# Quantum Feature Map¶

Many classical machine learning methods re-express their input data in a different space to make it easier to work with, or because the new space may have some convenient properties. A common example is support vector machines, which classify data using a linear hyperplane. A linear hyperplane works well when the data is already linearly separable in the original space, however this is unlikely to be true for many data sets. To work around this it may be possible to transform the data into a new space where it is linear by way of a feature map.

More formally, let $$\cal{X}$$ be a set of input data. A feature map $$\phi$$ is a function that acts as $$\phi : \cal{X} \rightarrow \cal{F}$$ where $$\cal{F}$$ is the feature space. The outputs of the map on the individual data points, $$\phi(x)$$ for all $$x \in \cal{X}$$, are called feature vectors.

A feature map can transform data into a space where it is easier to process.

In general $$\cal{F}$$ is just a vector space — a quantum feature map $$\phi : \cal{X} \rightarrow \cal{F}$$ is a feature map where the vector space $$\cal{F}$$ is a Hilbert space and the feature vectors are quantum states. The map transforms $$x \rightarrow |\phi(x)\rangle$$ by way of a unitary transformation $$U_{\phi}(x)$$, which is typically a variational circuit whose parameters depend on the input data.

For some more detailed examples of quantum feature maps, see the entry for quantum embeddings, and the key references Schuld & Killoran (2018), and Havlicek et al. (2018).