Where can quantum kernel methods make a big difference?

1 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 Grover ( 1996 ) and Shor ( 1994 ) . 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 an approach to train QNNs , but the complexity of its method is exponential . Subsequently , Lloyd et al . ( 2013 ) , Blacoe et al . ( 2013 ) , and Rebentrost et al . ( 2014 ) tried to introduce quantum computing into clustering , distributed semantics , and SVMs , respectively , but their approaches were too limited to theory . As researchers introduce quantum into various machine learning algorithms , Schuld et al . ( 2015 ) , Biamonte et al . ( 2017 ) , Kopczyk ( 2018 ) , Ciliberto et al . ( 2018 ) 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 , Arute et al . ( 2019 ) announced the achievement of quantum hegemony , a milestone event in quantum computing . 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 . 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 in 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 ( 1998a ) , Muller et al . ( 2001 ) , Scholkopf ( 2001 ) and Hofmann et al . ( 2008 ) . Inspired by the classical kernel approach , Rebentrost et al . ( 2014 ) proposed a quantum kernel approach based on SVM , but only theoretically feasible . It was not until Schuld & Killoran ( 2019 ) , and Havlı́ček et al . ( 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 . Later , Blank et al . ( 2020 ) , Wang et al . ( 2021 ) , Kusumoto et al . ( 2021 ) , and Peters et al . ( 2021 ) 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 ( 2021 ) 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 δ 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 a distance in the data space , i.e. , the distance between digital representations of data points . Specifically , the kernel method uses a feature mapping function fc to map data from a point in the original input space O to a higher-dimensional Hilbert feature space Fc , i.e. , fc : O → Fc , making separability 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 a large number of calculations . A nonlinear classification problem is one of the classical machine learning problems , and kernel methods can effectively handle such problems . Recall that in classical kernel methods , such as support vector machine , a data point xi ∈ Rn is mapped into a potentially much higher dimensional feature space Fc via a nonlinear mapping function fc , where xi is represented by ϕ ( xi ) , i.e. , fc : xi → ϕ ( xi ) . In space Fc , the nonlinear classification problem becomes a linear classification problem and simplifies the problem . The inner product of ϕ ( xi ) Tϕ ( xj ) is often seen as distances between xi and xj in the new space Fc . Quantum Kernel . The quantum kernel method is a 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 it maps the data point from the original input space O to the quantum Hilbert space Fq , i.e. , fq : O → Fq . The key to the quantum kernel methods is the quantum mapping function fq . We can view the feature mapping function fq 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 xi is mapped from the original input space O to the quantum state space Fq , i.e. , fq : xi → |ϕ ( xi ) ⟩ , where the |·⟩ denotes a vector and physically it represents a state of some quantum system . The ⟨·| is the Hermitian Conjugate of the vector |·⟩ . In practice , the feature map is realized by acting the circuit U ( xi ) on the initial quantum state |0n⟩ , i.e. , |ϕ ( xi ) ⟩ = U ( xi ) |0n⟩ . ( 1 ) The quantum kernel can be obtained by running the circuit U† ( xj ) U ( xi ) on the initial quantum state |0n⟩ , where U† is the Hermitian conjugate of U . Then estimate | ⟨0n|U† ( xj ) U ( xi ) |0n⟩ |2 by counting the frequency of the 0n output as a value of k ( xi , xj ) . Fig.1A shows the process flow of the quantum kernel method and classical kernel method . 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 is k ( xi , xj ) = | ⟨ϕ ( xi ) |ϕ ( xj ) ⟩ |2 = | 〈 02 ∣∣U† ( xj ) U ( xi ) ∣∣02〉 |2 . ( 2 ) By the definition of Havlı́ček et al . ( 2019 ) , the quantum circuit U is realized by U ( x⃗ ) = Uϕ ( x⃗ ) H ⊗ 2Uϕ ( x⃗ ) H ⊗ 2 , where the ⊗ is the Kronecker Product of two matrices . For the Secondorder Pauli-Z evolution circuit , Uϕ ( x⃗ ) = exp ( i ( x0Z0 + x1Z1 + ( π − x0 ) ( π − x1 ) Z0Z1 ) ) , where Z0 , Z1 are quantum Z-Gates , and H is the quantum Hadamard-Gate . 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 ) . 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 . In this paper , all references to quantum kernel methods refer to 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 as one of the most successful cases of the kernel approach . SVMs are dedicated to finding a 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 a mapping function . Then the distance between two data points can be calculated using the kernel function . Suppose we have a set of data points D = { ( x1 , y1 ) , · · · , ( xn , yn ) } , where xi ∈ Rd and yi ∈ { −1 , +1 } . According to Burges ( 1998b ) , the nonlinear SVM can be modified and expressed by an optimization problem as maximize : LD ≡ n∑ i αi − 1 2 n∑ i , j=0 αiαjyiyjk ( xi , xj ) ( 3 ) s.t . 0 ≤ αi ≤ C and ∑n i αiyi = 0 , where i = 1 , · · · , n. The decision function is f ( x ) = sign ( Ns∑ i=1 αiyiϕ ( si ) Tϕ ( x ) + b ) = sign ( Ns∑ i=1 αiyik ( si , x ) + b ) , ( 4 ) where si are the support vectors and Ns is the number of support vectors . 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 . Havlı́ček et al . ( 2019 ) refer to it as quantum kernel estimation . We show the specific estimation method in Section 4.2 . 3 METHODS . 3.1 QUANTUM KERNEL IS A PROBABILISTIC KERNEL . The kernel function is an equation for measuring similarity . In vector space , we estimate the similarity of vectors utilizing vector kernel functions . 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 . A question arises as to how to define kernels or what kind of kernel functions can be effectively used or not . There is no answer to this question . Mercer ( 1909 ) argues that a valid kernel function needs to 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 to some traditional kernel functions . It follows the Mercer theorem and is a 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 a statistical sense , we call it a probabilistic kernel function . For correspondence , we call the classical kernel function a deterministic kernel function . We try to clearly show the relationship and difference between different kernel functions by a diagram ( Fig.2 ) . 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 ) ) . 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 .

Summary: 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.