  We implement an all-optical setup demonstrating kernel-based quantum machine
learning for two-dimensional classification problems. In this hybrid approach,
kernel evaluations are outsourced to projective measurements on suitably
designed quantum states encoding the training data, while the model training is
processed on a classical computer. Our two-photon proposal encodes data points
in a discrete, eight-dimensional feature Hilbert space. In order to maximize
the application range of the deployable kernels, we optimize feature maps
towards the resulting kernels' ability to separate points, i.e., their
resolution, under the constraint of finite, fixed Hilbert space dimension.
Implementing these kernels, our setup delivers viable decision boundaries for
standard nonlinear supervised classification tasks in feature space. We
demonstrate such kernel-based quantum machine learning using specialized
multiphoton quantum optical circuits. The deployed kernel exhibits
exponentially better scaling in the required number of qubits than a direct
generalization of kernels described in the literature.
