cek et al.
Revised: Havl ´ ı ˇ cek et al.

Original: (2019) refer to it as quantum kernel estimation.
Revised: (2019) refer to it as quantum kernel estimation.

Original: We show the speciﬁc estimation method in Section 4.2.
Revised: We show the specific estimation method in Section 4.2.

Original:  M ETHODS 3. Q UANTUM KERNEL IS A PROBABILISTIC KERNEL
Revised:  M ETHODS 3. Q UANTUM KERNEL IS A PROBABILISTIC KERNEL

Original: The kernel function is an equation for measuring similarity.
Revised: The kernel function is an equation for measuring similarity.

Original: In vector space, we estimate the similarity of vectors utilizing vector kernel functions.
Revised: In vector space, we estimate the similarity of vectors utilizing vector kernel functions.

Original: Similarly, graph kernels describe the similarity of two graphs, and tree kernels compare the similarity of trees, which are often used in natural language processing.
Revised: Similarly, graph kernels describe the similarity of two graphs, and tree kernels compare the similarity of trees, which are often used in natural language processing.

Original: A question arises as to how to deﬁne kernels or what kind of kernel functions are valid kernels.
Revised: A question arises as to how to define kernels or what kind of kernel functions can be effectively used or not.

Original: There is no answer to this question.
Revised: There is no answer to this question.

Original: Mercer (1909) argues that a valid kernel function needs to satisfy symmetry and positive deﬁniteness.
Revised: Mercer (1909) argues that a valid kernel function needs to satisfy symmetry and positive definiteness.

Original: 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.
Revised: 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).

Original: Mix kernel function proposed by Smits & Jordaan (2002) tries to achieve better properties when combining different kernel functions.
Revised: Mix kernel function proposed by Smits & Jordaan (2002) tries to achieve better properties when combining different kernel functions.

Original: The mechanism of the quantum kernel function is similar to some traditional kernel functions.
Revised: The mechanism of the quantum kernel function is similar to some traditional kernel functions.

Original: It follows the Mercer theorem and is a practical kernel function that expands the family of kernel functions.
Revised: It follows the Mercer theorem and is a practical kernel function that expands the family of kernel functions.

Original: However, its implementation is based on quantum superposition states and entanglement.
Revised: However, its implementation is based on quantum superposition states and entanglement.

Original: Since the values obtained are based on probabilities in a statistical sense, we call it a probabilistic kernel function.
Revised: Since the values obtained are based on probabilities in a statistical sense, we call it a probabilistic kernel function.

Original: For correspondence, we call the classical kernel function a deterministic kernel function.
Revised: For correspondence, we call the classical kernel function a deterministic kernel function.

Original: We try to clearly show the relationship and difference between different kernel functions by a diagram (Fig.2).
Revised: We try to clearly show the relationship and difference between different kernel functions by a diagram (Fig.2).

Original: It is worth noting that there are no guarantees for one kernel to work better than the other in all cases, according to the No Free Lunch Theorem (Wolpert & Macready (1997)).
Revised: It is worth noting that there are no guarantees for one kernel to work better than the other in all cases, according to the No Free Lunch Theorem (Wolpert & Macready (1997)).

Original: Choosing different kernel functions in various subjects will achieve better results.
Revised: Choosing different kernel functions in various subjects will achieve better results.

Original: 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.
Revised: 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.

Original: 3. T HE PROPOSED PATTERNS AND CRITERIA FOR JUDGING QKM
Revised: 3. T HE PROPOSED PATTERNS AND CRITERIA FOR JUDGING QKM

Original: 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.
Revised: 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.

Revised: The

Revised: Mersenne Twister is a pseudo random number generator which was first proposed by Matsumoto & Nishimura (1998).

Revised: The Mersenne Twister is used as default pseudo random number generator by many software, such as Python, R, and PHP.

Revised: The Mersenne Twister random distribution is a distribution that be generated by the Mersenne Twister method.

Original: Theorem 3.1 (Advantages of QKM for random distributions)
Revised: Theorem 3.1 (Advantages of QKM for random distributions)

Original: In two dimensions, the Z-ZZ quantum kernel method has better learning ability than classical kernel methods for randomly distributed data patterns based on Mersenne Twister.
Revised: In two dimensions, the Z-ZZ quantum kernel method has better learning ability than classical kernel methods for randomly distributed data patterns based on Mersenne Twister.

