This chapter gives an example of quantum kernels and the swap test. Then, it describes quantum kernels and the inversion test. Quantum feature maps encodes classical data into quantum data via a parametrized quantum circuit. A quantum computer can estimate a quantum kernel and the estimate can be used by a kernel method on a classical computer. This is because the exponential quantum advantage in evaluating inner products allows to estimate the quantum kernel machines directly in the higher-dimensional space. Quantum feature maps encodes classical data into quantum data via a parametrized quantum circuit. Quantum feature maps encodes classical data into quantum data via a parametrized quantum circuit that are hard to simulate classically and is an active area of current research.
This chapter gives an example of quantum kernels and the swap test. Then, it describes quantum kernels and the inversion test. Quantum feature maps encodes classical data into quantum data via a parametrized quantum circuit. A quantum computer can estimate a quantum kernel and the estimate can be used by a kernel method on a classical computer. This is because the exponential quantum advantage in evaluating inner products allows to estimate the quantum kernel machines directly in the higher-dimensional space. Quantum feature maps encodes classical data into quantum data via a parametrized quantum circuit. Quantum feature maps encodes classical data into quantum data via a parametrized quantum circuit that are hard to simulate classically and is an active area of current research.
Kernel method in machine learning consists of encoding input data into a vector in a Hilbert space called the feature space and modeling the target function as a linear map on the feature space. Given a cost function, computing such an optimal linear map requires computation of a kernel matrix whose entries equal the inner products of feature vectors. In the quantum kernel method it is assumed that the feature vectors are quantum states in which case the quantum kernel matrix is given in terms of the overlap of quantum states. In practice, to estimate entries of the quantum kernel matrix one should apply, e.g., the SWAP-test and the number of such SWAP-tests is a relevant parameter in evaluating the performance of the quantum kernel method. Moreover, quantum systems are subject to noise, so the quantum states as feature vectors cannot be prepared exactly and this is another source of error in the computation of the quantum kernel matrix. Taking both the above considerations into account, we prove a bound on the performance (generalization error) of the quantum kernel method.
A basic idea of quantum computing is surprisingly similar to that of kernel methods in machine learning, namely, to efficiently perform computations in an intractably large Hilbert space. In this Letter we explore some theoretical foundations of this link and show how it opens up a new avenue for the design of quantum machine learning algorithms. We interpret the process of encoding inputs in a quantum state as a nonlinear feature map that maps data to quantum Hilbert space. A quantum computer can now analyze the input data in this feature space. Based on this link, we discuss two approaches for building a quantum model for classification. In the first approach, the quantum device estimates inner products of quantum states to compute a classically intractable kernel. The kernel can be fed into any classical kernel method such as a support vector machine. In the second approach, we use a variational quantum circuit as a linear model that classifies data explicitly in Hilbert space. We illustrate these ideas with a feature map based on squeezing in a continuous-variable system, and visualize the working principle with two-dimensional minibenchmark datasets.
It has been hypothesized that quantum computers may lend themselves well to applications in machine learning. In the present work, we analyze function classes defined via quantum kernels. Quantum computers offer the possibility to efficiently compute inner products of exponentially large density operators that are classically hard to compute. However, having an exponentially large feature space renders the problem of generalization hard. Furthermore, being able to evaluate inner products in high dimensional spaces efficiently by itself does not guarantee a quantum advantage, as already classically tractable kernels can correspond to high- or infinite-dimensional reproducing kernel Hilbert spaces (RKHS). We analyze the spectral properties of quantum kernels and find that we can expect an advantage if their RKHS is low dimensional and contains functions that are hard to compute classically. If the target function is known to lie in this class, this implies a quantum advantage, as the quantum computer can encode this inductive bias, whereas there is no classically efficient way to constrain the function class in the same way. However, we show that finding suitable quantum kernels is not easy because the kernel evaluation might require exponentially many measurements. In conclusion, our message is a somewhat sobering one: we conjecture that quantum machine learning models can offer speed-ups only if we manage to encode knowledge about the problem at hand into quantum circuits, while encoding the same bias into a classical model would be hard. These situations may plausibly occur when learning on data generated by a quantum process, however, they appear to be harder to come by for classical datasets.
Quantum kernels map data to higher dimensions for classification and have been shown to have an advantage over classical methods. In our work, we generalize recent results in binary quantum kernels to multivalued logic by using higher dimensional entanglement to create a qudit memory and show that the use of qudits offers advantages in terms of quantum memory representation as well as enhanced resolution in the outcome of the kernel calculation. Our method is not only capable of finding the kernel inner product of higher dimensional data but can also efficiently and concurrently compute multiple instances of quantum kernel computations in linear time. We discuss how this method increases efficiency and resolution for various distance-based classifiers that require large datasets when accomplished with higher-dimensioned quantum data encodings. We provide experimental results of our qudit kernel calculations with different data encoding methods through the use of a higher-dimensioned quantum computation simulator.
Quantum processors may enhance machine learning by mapping high-dimensional data onto quantum systems for processing. Conventional feature maps, for encoding data onto a quantum circuit are currently impractical, as the number of entangling gates scales quadratically with the dimension of the dataset and the number of qubits. In this work, we introduce a quantum feature map designed to handle high-dimensional data with a significantly reduced number of qubits and entangling operations. Our approach preserves essential data characteristics while promoting computational efficiency, as evidenced by extensive experiments on benchmark datasets that demonstrate a marked improvement in both accuracy and resource utilization when using our feature map as a kernel for characterization, as compared to state-of-the-art quantum feature maps. Our noisy simulation results, combined with lower resource requirements, highlight our map's ability to function within the constraints of noisy intermediate-scale quantum devices. Through numerical simulations and small-scale implementation on a superconducting circuit quantum computing platform, we demonstrate that our scheme performs on par or better than a set of classical algorithms for classification. While quantum kernels are typically stymied by exponential concentration, our approach is affected with a slower rate with respect to both the number of qubits and features, which allows practical applications to remain within reach. Our findings herald a promising avenue for the practical implementation of quantum machine learning algorithms on near future quantum computing platforms.
Quantum kernel methods (QKMs) have emerged as a prominent framework for supervised quantum machine learning. Unlike variational quantum algorithms, which rely on gradient-based optimisation and may suffer from issues such as barren plateaus, non-variational QKMs employ fixed quantum feature maps, with model selection performed classically via convex optimisation and cross-validation. This separation of quantum feature embedding from classical training ensures stable optimisation while leveraging quantum circuits to encode data in high-dimensional Hilbert spaces. In this review, we provide a thorough analysis of non-variational supervised QKMs, covering their foundations in classical kernel theory, constructions of fidelity and projected quantum kernels, and methods for their estimation in practice. We examine frameworks for assessing quantum advantage, including generalisation bounds and necessary conditions for separation from classical models, and analyse key challenges such as exponential concentration, dequantisation via tensor-network methods, and the spectral properties of kernel integral operators. We further discuss structured problem classes that may enable advantage, and synthesise insights from comparative and hardware studies. Overall, this review aims to clarify the regimes in which QKMs may offer genuine advantages, and to delineate the conceptual, methodological, and technical obstacles that must be overcome for practical quantum-enhanced learning.