12.3 Quantum Circuits for Feature Maps
Figure 12.1 displays a schematic representation of the feature map circuit. In this example, we work with an 8-dimensional dataset with features encoded in the rotation angles such that there is a direct mapping of a sample xi := (x1i,…,x8i) into the vector of adjustable circuit parameters 𝜃i := (𝜃1i,…,𝜃8i). The first section of the circuit implements the operator U(xi), creating an entangled state due to the layer of fixed two-qubit CZ gates, whereas the second section of the circuit implements U†(xj). Here, we use the fact that
It is easy to see that if the samples xi and xj are identical (so that 𝜃i = 𝜃j), then U(xi)U†(xj) = I and all K measurements will return the all-zero string 
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Figure 12.1: Schematic quantum kernel circuit.
The rest of the protocol is classical – the quantum computer is used to assist the classifier with the...