Original: Proof.
Revised: Proof.

Original: We assume that the Z-ZZ feature map can effectively simulate the efﬁcacy of the feature map proposed by Liu et al.
Revised: We assume that the Z-ZZ feature map can effectively simulate the efficacy of the feature map proposed by Liu et al.

Original: By Matsumoto & Nishimura (1998), for a k-bits binary number, the
Revised: By Matsumoto & Nishimura (1998), for a k-bits binary number, the

Original: Mersenne Twister algorithm generates discrete uniformly distributed random numbers in the range [0 , 2 k − 1] .
Revised: Mersenne Twister algorithm generates discrete uniformly distributed random numbers in the range [0 , 2 k − 1] .

Original: Solving this problem is a discrete logarithm problem (DLP).
Revised: Solving this problem is a discrete logarithm problem (DLP).

Original: For DLP, Liu et al.
Revised: For DLP, Liu et al.

Original: (2021) say no efﬁcient classical algorithm can achieve inverse-polynomially better accuracy than random guessing.
Revised: (2021) say no efficient classical algorithm can achieve inverse-polynomially better accuracy than random guessing.

Original: Therefore, the Z-ZZ quantum kernel method can demonstrate quantum superiority over the classical kernel methods for Mersenne Twister-based randomly distributed data patterns.
Revised: Therefore, the Z-ZZ quantum kernel method can demonstrate quantum superiority over the classical kernel methods for Mersenne Twister-based randomly distributed data patterns.

Original: Second, we demonstrate that whether quantum kernel methods have quantum advantages is related to the δ of the data set, where δ will be deﬁned in Equ.(7).
Revised: Second, we demonstrate that whether quantum kernel methods have quantum advantages is related to the δ of the data set, where δ will be defined in Equ.(7).

Original: Because any classiﬁcation problem can be transformed into a binary situation, we all base our study on the binary classiﬁcation problem.
Revised: Because any classification problem can be transformed into a binary situation, we all base our study on the binary classification problem.

Original: If the distance between classes is large enough, i.e., the simple case, the quantum kernel methods are not as good as the classical kernel methods.
Revised: If the distance between classes is large enough, i.e., the simple case, the quantum kernel methods are not as good as the classical kernel methods.

Original: Based on fourteen Adhoc-Modify datasets (Section 4.1 Qiskit package datasets (6)), we measure the δ of each dataset and draw a graph with δ as the horizontal coordinate.
Revised: Based on fourteen Adhoc-Modify datasets (Section 4.1 Qiskit package datasets (6)), we measure the δ of each dataset and draw a graph with δ as the horizontal coordinate.

Original: Fig.3 shows that when δ increases to a certain level, the advantage of the quantum kernel method disappears.
Revised: Fig.3 shows that when δ increases to a certain level, the advantage of the quantum kernel method disappears.

Original: To measure the degree of separation of two classes C l and C m , we ﬁrst deﬁne the inter-class distance
Revised: To measure the degree of separation of two classes C l and C m , we first define the inter-class distance

Original: [Equation]
Revised: [Equation]

Original: where N l and N m are the sizes of  class C l and C m , respectively.
Revised: where N l and N m are the number of observations belonging to class C l and C m , respectively.

Original: The x ( l ) n is the n-th sample in class C l and d ( , ) is the euclidean distance between two samples, i.e., d ( (cid:126)x, (cid:126)y [Equation]
Revised: The x ( l ) n is the n-th sample in class C l and d ( , ) is the euclidean distance between two samples, i.e., d ( ⃗x, ⃗y ) =

Original: = (cid:112)(cid:80) 
Revised: (cid:112)(cid:80) ni

Original: [Equation]
Revised: =1 ( x i − y i ) 2 .

Original: For the sake of uniformity, we also need to deﬁne the intra-class distance for class C, i.e.,
Revised: For the sake of uniformity, we also need to define the intra-class distance for class C, i.e.,

Original: [Equation]
Revised: [Equation]

Original: where the N is thesize of class C , and x n is the n-th sample in class C .
Revised: where the N is the of class C , and x n is the n-th sample in class C .

