  The kernel trick in supervised learning signifies transformations of an inner
product by a feature map, which then restructures training data in a larger
Hilbert space according to an endowed inner product. A quantum feature map
corresponds to an instance with a Hilbert space of quantum states by fueling
quantum resources to machine learning algorithms. In this work, we point out
that the quantum state space is specific such that a measurement postulate
characterizes an inner product and that manipulation of quantum states prepared
from classical data cannot enhance the distinguishability of data points. We
present a feature map for quantum data as a probabilistic manipulation of
quantum states to improve supervised learning algorithms.
