Quantum kernel model

A quantum kernel encodes a classical input $x$ through a data-encoding circuit $U(x)$ into the feature state $|\phi(x)\rangle = U(x)|0\rangle$ on $n$ qubits. The fidelity quantum kernel is the squared overlap of feature states, $$K(x,x') = |\langle \phi(x) | \phi(x')\rangle|^2 = |\langle 0| U^\dagger(x') U(x) |0\rangle|^2.$$ For any finite data set $\{x_i\}$ the Gram matrix $K_{ij}=K(x_i,x_j)$ is positive semidefinite, so $K$ is a valid kernel.