Original: We propose a criterion to evaluate whether quantum kernel methods will be better than classical kernel methods.
Revised: We propose a criterion to evaluate whether quantum kernel methods will be better than classical kernel methods.

Original: In a binary classiﬁcation problem, we deﬁne the degree of integration of C l and C m as δ lm .
Revised: In a binary classification problem, we define the degree of integration of C l and C m as δ lm .

Original: [Equation]
Revised: [Equation]

Original: Theorem 3.2 illustrates that the larger the δ , the greater the separation of the two classes.
Revised: Theorem 3.2 illustrates that the larger the δ , the greater the separation of the two classes.

Original: When the delta is large enough, the quantum kernel function loses its quantum advantage.
Revised: When the delta is large enough, the quantum kernel function loses its quantum advantage.

Original: Theorem 3.2 (Deﬁciencies of the QKM)
Revised: Theorem 3.2 (Deficiencies of the QKM)

Original: In the case of a balanced number of the two classes,  the quantum kernel method will not be better than the classical kernel method in handling classiﬁcation problems when δ > δ 0 .
Revised: In the case of a balanced number of the two classes, there exists δ 0 such that the quantum kernel method will not be better than the classical kernel method in handling classification problems when δ > δ 0 .

Original: In practice, δ 0 is usually taken as 0.6.
Revised: In practice, δ 0 is usually taken as 0.6.

Original: Proof.
Revised: Proof.

Original: Suppose our measurement independent identical distribution M 1 , M 2 · · · M R which have expectation E ( M ) = µ and variance D ( M )
Revised: Suppose our measurement independent identical distribution M 1 , M 2 · · · M R which have expectation E ( M ) = µ and variance D ( M )

Original: =
Revised: =

Original: σ 2 , where M is the random variable, R is the number of measurement shots.
Revised: σ 2 , where M is the random variable, R is the number of measurement shots.

Original: By the Central Limit Theorem, for any m , the distribution function
Revised: By the Central Limit Theorem, for any m , the distribution function

Original: [Equation]
Revised: [Equation]

Original: m
Revised: 1 √

Original: [Equation]
Revised: [Equation]

Original: satisﬁes
Revised: π (cid:82) m

Original: [Equation]
Revised: [Equation]

Original: 1 √ 2 π (cid:82) m
Revised: e −

Original: [Equation]
Revised: [Equation]

Original: 2 2 dt .
Revised: 2 2 dt .

Original: This shows that when R is large enough, the random variable Y R =
Revised: This shows that when R is large enough, the random variable Y R =

Original: (cid:80) Ri =1 M i −
Revised: (cid:80) Ri [Equation]

Original: RE √ RD obeys normal distribution N (0 , 1) .
Revised: RE [Equation] RD obeys normal distribution N (0 , 1) .

Original: So, (cid:80) Ri =
Revised: So, (cid:80) R i =

Original: 1 M i = √
Revised: 1 M i = √

Original: RDY R + RE = √ RσY R + Rµ
Revised: RDY R + RE = √ RσY R + Rµ

Original: obeys normal distribution
Revised: obeys normal distribution

Original: N ( Rµ, Rσ 2 ) .
Revised: N ( Rµ, Rσ 2 ) .

Original: So, even fully error corrected, the quantum methods still have the noise caused by measurement.
Revised: However, even fully error corrected, the quantum methods still have the noise caused by measurement.

Original: Two types of data are linearly separable or almost linearly separablewhen
Revised: Two types of data are linearly separable or almost linearly separable

Original:  D ( C l , C m ) (cid:29) D ( C l ) + D ( C m ) , i.e., δ (cid:29) δ 0 .
Revised: when D ( C l , C m ) ≫ D ( C l ) + D ( C m ) , i.e., δ ≫ δ 0 .

Original: It is easy for a deterministic kernel method to ﬁnd a boundary line with an inﬁnitely small error.
Revised: It is easy for a deterministic kernel method to find a boundary line with an infinitely small error.

Original: So, when δ > δ 0 , the quantum kernel method is not better than the classical kernel method in handling classiﬁcation problems.
Revised: So, when δ > δ 0 , the quantum kernel method is not better than the classical kernel method in handling classification problems.

