  In quantum and quantum-inspired machine learning, the very first step is to
embed the data in quantum space known as Hilbert space. Developing quantum
kernel function (QKF), which defines the distances among the samples in the
Hilbert space, belongs to the fundamental topics for machine learning. In this
work, we propose the rescaled logarithmic fidelity (RLF) and non-parametric
semi-supervised learning in the quantum space, which we name as RLF-NSSL. The
rescaling takes advantage of the non-linearity of the kernel to tune the mutual
distances of samples in the Hilbert space, and meanwhile avoids the
exponentially-small fidelities between quantum many-qubit states. Being
non-parametric excludes the possible effects from the variational parameters,
and evidently demonstrates the advantages from the space itself. We compare
RLF-NSSL with several well-known non-parametric algorithms including naive
Bayes classifiers, k-nearest neighbors, and spectral clustering. Our method
exhibits better accuracy particularly for the unsupervised case with no labeled
samples and the few-shot cases with small numbers of labeled samples. With the
visualizations by t-stochastic neighbor embedding, our results imply that the
machine learning in the Hilbert space complies with the principles of maximal
coding rate reduction, where the low-dimensional data exhibit within-class
compressibility, between-class discrimination, and overall diversity. Our
proposals can be applied to other quantum and quantum-inspired machine
learning, including the methods using the parametric models such as tensor
networks, quantum circuits, and quantum neural networks.
