This paper explores the connection between quantum computing and kernel methods in machine learning, showing how quantum systems can be used to perform efficient computations in a high-dimensional Hilbert space. The authors interpret the process of encoding inputs into a quantum state as a nonlinear feature map that maps data into a quantum Hilbert space. This allows quantum computers to analyze input data in this feature space, enabling the development of new quantum machine learning algorithms. Two approaches for building quantum models for classification are discussed. The first approach involves using a quantum computer to estimate inner products of quantum states, which can be used to compute a classically intractable kernel. This kernel can then be fed into a classical kernel method such as a support vector machine. The second approach uses a variational quantum circuit as a linear model that classifies data explicitly in Hilbert space. The paper also introduces a feature map based on squeezing in a continuous-variable system and illustrates these ideas with a feature map based on squeezing in a continuous-variable system. The authors show that the idea of embedding data into a quantum Hilbert space opens up a promising avenue for quantum machine learning, where quantum devices can be used for pattern recognition. The implicit and explicit approaches are hardware-independent and suitable for intermediate-term quantum technologies, allowing them to be tested with the generation of quantum computers currently being developed. The paper also discusses the theoretical foundations of quantum machine learning, including the relationship between feature maps, kernels, and quantum computing. It shows that every feature map corresponds to a kernel and that every kernel corresponds to a reproducing kernel Hilbert space. The paper also discusses the representer theorem, which states that the solution to a common family of machine learning optimization problems over functions in an RKHS can be represented as an expansion of kernel functions. The paper concludes by highlighting the potential of quantum machine learning in feature Hilbert spaces, noting that the use of quantum computing for machine learning is still in its early stages. The authors suggest that further research is needed to explore the potential of quantum machine learning in feature Hilbert spaces, particularly in the design and training of variational circuits and the development of new kernel functions.This paper explores the connection between quantum computing and kernel methods in machine learning, showing how quantum systems can be used to perform efficient computations in a high-dimensional Hilbert space. The authors interpret the process of encoding inputs into a quantum state as a nonlinear feature map that maps data into a quantum Hilbert space. This allows quantum computers to analyze input data in this feature space, enabling the development of new quantum machine learning algorithms. Two approaches for building quantum models for classification are discussed. The first approach involves using a quantum computer to estimate inner products of quantum states, which can be used to compute a classically intractable kernel. This kernel can then be fed into a classical kernel method such as a support vector machine. The second approach uses a variational quantum circuit as a linear model that classifies data explicitly in Hilbert space. The paper also introduces a feature map based on squeezing in a continuous-variable system and illustrates these ideas with a feature map based on squeezing in a continuous-variable system. The authors show that the idea of embedding data into a quantum Hilbert space opens up a promising avenue for quantum machine learning, where quantum devices can be used for pattern recognition. The implicit and explicit approaches are hardware-independent and suitable for intermediate-term quantum technologies, allowing them to be tested with the generation of quantum computers currently being developed. The paper also discusses the theoretical foundations of quantum machine learning, including the relationship between feature maps, kernels, and quantum computing. It shows that every feature map corresponds to a kernel and that every kernel corresponds to a reproducing kernel Hilbert space. The paper also discusses the representer theorem, which states that the solution to a common family of machine learning optimization problems over functions in an RKHS can be represented as an expansion of kernel functions. The paper concludes by highlighting the potential of quantum machine learning in feature Hilbert spaces, noting that the use of quantum computing for machine learning is still in its early stages. The authors suggest that further research is needed to explore the potential of quantum machine learning in feature Hilbert spaces, particularly in the design and training of variational circuits and the development of new kernel functions.