Original: In fact, as a probabilistic kernel method, a quantum kernel method has more error than a deterministic kernel method caused by the measurement process.
Revised: In fact, as a probabilistic kernel method, a quantum kernel method has more error than a deterministic kernel method caused by the measurement process.

Original: It is especially evident in some simple cases because we assume that deterministic kernel methods must ﬁnd a boundary line.
Revised: It is especially evident in some simple cases because we assume that deterministic kernel methods must find a boundary line.

Original: However, errors in probabilistic methods are inevitable.
Revised: However, errors in probabilistic methods are inevitable.

Original:  E XPERIMENTS 4. D ATA PREPARATION
Revised:  E XPERIMENTS 4. D ATA PREPARATION

Original: Qiskit package datasets.
Revised: Qiskit package datasets.

Original: Five datasets from qiskit.ml.datasets were used in the experiments.
Revised: Five datasets from qiskit.ml.datasets were used in the experiments.

Original: Digits datasets.
Revised: Digits datasets.

Original: This dataset consists of 1797 8 × 8 images, and each image is a handwritten number
Revised: This dataset consists of 1797 8 × 8 images, and each image is a handwritten number

Original: 0 ∼ 9.
Revised: 0 ∼ 9.

Original: We perform binary classiﬁcation for every two digits.
Revised: We perform binary classification for every two digits.

Original: Thus, 45 small datasets were generated.
Revised: Thus, 45 small datasets were generated.

Original: (2) Breast cancer dataset.
Revised: (2) Breast cancer dataset.

Original: (3) Iris dataset.
Revised: (3) Iris dataset.

Original: (4) Wine dataset.
Revised: (4) Wine dataset.

Original: (5) Adhoc dataset.
Revised: (5) Adhoc dataset.

Original: Note that (1) 4) are copies from UCI ML Hand-written Digits Dataset, UCI ML Breast Cancer Wisconsin (Diagnostic) 
Revised: Note that (1) ∼ (4) are copies from UCI ML Hand-written Digits Dataset, UCI ML Breast Cancer Wisconsin (Diagnostic) Dataset, UCI ML

Original: , UCI ML
Revised: Plants and UCI ML

Original: Recognition respectively.
Revised: Recognition respectively.

Original: For
Revised: For

Original:  Adhoc
Revised: 5) Adhoc

Original: , we separate the two classes and translate all the data points in one class by the same distance, leaving the other class unchanged.
Revised: , we separate the two classes and translate all the data points in one class by the same distance, leaving the other class unchanged.

Original: In this way, we get 14 datasets, denoted as (6)
Revised: In this way, we get 14 datasets, denoted as (6)

Original: Adhoc-Modify datasets.
Revised: Adhoc-Modify datasets.

Original: Kaggle datasets.
Revised: Kaggle datasets.

Original: Five datasets from Kaggle were used in the experiments.
Revised: Five datasets from Kaggle were used in the experiments.

Original: (7) Email spam dataset (Balaka Biswas), (8) Heart disease dataset (Zeeshan Mulla).
Revised: (7) Email spam dataset (Balaka Biswas), (8) Heart disease dataset (Zeeshan Mulla).

Original: (9) Giants and dwarfs dataset (Vinesmsuic).
Revised: (9) Giants and dwarfs dataset (Vinesmsuic).

Original: (10) Star type dataset (Baris Dincer).
Revised: (10) Star type dataset (Baris Dincer).

Original: (11) Drug dataset (Pratham Tripathi).
Revised: (11) Drug dataset (Pratham Tripathi).

Original: Geometric toy datasets.
Revised: Geometric toy datasets.

Original: 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.
Revised: 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.

Original: (12) Geometric non-random datasets.
Revised: (12) Geometric non-random datasets.

Original: We designed 2 ∼ 5 layers of concentric circles and four squares with different mixing degrees.
Revised: We designed 2 ∼ 5 layers of concentric circles and four squares with different mixing degrees.

Original: Fig.4A shows the four circular datasets and four square datasets.
Revised: Fig.4A shows the four circular datasets and four square datasets.

Original: (13) Geometric random datasets.
Revised: (13) Geometric random datasets.

Original: Based on the Mersenne Twister random distribution, we designed datasets with random distributions in three geometric patterns: Circular, square, and equilateral triangle patterns.
Revised: Based on the Mersenne Twister random distribution, we designed datasets with random distributions in three geometric patterns: Circular, square, and equilateral triangle patterns.

