  Quantum machine learning is an emerging field that combines machine learning
with advances in quantum technologies. Many works have suggested great
possibilities of using near-term quantum hardware in supervised learning.
Motivated by these developments, we present an embedding-based framework for
supervised learning with trainable quantum circuits. We introduce both explicit
and implicit approaches. The aim of these approaches is to map data from
different classes to separated locations in the Hilbert space via the quantum
feature map. We will show that the implicit approach is a generalization of a
recently introduced strategy, so-called \textit{quantum metric learning}. In
particular, with the implicit approach, the number of separated classes (or
their labels) in supervised learning problems can be arbitrarily high with
respect to the number of given qubits, which surpasses the capacity of some
current quantum machine learning models. Compared to the explicit method, this
implicit approach exhibits certain advantages over small training sizes.
Furthermore, we establish an intrinsic connection between the explicit approach
and other quantum supervised learning models. Combined with the implicit
approach, this connection provides a unified framework for quantum supervised
learning. The utility of our framework is demonstrated by performing both
noise-free and noisy numerical simulations. Moreover, we have conducted
classification testing with both implicit and explicit approaches using several
IBM Q devices.
