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Kernel methods are a class of machine learning algorithms for pattern analysis (e.g. SVMs). Any linear model can be turned into a non-linear model by applying the kernel trick to the model, i.e. replacing its features with a kernel function. Quantum computers are expected to improve existing classical kernel-based ML methods through their ability to efficiently access and manipulate data in large quantum feature spaces, which is classically intractable.

Let us consider a quantum model of the form

$$f(x) = \langle \phi(x)|\mathcal{M}|\phi(x)\rangle,$$

where $$|\phi(x)\rangle$$ is prepared by a fixed embedding circuit that encodes data inputs $$x$$, and $$\mathcal{M}$$ is an arbitrary observable. This model includes variational quantum machine learning models since the observable can effectively be implemented by a simple measurement that is preceded by a variational circuit. For example, applying a circuit $$G(\theta)$$ and then measuring the Pauli-Z observable $$\sigma^0_z$$ of the first qubit implements the trainable measurement $$\mathcal{M}(\theta) = G^\dagger(\theta) \sigma^0_z G(\theta)$$.

The main practical consequence of approaching quantum machine learning with a kernel approach is that instead of training $$f$$ variationally, we can often train an equivalent classical kernel method with a kernel executed on a quantum device. This quantum kernel is given by the mutual overlap of two data-encoding quantum states,

$$\kappa(x, x') = |\langle\phi(x')|\phi(x)\rangle|^2.$$

Kernel-based training, therefore, bypasses the processing and measurement parts of common variational circuits, and only depends on the data encoding. If the loss function is the hinge loss, the kernel method corresponds to a standard support vector machine (SVM) in the sense of a maximum-margin classifier. Other convex loss functions lead to more general variations of support vector machines [1].