Original: The three graphs on the right side of Fig.4B show a sample of these three datasets, respectively.
Revised: The three graphs on the right side of Fig.4B show a sample of these three datasets, respectively.

Original: 4. P ROCESS OF TRAINING AND TESTING OF QUANTUM KERNEL METHODS
Revised: 4. P ROCESS OF TRAINING AND TESTING OF QUANTUM KERNEL METHODS

Original: We brieﬂy introduce the training process of the quantum kernel method here, following the idea of Liu et al.
Revised: We briefly introduce the training process of the quantum kernel method here, following the idea of Liu et al.

Original: Suppose a training dataset D train = { ( x 1 , y 1 ) , ..., ( x n , y n ) } .
Revised: Suppose a training dataset D train = { ( x 1 , y 1 ) , ..., ( x n , y n ) } .

Original: Since the only difference of the quantum kernel method in an SVM from the classical SVM is kernel calculation, here we only show the kernel calculation process in a quantum computer.
Revised: Since the only difference of the quantum kernel method in an SVM from the classical SVM is kernel calculation, here we only show the kernel calculation process in a quantum computer.

Original: Training:
Revised: Training:

Original: For each pair of data points x i and x j ( i (cid:54) = j ) in D train , we apply U † ( x i ) U ( x j ) on the input (cid:12) (cid:12) 

Revised: For each pair of data points x i and x j ( i ̸ = j ) in D train , we apply U † ( x i ) U ( x j ) on the input (cid:12) (cid:12)

Original: (cid:78) 2 (cid:11) .
Revised: (cid:78) 2 (cid:11) .

Original: After R repetitive runs, we record the number of times the output results in (cid:12) (cid:12) 

Revised: After R repetitive runs, we record the number of times the output results in (cid:12) (cid:12)

Original: (cid:78) 2 (cid:11) , denoted as R 0 and k ( x i , x j ) =
Revised: (cid:78) 2 (cid:11) , denoted as R 0 and k ( x i , x j ) =

Original: R 0 2 R , where i (cid:54) =
Revised: R 0 2 R , where i ̸ =

Original: j and k ( x i , x i ) = 1 .
Revised: j and k ( x i , x i ) = 1 .

Original: In the end, we apply Equ.(3) directly.
Revised: In the end, we apply Equ.(3) directly.

Original: Testing: When there comes a new sample x new , for each data point x i in D train , we apply
Revised: Testing: When there comes a new sample x new , for each data point x i in D train , we apply

Original: U † ( x i ) U
Revised: U † ( x i ) U

Original: ( x new ) on the input (cid:12)(cid:12) 

Revised: ( x new ) on the input (cid:12)(cid:12)

Original: (cid:78) 2 (cid:11) .
Revised: (cid:78) 2 (cid:11) .

Original: After R repetitive runs, we record the number of times the output results in (cid:12)(cid:12) 

Revised: After R repetitive runs, we record the number of times the output results in (cid:12)(cid:12)

Original: (cid:78) 2 (cid:11) , denoted as R 1 and k ( x new , x i ) =
Revised: (cid:78) 2 (cid:11) , denoted as R 1 and k ( x new , x i ) =

Original: R 1 +2 R 2 R .
Revised: R 1 +2 R 2 R .

Original: In the end, we apply Equ.(4) directly.
Revised: In the end, we apply Equ.(4) directly.

Original: 4. E XPERIMENTS R ESULTS
Revised: 4. E XPERIMENTS R ESULTS

Original: The quantum kernel method is good at complex data patterns.
Revised: The quantum kernel method is good at complex data patterns.

Original: In this part, we explore the ability of quantum kernel methods to solve classiﬁcation problems under challenging patterns.
Revised: In this part, we explore the ability of quantum kernel methods to solve classification problems under challenging patterns.

Original: Firstly, we increase the learning difﬁculty by increasing the complexity of the geometric patterns and get four patterns, i.e., P1 ∼ P4 in Fig.4A. Experiments show that from P1 to P4, the classical kernel approach, which initially performs better, gradually loses its superiority.
Revised: Firstly, we increase the learning difficulty by increasing the complexity of the geometric patterns and get four patterns, i.e., P1 ∼ P4 in Fig.4A. Experiments show that from P1 to P4, the classical kernel approach, which initially performs better, gradually loses its superiority.

