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: 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.