Original: In contrast, the superiority of the quantum kernel approach begins to emerge.
Revised: In contrast, the superiority of the quantum kernel approach begins to emerge.

Original: To further increase the learning difﬁculty, we hypothesized the existence of a pattern in which the data are randomly distributed over the geometry according to the Mersenne Twister.
Revised: To further increase the learning difficulty, we hypothesized the existence of a pattern in which the data are randomly distributed over the geometry according to the Mersenne Twister.

Original: The right three graphs in Fig.4B show this kind of data pattern.
Revised: The right three graphs in Fig.4B show this kind of data pattern.

Original: To explore the learning ability of quantum kernel 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.
Revised: To explore the learning ability of quantum kernel 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.

Original: Considering the randomness of the random distribution, we take the average result of 50 times as the result of each experiment.
Revised: Considering the randomness of the random distribution, we take the average result of 50 times as the result of each experiment.

Original: The experiments demonstrate that the quantum kernel method has a more robust learning capability than the classical kernel method in this complex pattern.
Revised: The experiments demonstrate that the quantum kernel method has a more robust learning capability than the classical kernel method in this complex pattern.

Original: The quantum kernel method fails at large values of δ .
Revised: The quantum kernel method fails at large values of δ .

Original: In this part, we will show that our criteria can effectively determine whether quantum kernel methods can demonstrate advantages over classical kernel methods or not.
Revised: In this part, we will show that our criteria can effectively determine whether quantum kernel methods can demonstrate advantages over classical kernel methods or not.

Original: First, taking the Digits dataset as an example, we perform binary classiﬁcation for every two digits.
Revised: First, taking the Digits dataset as an example, we perform binary classification for every two digits.

Original: Thus, we generate 45 small datasets.
Revised: Thus, we generate 45 small datasets.

Original: We then use PCA to reduce the dimensionality of all data to 2 dimensions, as shown in Fig.5.
Revised: We then use PCA to reduce the dimensionality of all data to 2 dimensions, as shown in Fig.5.

Original: We recorded each small dataset’s training and test accuracy after applying four classical kernel methods and a quantum kernel method.
Revised: We recorded each small dataset’s training and test accuracy after applying four classical kernel methods and a quantum kernel method.

Original: The bottom left panel in Fig.5 shows the prediction accuracy of each method.
Revised: The bottom left panel in Fig.5 shows the prediction accuracy of each method.

Original: According to the experimental results, we attach a corner mark to each data set.
Revised: According to the experimental results, we attach a corner mark to each data set.

Original: 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 it does not.
Revised: 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 it does not.

Original: The blue corner markers indicate that the quantum kernel method is indistinguishable from the classical kernel method.
Revised: The blue corner markers indicate that the quantum kernel method is indistinguishable from the classical kernel method.

Original: Intuitively, the quantum kernel approach for linearly divisible data sets does not show superiority.
Revised: Intuitively, the quantum kernel approach for linearly divisible data sets does not show superiority.

Original: However, for complex models with relatively high fusion, the quantum kernel approach has the potential to be superior.
Revised: However, for complex models with relatively high fusion, the quantum kernel approach has the potential to be superior.

Original: To validate our criteria δ , we prepared 81 datasets: 45 small datasets from Digits, 14 datasets from
Revised: To validate our criteria δ , we prepared 81 datasets: 45 small datasets from Digits, 14 datasets from

Original: Adhoc-Modify, Breast cancer, Iris, Wine, Email spam, Heart disease, Giants and dwarfs, Star Type,
Revised: Adhoc-Modify, Breast cancer, Iris, Wine, Email spam, Heart disease, Giants and dwarfs, Star Type,

Original: Drug, 4 datasets from Geometric non-random, i.e., the top-left, bottom-left, top-right and bottomright four datasets in Fig.4A, 10 datasets from Geometric random including ﬁve circles of size thirty and ﬁve squares of size
Revised: Drug, 4 datasets from Geometric non-random, i.e., the top-left, bottom-left, top-right and bottomright four datasets in Fig.4A, 10 datasets from Geometric random including five circles of size thirty and five squares of size

Original: forty.
Revised: forty.

Original: The difference in prediction accuracy between classical kernel methods and quantum kernel methods on each dataset is calculated.
Revised: The difference in prediction accuracy between classical kernel methods and quantum kernel methods on each dataset is calculated.

Original: It is shown by the relevant vertical lines in Fig.6.
Revised: It is shown by the relevant vertical lines in Fig.6.

Original: The magnitude of δ on each dataset is shown with an asterisk in Fig.6.
Revised: The magnitude of δ on each dataset is shown with an asterisk in Fig.6.

Original: According to the experiments, when the quantum kernel method has an advantage, the asterisks appear below the 0.
Revised: According to the experiments, when the quantum kernel method has an advantage, the asterisks appear below the 0.

Original: level line all the time.
Revised: level line all the time.

Original: We can get the conclusion that (i) the availability of quantum superiority correlates with δ and (ii) the quantum kernel method can be better than the classical kernel method when δ < 0 .
Revised: We can get the conclusion that (i) the availability of quantum superiority correlates with δ and (ii) the quantum kernel method can be better than the classical kernel method when δ < 0 .

Original: 6 , although it is not a sufﬁcient condition.
Revised: 6 , although it is not a sufficient condition.

Original: Simple quantum kernel methods do not offer quantum advantages.
Revised: Simple quantum kernel methods do not offer quantum advantages.

Original: 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.
Revised: 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.

Original: So, quantum kernel methods are preferred over classical kernel methods for classiﬁcation problems with complex patterns.
Revised: So, quantum kernel methods are preferred over classical kernel methods for classification problems with complex patterns.

Original: However, a simpler quantum kernel function can be simulated classically.
Revised: However, a simpler quantum kernel function can be simulated classically.

Original: For a simple-kernel-based quantum kernel method, it no more has the superiority.
Revised: For a simple-kernel-based quantum kernel method, it no more has the superiority.

Original: Its ability to handle complex pattern problems is signiﬁcantly reduced and even inferior to classical kernel methods.
Revised: Its ability to handle complex pattern problems is significantly reduced and even inferior to classical kernel methods.

Original: Interestingly, sometimes this simple quantum kernel function works well with simple pattern problems.
Revised: Interestingly, sometimes this simple quantum kernel function works well with simple pattern problems.

Original: We illustrate the above opinion experimentally.
Revised: We illustrate the above opinion experimentally.

Original: In this part, we 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 [Table] 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.
Revised: In this part, we 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- 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.

Original: By comparing the performance of the best classical kernel method and the three quantum kernel methods on 68 training datasets, we 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.
Revised: By comparing the performance of the best classical kernel method and the three quantum kernel methods on 68 training datasets, we 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.

Original: However, when quantum superiority exists, it is often achieved by Z-ZZ kernel methods.
Revised: However, when quantum superiority exists, it is often achieved by Z-ZZ kernel methods.

Original: The experiments show that the quantum kernel method superiority on only one dataset is achieved by the Z quantum kernel method.
Revised: The experiments show that the quantum kernel method superiority on only one dataset is achieved by the Z quantum kernel method.

Original: Correspondingly, are used to represent the value of δ when the Z-ZZ quantum kernel method and the classical kernel method are best, respectively.
Revised: to represent the value of δ when the Z-ZZ quantum kernel method and the classical kernel method are best, respectively.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: The classiﬁcation problem is one of the most common problems in machine learning, and both classical kernel methods and quantum kernel methods can effectively handle the classiﬁcation problem.
Revised: The classification problem is one of the most common problems in machine learning, and both classical kernel methods and quantum kernel methods can effectively handle the classification problem.

Original: However, it is difﬁcult to determine what situation each of them is suitable for.
Revised: However, it is difficult to determine what situation each of them is suitable for.

Original: This paper explores when quantum kernel methods can take quantum advantage by comparing different kernel functions.
Revised: This paper explores when quantum kernel methods can take quantum advantage by comparing different kernel functions.

Original: Moreover, a judgment criterion is proposed to help one decide when quantum kernel methods can achieve better results than classical kernel methods.
Revised: Moreover, a judgment criterion is proposed to help one decide when quantum kernel methods can achieve better results than classical kernel methods.

Original: Experiments show that our method is effective.
Revised: Experiments show that our method is effective.